(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 277968, 6413] NotebookOptionsPosition[ 270760, 6193] NotebookOutlinePosition[ 271269, 6213] CellTagsIndexPosition[ 271226, 6210] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{ RowBox[{"Quit", "[", "]"}], ";"}]], "Input", CellChangeTimes->{{3.7311072152584*^9, 3.7311072163036003`*^9}}], Cell["Mec\[AAcute]nica Cl\[AAcute]sica 2018 1c - Turno A", "Chapter", CellChangeTimes->{{3.7311487609214*^9, 3.7311487993634*^9}, { 3.7311557874128*^9, 3.7311557877404003`*^9}}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Algunos sistemas de la Gu\[IAcute]a 1 resueltos \ num\[EAcute]ricamente.", FontSize->36], " ", StyleBox["(Versi\[OAcute]n h\[IAcute]per-comentada).", FontSize->24], "\n", StyleBox["\nSi ya usaron el ", FontSize->24], StyleBox["Mathematica", FontSize->24, FontSlant->"Italic"], StyleBox[", si son buenos entendedores o gente de poca paciencia", FontSize->24], StyleBox[",", FontSize->24, FontSlant->"Italic"], StyleBox[" la explicaci\[OAcute]n detallada que se da en este notebook les \ va a resultar cansadora: vean los notebooks ", FontSize->24], StyleBox["m\[IAcute]nimos", FontSize->24, FontWeight->"Bold"], StyleBox[" subidos aparte para cada problema.", FontSize->24] }], "Title", CellFrame->{{0, 0}, {0, 3}}, CellChangeTimes->{{3.7310896361156*^9, 3.7310896501706*^9}, 3.7310995396004*^9, {3.7311557931303997`*^9, 3.7311558014768*^9}, { 3.7311831648501997`*^9, 3.7311833420741997`*^9}, {3.7311872335882*^9, 3.7311872905902*^9}, {3.7311922905912*^9, 3.7311923145752*^9}, { 3.7312323139576*^9, 3.7312323537472*^9}}], Cell[TextData[{ StyleBox["\[FilledSquare] Recomendaci\[OAcute]n general: la ayuda de los \ comandos del ", FontSize->18], StyleBox["Mathematica", FontSize->18, FontSlant->"Italic"], StyleBox[" es complet\[IAcute]sima. Incluye muchos ejemplos, y siempre se \ descubre una manera m\[AAcute]s eficaz o prolija de hacer las cosas. Adem\ \[AAcute]s es navegable. Nunca saben d\[OAcute]nde van a terminar. Pueden \ apretar F1 para el men\[UAcute] principal de la ayuda o F1 luego de \ seleccionar una palabra o comando. Por ejemplo, ubiquen el cursor sobre la \ palabra NDSolve y tecleen F1.", FontSize->18], "\n", StyleBox["\n\[FilledSquare] Aquellos que no hayan usado nunca el ", FontSize->18], StyleBox["Mathematica", FontSize->18, FontSlant->"Italic"], StyleBox[" deben saber s\[OAcute]lo esto: para evaluar una celda se usa el \ Enter del teclado num\[EAcute]rico, o bien la combinaci\[OAcute]n de teclas \ Shift+Enter. Para que eso tenga efecto, el cursor tiene que estar dentro de \ la celda que quieren evaluar, o la celda tiene que estar seleccionada con el \ mouse usando las barritas que aparece en el extremo derecho de la pantalla. \ Pero no se apresuren a\[UAcute]n.\n\n\[FilledSquare] No todas las celdas \ pueden evaluarse. \[CapitalEAcute]sta, por ejemplo, es una celda de texto. \n\ \n\[FilledSquare] Cada celda est\[AAcute] abarcada por un corchete en el \ costado derecho de la pantalla. As\[IAcute] saben qu\[EAcute] l\[IAcute]neas \ incluye. Las celdas pueden agruparse jer\[AAcute]rquicamente.\n\n\ \[FilledSquare] Un \"punto y coma\" luego de una serie de comandos hace que \ el resultado no se muestre en pantalla. Eso es \[UAcute]til para pasos \ intermedios que no estamos interesados en ver, por ejemplo asignarle cierto \ valor a una variable. No es \[UAcute]til si se trata de mostrar en pantalla \ el gr\[AAcute]fico de una funci\[OAcute]n.\n\n\[FilledSquare] En una primera \ lectura, eval\[UAcute]en las celdas con los par\[AAcute]metros y condiciones \ iniciales tal como figuran. Est\[AAcute] chequeado que as\[IAcute] las \ soluciones num\[EAcute]ricas son bien comportadas y los resultados no \ producen mensajes de error. Luego experimenten.\n\n\[FilledSquare] Para los \ que no usaron antes el ", FontSize->18], StyleBox["Mathematica,", FontSize->18, FontSlant->"Italic"], StyleBox[" antes de pasar al par de problemas de la gu\[IAcute]a 1, abajo \ tienen un ejemplo de celdas evaluables con unos cuantos comandos \ t\[IAcute]picos. Prueben ejecutarlas una por una. F\[IAcute]jense en la \ figura d\[OAcute]nde tienen que hacer doble click para desplegar estas \ celdas.\n\nNoten el formato que se usa para incluir comentarios: (* blah \ blah blah *). Las celdas est\[AAcute]n agrupadas y ocultas, para no ocupar la \ pantalla.\n\n", FontSize->18], Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJztnQmcFOWd9xtRIxolxlxOgomGuBgjOca8xkiQHAjRDbLEI0iIGzO+8cJs AEPciKAGcJXwgtEIzoomThLEoKgrXqhRRCCLCCIooCIOyBFGHG4Y4P8+T3V1 T3V33V3dVd39/X4+3wRnpuo56umq+nVVPXX8Jb8Y8H8PSqVSww5T/zPgpyO+ PXToT6/94cHqP75z7fBLL+mo/rFJOV/9Q/9QRBARERERERFrQoVOzCOUf1Z+ Pe76ICIiIiIiIpZbRSflTH2p2HRd3HVCRERERERELLeKWyzZWLst7johIiIi IiIillPFJ5V78/JxY9z1QkRERERERCyniqvzsnGz8qi464WIiIiIiIhYLs3n jt/Ky8c/iLteiIiIiIiIiOVU8cu8bDw/7johIiIiIiIilludhy3Z+D3lcXHX CREREREREbGcKj6hPGDJx73jrhMiIiIiIiJiuVV835KN/xR3fRARERERERHj 0PLs8Xp9LTnu+iAiIiIiIiLGoeJC5TvKr8VdF0RERERERIxfxSHKE5V9lD3M fx8ad70QERERERGxulT8zJwb+TXl2XHXx6zTp5RXKZ9XtuW940jbqvyj8ntF lvNd5UjlbcqblEOU5ypPUX5E2UF5uPKrPtZ1hPLbyl/p54WVjynnKV9UPqgc r/xh3jIHK+9UzlTeo5xsOkV5r3K68jnzHcg7lK8rFytfVv5deYfy03FvL0RE RERExEpX0Tcvd+os2j3G+nzazHx7bDKxk48qjw9RVv67he1ss+RznW+vV65W dslb1yXKjT7rO9iy3CMB2unkk3GPI0RERERExEpX8ZJN3podQz30ddprlLtD ZsRdyp8GLO/dIjLp9/PW988Ay460LLcygnw8I+5xhIiIiIiIWMkqTnXJXKeW sR51ymcc6rEtlb7P+DvKzym/przZ/Hn+3+p3BV/is8zTisyk3fLW96sAy/7M spxu11sh66Dbq+8xPzzusYSIiIiIiFjJKsa4ZC/93GyHMtShW8r5Ou4k5REO y+ms/JTNMvq+7C/7KLdTKj03dJA8ujSVvvf7cod1Djczq92y+5VPpNLPFh+d t9yHzLz+f1Ppe7f91OUF5TfiHkOIiIiIiIjVoJn33DLYj0tcvr4WbHdfsr5X +ic+1/F7m+WXKY/yseyhqfQ7lP7uI4/+RnmQj3Wel7KfT2y0z/Z8VPm0R11G +lkXIiIiIiIiequvYbpc68z4agnLP165yeH675kB1qPf/TTPZj36Wm3HAOv5 i0s/TA/Ytt/YrOOUAMuf6VKXncpOcY8fRERERETEalFxlo9rptq6EpTd2bzG a1fev4dY38kO67ozwDqOdemDCwPWR7+zaV/eOmzvE3dY/hSXujwb99hBRERE RESsJlP+55M6swRl/9WhrDuKWOcLDusc7nP5I1z6oE+I+lifqW4JuOynXery SNxjBxERERERsZpU3O0zHw+OuNx/dShH3+v9+SLWe6nLenv7WL5Dyvl+88B9 kMq953tlwGUPc9keD8U9dhAREREREatFMwu+5jMf/2eE5R6Wcp6ruqj7hhVf dmnDAh/LH+Wy/LAQ9ZltWf61EMvv5PoxIiIiIiJiaVX82Gc2LuqeZ5tyL3Mp 55dFrlvP07XHZf2u15AVJ7kse3OI+jxmWX5RiOWbHeryTNzjBxERERERsRpM pd/7a81eLyrfdMmGsyIqt6NHOedGUIZTptTe67Hs91yW/VOIusywLB94HnDF Eoe6eF4LR0RERERERG8V/2G9rplK32ut3/U03yGPrY2oXK/5srtHUIbbPeOu 9zgrLnJZ9oUQdbG+L+qtEMs/61CXkr1zCxERERERsVY0s/BKS9b6ruV3P3XJ h5+KoOxJHvn4qAjKsHsPckb9vqXDXJa93GXZNSHq8mfL8utDLP+gQ11WxD2O EBERERERK928a7iv5f1Ov/93v0Mm+3UEZa9yyZ97I2rfQo8M/hGXZUd4ZOuD A9ZlmmX51hBt+ZNDXRbHPY4QERERERErXcXDlpw1xeb3Tu8QXq08qIhyD/HI rVHl41c9yjnSZdmxHst+PGBdHrAs2xaiLX9wqMdMh2VGIyIiIiIievniS2sN /f6ura1tdEtLy+h169aN3rRp0+j3339/9JYtW2z94IMPcmxtbTX+Xi3XSxKQ iTMquuRdHy54p69ivEs+PLuIsj9Tpnzs9vxxm87pLsv+3qOOge4xTxXeHx30 +vMtDvWY5LAMAAAAAACAJ3NfWmvo93f79u0TlXFF5WNROdf4t8rCWVUOtlVl Y9m6davx9zpXSwJycUbFf1oylr5f+DM2fzPOJR86Xbf0U7bbu4mjzMdLXcrw mp/rTo86fjpgXR7JW/7wgMtf71CP4Q7LAAAAAAAAeEI+NvLWG5aM9aLN709U bnHJh/rdwqHm0FJ8ziN77ouojU7vRNI2eSx7l0cdjw1Ylyfylnd89tlh+eEO 9fiRwzIAAAAAAACe1Ho+VnwjL2PtVP5KearyZOU1yvc88qFbNvMq/3DlAY91 HxNBOxe7rP8Sj2WnetQv6PPHL+Yt/4mAy1/jUI8eDssAAAAAAAB4Qj72vHfY r/cXUQe37KqN4v3HixzW3eaVvxV/9Khf0Ou/r+QtX3A/u8fyv3aoR1eHZQAA AAAAADyp5XysOCzlft90EFtTLnNcedTjZo91fz+Ctj7msO7HfSx7t0f9At1b nsp9z7T2hIDL/8ahHkc7LAMAAAAAAOBJjefjgRFl44zfC1mPzyr3uqy3IYK2 Xu6w7rN8LPs7j3YHff54Xd7ynwy4vN38XHpetQ4OywAAAAAAAHhS4/n4aYec 9az5u0eV5yq/q+yvHJlynwf6tiLq0uiy3v8XQVvt5sle5HNZp/miM54YsC67 8pY/NODyN9jU4Z8uywAAAAAAAHhSq/nYvGabPy+Wfg7Yda4ofY1SOdMhJ64p oj7Hp9LPAtutd34E7e1ls94LfC77C498fGqAehyTt+yWEG2ZbFOHzS7LAAAA AAAAeFLD+Tj/GuRm5XE+lz0y5Xw/tO+saLNeu9yn1e+POqzI9l6Wt843lR19 LvsTj3x8XoB65F/HnhOiLbNs6rDDZRkAAAAAAABPajEfKzopN+Xlq9sDrmOZ Q1YcV0S9jlK+47Dennl/++lUep6qB5ULU+lnevV94Tcqv2Gz7il567s0QL1O 88jHvw+wrnOK6XeXvt/vsgwAAAAAAIAnNZqPr7TJV30CrmOGQ1Z8W3lQEXXr qbOezXr/av6+TvlIKv2ctP7Bq6n0+5qvUK62/L2+B/wjlvWuzKuj77m2U+l3 NO+zqVPGVwOs6z/zlu0Xoo+2OdTjow7LAAAAAAAAeFJr+VjR0cyH+dnquwHX 87BLXjy/yDr+l931UeWZyjfM/9bPSn8/b7nP5C3TrLxEOTjv55eEqNNyl/bq 57iP97meFy3LbQ6S083lP+dSj39zWA4AAAAAAMCTGszHQx2y1c8DrmeBS077 35Tzu4b8rPvQVPqeabuMrP9/RMrhGnUq9xqynauUB4eo030e633AY3k9r9k1 qdxr4xNC1MPpPVXaOxyWAwAAAAAA8KSW8nEqPYez0xzRO5T9fa7n48otHnnx riIzclflVpv1/lP5RZflHveoV+Brx+Z6T/dYr/Z3dtk7lZ6z+pG8v9X9d0zA OhyifN2lfH29/AibZQEAAAAAADyphXycSs8v9adU4Xt37Zym/GTe8vrdSz9S /kw5SrnIx3q028zMttb8f31PtH7+1tc9xYqLXNb7Y4dl7nKpT0uqiHmwFU/5 aPMqs4/0nNf63u7fp+yfF/6PAOWeqrxJOd9H+dN1RlZ2sawDAAAAAADAk2rO x4rvK+f4zLJW9XOx55vr0PcF2z2vXIzfDNCGO13W8xfl0Xl/f5vL3/+5yP48 JeV9/7Yf9XcQvq6tK34bYv36exB9n8D/EfIxAAAAAAD4pBrzcSr9nqSnI8hx 1ypPjjgbt6Z8vmfZbMvByidd1rchZXn/sOIZl7+9IYK+1e/GGqJcE7L9c1MB rmGn7N91HPR7CAAAAAAAAE+qNB/fGCBD7Uyl3x28RPlYKn1/8k1mBtTvTPqu 8v5U+p7qveYy+v+fUP5SebbyUuUtqfS7iJea68ysf73yPfPf+t7j74Roz6Fm m+yeR874srLRo62TIuxjPQ/4gFT6nmu3dz9Z1XN8HR6wnDPMdj1vZnL93uoV ypHKC5Q3m+t9KJV+p5VWX58+z7IeAAAAAAAAT6o0Hx+WSj+7q5/1Ha28Snmh 8tvK7qn0O4I+mgo4j7PiaHN51+ufqfQ92ccpT7L87MOpIt6LLO2ZtF75H8pb dd41s6Oe69pp3jGri0rU3580v0vQ3y/o6+PbU+nrxFr9Pip9f/rlpdzmHgai ZcVzMmNak8xa1Cw7gy5st64po2XKjOdkRYvnX8uK52bIlNFTZMZzK8Tzzz3K nNA0SxY1h28BdQ9XZrF192Rnsyya1STTfLXNL9H0QfUQsD9aVshzM6bI6Ckz 5LnoNkri2dm8SGY9M9u7zZn+GT1BmmYtklJ+PCAgNTp2y4v9/sT35wcAyk41 5uNaVF+XNfOyV0b+fhnqEnre7hLpH3WuMGPcZTLg4qHSOLf4fLzq0dEyqD4l 9YNGy6OrPP9aHh09SOpT9TJo9KPi+eceZXbpe7XcPT/8cZe6hyuz2Lp7ovLx 3MahcvGAy2TcjKiybDR9UD0E7I9Vj8roQfWSqh8ko70HXHWg9pWzJl0mp114 hUebd0rz7InS0KNO6rr3k5FN5ONEUYtjt+zY7E98f34AIA7Ix9WlYoJHPtbX d/W80r7m0K4SfdIiK2aMk4F9LpCGibMjOYer5JxG3cOVWfJ8LDoiz5aJDRdI n4HjZEYk1x7Ix7mQjz3x2+aWxdJ0XX+prz9NLps0K8J7HiASanHslh2b/Qn9 DpBoyMfVpeJYn88Db1Q+oByql4m73iXWF+nMcY707H+dNC2O6KpcBec06h6u zHLkY/1dzuKm66R/zx4ycNyMCDIH+TgX8rEnvvNx+v7dkj93AOGoxbFbdsjH AJUG+bj6VDznc76sjMvirnOJ9UEmb5wT2bVjTU5Om595Bm+0YeH5oss5ufnM 6QRzWed15OW055bkLjfB/3PV1D2euvtl58pHZIw6v+rWo0Emzi523UXk4wB9 lCbzLF7u3/t7Vty7bPf17JTmRbOkaYK1bLtnjJ37w3hmsGlCbnnzXc514+wf 3+0NUra5zusGSq/udZKq6y69Bg4pfH41QLtz5gtYlft5DfXZCdzn7XWw9tFa c1vnLGfN+0sW59RV13/ePOd5CLJjx9JXOW1fm1tv1+8UQrQx6WPXrn9yVx/0 WWmb8e81nnzuU9zmmyhsh3V/Ml2ei/jzY9Yowv2GfVtLOkY9PlcrWkKs0097 Au1vQownR5IyNhO2fRME+bj6NOchC5KPX467ziXWk/S14x5S32eITJkX3Qc1 m9P6DJRh1wxLHw/Nfu/c9TTpN7LJcky0Pyc39l0TG6TfaV2lc962K1yHJaf1 7C9XDh8ug63Lde4qpw2+WWb42J9T93jq7pudK+WRMareXVS9xzwiK4tacbh8 HLSP0sfTSXJZr+5Sl/f3dd17GdfC/c5VYxznG4cWlK3Xo+/jLTg+G2U3ytB+ p0nXztay66R7r4EyLucY7Twmmkb2k9O6ds4pT4+xgX0KM0ac/ROsvUHKbpH5 d18tfbvkHUcsbQ/72enWo59cdsVlOZ9X47PTb6Q0+Zw3LnCfm+epky7rJd3r rH10mVw/8krp37NL7j0h5nU/43N+5RXSL1NXVc/+1zXJA/c630fSMv9uubpv l5y+yu6rBjTIDTffnLPfsK9vmDZWxtjNfOfXua6PDJkyr2B7t8ybIkP61Emd n+O0cT4+URryx78xnhpkos05eZB9itv9QoXb2bo/+aPcF/HnJ9L9hg1lGaMe nyt9S1+Yce/WnkD7mxDjyZFEjc3kbN+kQT6uThU/V+7xkY31u6rOjru+JdaD nbLykTFq/9BN+gyZIhHG4+x+p65OHdO79TT2fSP0d2ZD2r87br8/1n4Ojxnj Bqrzts7S9bR+0jDC8j1ddh09cq55Z8rsrMrsosrsnSlzhLkv87k/ou7x1N0/ mXHbReoHjZFHigrIIfJx4D5SOWSuymu9u0oXffxsGNH+95k+yutT56an5yjr rc71O1vXZZbbuWt/ua5pseX8pkVW6PNHowzzeo1Z9oiGdGao6z7QMt+Z25gw zzmHpJcfMrCX9KmvM8ZaTsaIs38CtzdI2R7Xj4v57HRW27Nbj/b6Zv5e5VVf ++YQZe9sniuNQ3ur81TrMkNkYK8eUl/XRW3Xzrb5uOBzbl77eTXkuWld167S rb5v+/ps91Xh2lgxYzfznV/nbjJgxDRZkvO3LTJvyhDpU+djLOi82DQyfQ6e N/51m9P9kDdXXMB9Svh87HH9ONb9hj1lGaMen6udYdbp0R7f+5sw48mJxI3N hGxfj26LA/Jx9ao4XfmuSzbW72v+Qtz1LIPuZK/D9ZSGCc+I/achHJn9TsH1 w8zclfo8NXt/bOE5eeb78s5de8vQxrm5+15z3ht9XtezYYI8s9ajTLWPnT2x QXqo/W+3ASNk2hL3PRJ1j6fuQchcb+mijnVjHllZxDEmeD4O3kdr5ZkJDdKz c530aJgos60LZO/T8vf+n/brTN2N76mt976mj9u55zeZ+0Pq9PnIsMac77X1 9/Izbh6sxkQ3S73cxkTuPFMtK/S1R/M6gPUaUJz9E7i9Icp2eH6y6M/OZZNk VnvnZnOdn89O8LIz3zF1zjsntXy/kErZ5uNUSp0b9i58z0HYc1PjmrW+buW6 rwrXv5Uzdl2+82uZJ1OG9FFj2v7acs5aLPuH/PGvr3s1DuuVPre3zDUSdJ8S Ph+7P38c537DiXKMUa/PVah1erXH5/4mzHhyInljMxnbN4mQj6vbVPo9z/co H1E2KW9Unq/sGnfdyqg7a5+RCQ09pXO3ATJi2hLnz6z5/ESQZyWy31Pa5Jfs OWznTC63OSd3eI4k/fMR2ftzrPNQZfd1Nu1Z+8wEaejZWTr3bJAJHgcR6h5P 3S2FeY+3nUtk2ogB0q1zsd/thMjHgfuo/Tyue69h0hj6PcuWTGNzn2Xhu6/b z7mdzl8Kv2fI74/MtavONtfq2+tTcI9krP0TpL0hynY6v4/xsxO47Ox3ozb3 dli+17LNx53tn2sIe25aeM+D3b4qTP9W0th1/s7P/73V7vsH67rax1vQfUoJ 83GMfe9E6ceoeH6uQq3Toz3+9jdhxpMTSRybydi+SYR8jDWgKzuXTJMRA7o5 nH+pvdNOrZg5+l/lX83vxNI/93cd0/7aR+a4lrmXzCmjFM67kbkPJ3MvgN25 pl177PaN1D1JdQ863vLb4totbj0Wcn6uIH1kORc3n4EdEmp+jkyb/V0rkOzz sjb3guXfh+j4nUnmv+37ObMPKdy+cfRPmPaGKNt1/t14PveBy/b4bjRzrmyb j7v0lavvnl+QRcKem/rbV4Xp30oau+Iwr0KAe6v97B8y16Kz4z/oPqV0+dhc Qzx979HW0o1R8fxchVqnR3v87W/CjCcnkjg2k7R9kwX5uDJVfFl5Qtz1qBBd sT0HyqCO1XNnNMqYEZlzym/Lt3sPlqFjRsuIEWOkccZc13uW3N8zlDmP7SJ9 r75b5rfYHEOd5t0wnltpcL0W43ffSN0TVPfA4609C/k9djr0WPB8HKKPjDlG JuXNh5KyPBPpa15a9/P9QjLHd8+5GJT5edj878x1eqf+sTvXja1/wrQ3RNlO 5/cxfu4Dl+3xjp1M2bb52CFTh54bx9e+KkQbK2rsGmsqvMc6wL3V/vYPwb7/ sl1DqfJxrH3v3taSjVFrfzhdgw2zTo/2BNt2QcaTY8kJHJsJ2b4JhHxceSp+ oNymfF35kbjrUwG64vreWvP5nTENmXk/OxtzOnSp13MkjPF8pqe4nNY+70b+ HDvp492q7DyyycuY1D1U3QOPtxbbugQnaD4O10fpKlvep5GdD6X9+JrzfJVr Xf2eL2T+3uV6atbMu4+KzcdJ6J8g7Q1Rtu35fZyf+xBle5yzueZjh/qU9tw0 RBsrauyaWzLvHusW8xq3r3mrE5lB/Obj+Pvera0lG6OO/ZGhiH5xaQ/5uEz7 oHQhFfe+b/JxZan4pXK/ud9boTw+7jpVgK64Xj82yd6Drec77Ox/vmDX+1YK 5urM3y+231dj/xxh+3WiUmZM6l7euhtV9D3e4rp+HK6PCqvffk6XmVfZ7plx p7L9tTn4/WlBz1EK71GtpP4JWbbt+U6cn50QZWfur3a458/1/uqI87G/fVWY /q2ksZvXbuMe6/tl1v1B3i+RxHtY/ebjBPS9S1tLN0ad+iN/mSL7Ja89/rZd jd9fXZbtm0zIx5Wh4mDlZMu9M88qj467XhWiK5n9heP8L/pY0zhMevXoKb0H Xy6XDz5P+vbWc516P9vjOk9Uwbsenc7JLfe15NTL3I+mcvdrUWdM6l7eugcb b3E9fxyij7JzqeZdszTJbhfP55Pc5ysx7jVsmmC5p9Dy95Y5OHPra55PZt81 Uez8XAnpH7/tDVO27flOnJ+dMH3uMj9Xdps7zM/lUB/n+cTs58IKt68K0r+V NHbz69VF6vtfLJde3F/qu/m5tzq3TVHNgVS4Twm6nf3m4yT0fSGlH6NO/ZGt Qbh1erTH3/4m2fNzFT82k7B9kwn5OPnqe6iVT1uy8X8rD4m7XhWkK9nPv9O+ zcgrI6Vh2M3SOHelrJzbKDcPa5CRjf7zsd17BLLz5mf3ef6vY+Y/b2Tdz0ed Mal7eeseaLzFNn91iD7Kfg+d9w6J9FLt92z5eD7J8X0XlvdEWs8l2ufgzHv3 libzfYSqr+s7vxY3yXX90+8YHXzzjGyZ9u/Iibl/grY3TNm29ybH+dkJU7bL +52ea5Rh5vN1QfJx9npszjtD9TujZ8jNgwvfpRRsXxWufytp7GaXytxjrd+X quzm697q3GUjeYeO0z4l0HZ2ycfFfn5K0Pf5lGOMhr1+7LpOj/b43d8k9v1O kYzNJGzfZEI+TrZ6Di7zOWMx76v+Vdx1qkDd8ZMxWpqleW1z+txpp/lvH4fq 7HvX1fG9rr5P4Xvoc/ZfNs/Bzp4oDT0K3yNv3DNVXy/dtV1zvyeMOmNS9/LW Pb2Az/HmNP9u4PdeZvogd/5TO9NzoobpI8u5mj4O92uQEdn1Zt4JocofOE5m eH3xZDkv0O907dcwInde5pzta3SorJgxTgbqbZ9ftq8x4VymnrOzT32dMdZy nuGMs38CtzdE2Zmxl/l741r0SlkZ22cn3Od2Z/NcaRza23hfZ/vz2kNkYK8e Ul/XRW3XYPdXt7//0zI/ktGHvaVvfTfp2rXO9tzUaV+V+87icG2srLGb2ZyZ 3Jd+L6u/e6szy6r9X9NI6de9sA2ZOXZzvw9x7iPHfUqg7WyzP4ns8xOi7wMe H8oyRr2ePw6zTo/2+N7fhBlPTiRubCZh+yYT8nFyVfRQ/tPMxjuU/eOuU4Xq QZB3RwQjO+9Bn4Ey7JphOXNmGPutcdZ5JW2Oofpaj96X6n1g/ryU1zXJjHvH yxD9PaElH0U+xxV1L2vdg+D4TtCW+XL31X2li9M9UIU9ZvaB95zH2WeKQvRR +rxMncuZ81ta15s+7k+0vUfQDuO7anXOYFt+zvbN9Im+p3hidm7N9rLNXDTR Y0xI4ffjmfL0GBvYJ+/YH3P/BGtviLIt52zG32Xu34zzsxOmz43rKvo6qvXd JPp7osvkmmGDVX4MmI/zr9NY1jdp0g3SMKDe9tzUaV+lz0tz5lUK1cYKG7tm P2bvBfU1b7XNWLAb/2aGnGjz7qNg+5Qg29n9+7ZiPz+B+z7g8aEsY9Tze6dw 496tPYH2NyHGkyOJGpsJ2b4JhHycTBWDlbvNcbdO+bW461TBepK5/yzIPVy+ 1rvC8u72VZb5JW3fTZh5n9yUgvPQ9He9+dfwxOa5yfYy89/dnl5V4bMq1D1Z dfeP+b1ON5tnJwNfPy58l6H79eNwfWQuZOSRpgm567XrN0/cyrdfwKZsuz5y GBNiaZe1vMwYy9++cfeP7/aGKzs7rvXfWdsR52cnVJ+318E6r/d8u3lrWhy2 tWs/muPIZtmcuWPne+2rimtjZY3d9qzib95qO2zqY9cvOYsE2af43c72+5Oo Pj+ObXXq++z7ec6RS4Lk41KOUT+fq7DjPo/w+5sQ48mRpIzNBG3fhEE+TpaK DsqbLN//LVJ+Ou56VbjeqM/67IkN0qO+hzRMnO0jTwDEjDkPRr3THEwA4I7r 3Ebt81oVNze8O+7vVqllMteP/c5bDUFIz1HQIFf6mNeRMVrdsH3tIR8nR8Vh yvst2Xim8oi461UF+iJ9Dblevu3z3U0A8WE+A3ROT5t5dwHAF9l3Axc+G9uy YoaMG9hd6jrrdww9IqU6JHBu6oD5DGV9fX+5rmlxqHmXwQn93cPtMuryX8mk Wd73vDNGqxu2rz3k42So+KRyviUb36o8KO56VYn+MJ8HOq/vxRyPIdkY544X S9/zhkrjXK4dA4SjRRY3XSf99TOgtnPldJauvUv7GePc1IrlnlBjbqB66dEw UWZzQ1fE6H6eK/Pn+nn2hjFa7bB97SEfx6/iFOUaMxfvVTbEXacq0zfpuUwv lgGD89+TAJAQzPkvBw+4WIY2zuVZAIAicJr3Jj1XjsrGvp7fDw/nplZ2ypJp I2RAt8w8cpf5ur4JpYUxWt2wfe0hH8er4mzlVvOY/L7yO3HXqQoNgPn99d1h 51wAKDHmM5N3h5zzBgDysJlrJvy8O8HImUuQIGiZJ83v3IJQahij1Q3b1x7y cXwqrlbuM7PxKuW/xF2nKhUAAAAAAMAT8nH5VXRU3mF51vh55TFx16uKBQAA AAAA8IR8XF4VRymfsGTje5WHxl2vKhcAAAAAAMAT8nH5VHxO+ZqZiw8or427 TjUiAAAAAACAJ+Tj8qg4XbnRzMY7lefFXacaEgAAAAAAwBPycelVDFTuNrPx e8qvx12nGhMAAAAAAMAT8nFpVYyyPGu8WNkl7jrVoAAAAAAAAJ6Qj0uj4kPK v1iy8aPKD8ddrxoVAAAAAADAE/Jx9Co+oXzJko0nKA+Ku141LAAAAAAAgCfk 42hVnKxcbebiNuXP464TAgAAAAAAeEM+jk5FH2WrmY0/UPaOu05oCAAAAAAA 4An5OBoVVyj3mdn4LeVJcdcJswIAAAAAAHhCPi5ORUflJMuzxi8qPxZ3vTBH AAAAAAAAT8jH4VUcqXzMko3v0/NWx10vLBAAAAAAAMAT8nE4FccpXzVz8QHl yLjrhI4CAAAAAAB4Qj4OruI05QYzG+9SXhh3ndBVAAAAAAAAT8jHwVRcYGZi MTPyaXHXCT0FAAAAAADwhHzsX8V15r3U+j+WKj8bd53QlwAAAAAAAJ6Qj71V HKr8k2Uerll6bq6464W+BQAAAAAA8IR87K5+V5NyjiUb36bf6RR3vTCQAAAA AAAAnpCPnVV0U75l5uJ9yivjrhOGEgAAAAAAwBPysb2K7ym3mNm4Vdk37jph aAEAAAAAADwhHxeq+LmyzczGq5Unx10nLEoAAAAAAABPyMftKg5STrA8a/yS 8hNx1wuLFgAAAAAAwBPycVrFh5WPWLLxX5SHxV0vjEQAAAAAAABPyMdGNv6M crElGyeqfli0AAAAAAAAuaj8Kv/7vyLTpomMHy9y333yauMj8o9ZS2z/vBby seJU5XtmLt6tvCjuOmHkAgAAAABALbN3bzoHX6Ti3te/LvLRj+o06OxRR4l8 9asi558v0tgosmNH1edjxQ+VO8xsvEn5zbjrhCURAAAAAABqEZVb5frrRY49 VmxzcIcOInV1IqedJnLccSIHHWT/dx/5iKz70aXy8v1zclZfLflY8WvlATMb L1MeH3edsGQCAAAAAEAt8cILIuedJ3LwwZLNuIceKvKjH6XvpZ45U2TpUuO6 sJV5z78ti/76d5H/+R+RSZNELrlE5MgjJSdP9+kj8uijxt9Xej5WHKq8x/Ks 8ZPKznHVB8siAAAAAADUArt3i1x1leRc++3SRWTMGJGNGwv+fOOmHbLw5Q3Z e6fvf+B1eWn+utw/UjlW/vAH2XHCv+Su9/zzZd/mzYnOx4oOLr/7qPLvlmz8 B+XBpaoLJkYAAAAAAKh2li8X6d5dsvm1Vy+Rhx7SF3kdF8nPxzobr1271fZv 9e+X3jlD5MIL09eRdRmf/axsfeKJJOfjSx1+fqJylZmL9yl/Uao6YOIEAAAA AIBq5q67RA4/XIzM2qmTyJQpkReRMz/XY4+JfOxj6fI6dpStv/qVbNqwIVH5 WHGKconNz7+tfN/MxluV55SifEysAAAAAABQjaicacwxnblmrK8fL1tWkqIK 5q9eu1YO9OyZLXvvGWfIB2+8kYh8rO+TVi5XLsr7+c+Ue81svEbZPeqyMfEC AAAAAEC1oZ811vdQZ7LxkCHpn5UI2/c77dkju0aMyM57ve9LX5IP1qxJQj4e ZWbgWeZ/H6S8xfKs8QLlp6IuFytCAAAAAACoJg4cELngAsnc3yzTp5e8SLf3 H29+4AE5cMQRRn3aevSQLRs2xJaP9TVhyzXiu5RHKB+yZOPpyk5RlokVJQAA AAAAVBO/+IVkrxs3NpalSLd8rOfn+kBn9EMOMeq0t39/2dLSUvZ8bLmvWs+5 1ab8nfJlSzb+rduc1lgTAgAAAABAtXDLLZLNxjfdVLZivfKxnr96++TJ2bmt d196aRz5WN9XvdPMwruUW8x/71EOjqocrGgBAAAAAKAaaGpqf7fSFVeUtWg/ +Vj/e+eNN0omv++6/vqy5WPzvuo9lmvFB8z/b1Vepuyq7BxFWVjRAgAAAABA pfPWWyKHHSZG9vzhD0X27y9r8X7zsX7ueLfO7rqeKstve/LJkudjy33V+y35 WMz/3mHaZv5MP5u8RHke91rXpAAAAAAAUOmcfbYYmfPEE0V27Sp78e82bzW0 4pSPt6j/bzvjDKO++04+WT7YvLnU+Xi05b5qOw+Y91u/q/w583PVru+8844g IiIiImLlukk/12ves7yhqSn2+mTcrHKvbT5Wbl2wIDtf164xY0qWj23uq7a6 z7xerOfo6q/f8xR3PsN4jfszg4iIiIiI4V2zfLm0HXus6Jy5/Qc/iK0eL8x5 1dD6M7d8bNxnPXSoUW/97qetr78eeT4276t+0yYX7zGz8aPKb8SdyTBRAgAA AABApTJ8uBjXjo86SuS992KrRpDnjzO2rl8v+487zqj/3nPPLUU+nmqZhyuT i7VTlJ8Pu16sagEAAAAAoBJZulTk4IPFyMe33RZrVcLkY52Fd0ybJpl7w7f/ 7W+R5WPzvurMfFw6I28z3+90TJj1Yc0IAAAAAACVSL9+YmTLr31Nh9FYqxI2 H2v3mnOLGXN1RZCPLfNV6//YpLxGeVjQ9WBNCgAAAAAAlca774p07ChGPn76 6bhrU1Q+3jZ/vmSuIW9TbYkgH49UvqjsxzuaMKAAAAAAAFBpXH+9GJmya1eR Awfirk1R+VjbdvrpRnv2DBpUVD5WdFR+OcgyiBYBAAAAAKCSaGsTqasTIx/f ckvctTEoNh/vaGw02nOgUyfZ2twc6fuPEQMIAAAAAACVxIMPipGNDz1UVICM uzYGxebjDzZulAPHHGO0a9ett5KPMS4BAAAAAKCS6NNHjHx84YVx1yRL0flY uXvIEKNd+08+mXyMcQkAAAAAAJXCW2+JdOggRj5+9tm4a5Mliny8ddGibNt2 zJ5NPsY4BAAAAACASmHsWDGy8Re+kIh5uTJEkY+NebrOPNNo396GBvIxxiEA AAAAAFQK55wjRj6+5pq4a5JDVPl417hxRvv2f/GL5GOMQwAAAAAAqAT09eKj jxYjH8+cGXdtcogqH2/T94zr9nXoIFtWryYfY7kFAAAAAIBK4LXXxMiO2oTM W50hqnz8webNcuDww402bnvgAfIxllsAAAAAAKgEJk+W7LPHZWD79r2y7r1t 8m7zVsPmtVtl67Y9tn8bWT7WzyD37Gm0c9fw4eRjLLcAAAAAAFAJ/PjHYuTj f//3yFa5cdMOWfjyhmy+/cfC9bJ+w3bjd//85055+ZUN8qL6ufbxp96WZcs3 264nyny8e8QIo506J5OPscwCAAAAAEAlcPzxYuTjxsbIVpmfj2c8tMLIyJqd O9uMrJy5fjzribcKMnCGKPPxjoceMtqp77PetG7dDRJ/ZsLaEQAAAAAAko7K mNlnj5cvj3TV1ow8bfrrgTKw2+/C5uOtuq0dOxptbb377ikSf2bC2hEAAAAA AJLO88+LkY07dYr8vcdJy8etra2y/8QTjfbuuOqqhyT+zIS1IwAAAAAAJJ0H HxQjHx93XEmLCZqB3X5XTD7ed8YZRnt3DRr0hMSfmbB2dGCL7Fz5pGyYNVU2 LXtNdu/x+nkItr0tra9Mk+ZZ06Rl5dvSVsSqkkeE/eSXPetl27KHZd2s21Wf au9RZb8oW5aVuR75hN7OXn24Rdq2aSOtbTiyfV+Ofo5hbLlR1rbHQKXtp2p5 LCaOKPdRZTgmA0DyuesuMfLx175W0mKSko/b+vUz2rv77LNfkPgzE9aODqyW llnDZMlVPWXZ9PuldbvXz0OwYbY0T/03mXfVv8nKWbNlZxGrSh4R9pMf9Dn8 gony+q1nyYKrTlJ9qu0rb8y8TdbMLGM97Ai9nZ37sG39AmmZ81tpnvOktG4o QZ2Dsn2RbJj+M1lYln4u89jyoqxtj4FK20/V8lhMENHvo8pwTAaA5DNunBj5 +KyzSlpMUvLx3p/+1Gjv3jPOWCjxZyasHR1wOuZG+F11pZ13BqK85yxtb06T 1bd9S/Xlt2TJpGGy2rh+rK93PaO2VaXmY+frJa1zbpJl154kS6beLi3k43gh HyeLWh6LiaEU+yjyMQAohg8XIx9fdFFJi0lKPt5jtrete/dlEn9mwtrRgTIc cyvtvDMQ5T1n2bn4dlk59iSZd+0lsnrOy5Z7QBNw7hT5ds60iXyciPNh8nGy qOWxmBhKsY8iHwOApN95rPPx1VeXtJik5OPd5vXyfSecsFriz0xYOzpAPi6O mPLx2OHSvHh1bPWwhXwcMQnYplbIx8milsdiYiAfA0CJ+MEPxMjHN95Y0mKS ko93mc9b7//kJzdK/JkJa8TmOYtky5r1sl/yCX9/dfq5q8wcUcrZD9uX4Xre mSnn9lDz4ux//zXZsuCe9joo1y14Qba9v6vwj/Pn38n+t3Weq6D3knuds+yS 3Wuel02zb8+pY9CyMn29eupPZMm1+vpxP1k29bfGuja88g/Zuc373ClQX9mU HXg7++7fwrGWeaZv5aR+6vz/JFk45mJZOf329DN+67dE3r/efWWu733vTBK2 nwspz9iKr+2Z7a4/j8tle15b0uM67LrbKf34zVDp28ONMo7FgnnHrMcIv20I WJ8Qx4bS7aPC5uPSjD8AiInTTxcjH99xR0mLSUo+3vm3vxntPfDhD+sjf+y5 CWvDedf+TJbeOVE2FeTPEMdi8/xlTSarZeaJGnaWfRmO+Vif99wvq43zC/08 7TjZ4Dsfp8+Z3mu6XJaOOrW9DsoFoy6U15vuLczalnosmzpWmh8fKsvGfMuy 7Kmy6NbLZc2C3PNyd5z7KX1ueqe8eecFsmjYSTl1zJT15jP551L2ZK8bX1Vo +rqF27lTiL7SFLGdg/VvYd0d22u5bh5l/3r3VXp9bz8+Ud6cfJFDJgnZz46U Z2zF1/ZM+/R4uV7ebLK25VuqjGnSuq2Ifi3b+K2W7eFGGcei9br41JtV3pyc zaB+2hCqPiG2fen2UcGPyaUZfwAQKyefLEY+bmoqaTFJycc7nnoq3d5DDtkr CchNWBvq+7+MDDp1srQUdSxW513L/ihvGvMnW+eIynyPblOGbT7W33Wrc9c7 w2Rj67L16WP/zMx35ZZ66HNg63WibD2+JQuv/ZYsGn+BOscan7ecvg4wTJpf 8VsXh35S5+atc8bKslH15rnceMv3+e1lzbt2oDpvedHzu/3w149D9lVR2znd vwvHttcxp3/HXyfvLbOWVVh3z2szEfdvYV9ZyrSsc9G16bYtiKyf3SjP2Iqv 7Zn2FX4e18y8VzYtW64+gxUwfqtme7hRxrGYzceq7qP6prW2YfowM7vatCFs fUIcG0q3jwp4TC7J+AOA2PnmN6Umrx8feeRWSUBuwtqwRR37V+pzimKPxdte Vucul9hkWnU++spk+zIK8rE+f3tS/Ux/1x00G+s6LJdNM69Qy+pzv+HqfCX/ HZHmNelhZ8kbMx+W7ZlrPdl6pPPl6w8/bLlHz1J/431Jj8p2X+cR9v203yir vyxQdVj2gPp5zr2A1nP3elnadI9sed9f0wM/fxy2r4razmZ+1+3O5g/LcsPM ceB7DBY+21eS/s3pq/zrhNZ7HfQ1mYj62ZUyjq1Y2t6+ffM/j/v37DKshPFb PdvDjTKOxWw+Psmcp3+kvGe9P9j6jru8NoSuT+hjQyn2UcGOyaU61gBAzGSe P77hhpIWk5R8vGvKFOH5Yyy7G/7bPF8s7licfb9QwfzJYjkndTr30OeV/yNb isnGOlsvu1feHG+X9TNske0LJsgbo+pl4W2/lQ1vbsmrx6myaPIE2fRu3r1m 78+VdU0X2VyPccPhnCXz3J/Ts4579DnslbIo4LwuwfJx+L4qbjsH7d8Q+bgE /dveZu++mhdRP7tTvrEVT9vbt++C8dfJuuXr85arjPFbPdvDjTLu5yz5eMGo K2XNguW2985veeY3svTak3LaELo+Re+7otxHBczHJTrWAEDMZOavHjKkpMUk JR/vHjtWmL8ay+/q7BwnufPeBDkW71LnVePljdFO57Nb7MvInnv0laV3XiMr p6afkXI893FFnRfNVudFw05yPbfb/+6jsmZyX5k3Ol3G/rx62F4fzp5HBPme Peicopk+ar/vrXT5OGxfFb+dg/Vv8HzsTNj+9Wqz2VeZazWR9LMX5RpbcbW9 ffsumjxeZZL8+ZYqZfw6UWnbw40y7ucs91cvnnqnbN5gPw/X7uVT5c3x9T7b 4FGfovddUe6jopq/urhjDQDETOb9xwMHlrSYpOTjPcOGCe8/xhh0IMixuP3c y/581gHLd/MLhtUrM3OshMnH7ecjuc/k5Zl5Rm3YRfLm7LmyO6ceTu9vCXOu 43bOYjefqOV5sJz5tfy1Plg+DttXxW7noP0bNh9H2b9bpHXOb2XZtR5tzp4n R9HPXpRrbMXVdq/tWynjV1MN28ONMu7nsvm4j+tzLtmMX9CGEPUpet8V5T4q TD6O/lgDADFjvg9YevcuaTFJycd7zevle884Y6HEn5mwdnQgyLE45Lserc92 Gc+TXSLLnObI8cTyzKIvLec75czHxty598vbBfOJmnOJTr8u/fxcmfJxsL4q djuXIR9H3r/tmcT976PsZy/KNbbiarv/fJzo8Vs128ONMu7nsvnYo2522yts fUqRj0P3TfD5uUpxrAGAmGlsFCMff/WrJS0mKfm4zXzeevc557wg8WcmrB0d KOf148zzxi9La3aOnH6y8vEnQ7xPyeP6SFbLO5XLmI/b3n1Y1kzOnzvX8i7K 99ufsYv9+nFBXyX/+nH0/VuGa3Z2Y9KVco2tuNoe0fXjmMdv9WwPN8q4nyvi +nHo+pQgH4fvm4BzgpToWAMAMfPgg2Lk4y5dSlpMUvLxPnO+7l2DBj0h8Wcm rB2z7wjKvn/CIMrnjx3KsDv3yM6xGvR9SiHPfZ3qkUNU+Vif395knN86ti07 30t5nj8O1lcRbuccosrHpehfS5t1376y2v55Rtt5eooYk66Ua2zF1Xavz1sl jN9q2h5ulHE/Z3n+2O0578Lnj4uoT+T5uJi+CXJMLt2xBgBi5vnnxcjHnTqJ HDhQsmKSko/3f+ELRnt3DBnyoMSfmbBmXJ09juaeM0Y4f7XlWJ1ThtP7j1fd L2/flv7ee2nTveo8yM+53C7Z+cofVEasd31+2XZOz7LlYx/PpGXmmC1pPg7f V9FtZ7u+KjYfl6Z/s9dhHN97Y11vNP3sTvnGVjxt925L8sdvNW0PN8q4n7PM X134rnSTbW9Ly+NDZcm19Zb5pouoT+T5uJi+ifKZp/DHGgCImfXq2KTzsXb5 8pIVk4R8vFX9vXTsaLS19Z57JkvsmQlrxdY5w2TNnelnfnOzaMBnnbLXffPf zaTnB3nYvgync48969V56lhZNkqfy/1U3p6z0OZdJIXsf/9Fea9poPlu07x3 Y0r62tCmhy+XpaNOzT2/Kls+tswhW/Cdvp5TVPdT+7NipcvHRfRVlNvZs3+9 83HutbAS9a/1nTFjLla5Kv+ds9b1RtTPrpRxbMXSdu/PW/LHbzVtDzfKOBat 7z8edpYsvXOibLLeA67ffzxnXHp+sZx3WBVRnwjycXT7qHDPPEV9rAGABHDC CWLkY/0scolIQj7e8dBDRjsPHHGEbFq3Tr/wOfbchLWh8Syb7TE06FyZudd9 2593Gi9vmsfhgjJczj3az+XqZdFtY2XDKj/navq4f7+sNub4qk/PQTKz/Zm6 1VN/Ikv0taFRF6pzpxds3t9S6nzs1Eftc4ouGtVPFo/tZ/RXkPsgg+bj0H0V 8XZ271+nuref++n6vd403rwfdn3J+rdt/fNqPGbaZ31+M73eJaPOUus+y+Y9 qGH72Y3yjq3yt93P5y3p47eatocbZRyLlvurF17bU9k3d93ZubdVdm76o2zJ PitURH1C77tKsY8K9sxT+HIsc177vo8AAMrKT34iRj7WczuXiCTk490jRhjt bDvzTL3caElAbsLaUB/39bGzOefdx5oQ75Iw5st8WNaY51nt82V+y74M13MP dS6XmatLne8se0CV5es+6/S7Hd9rSl8HyZ2TNV2PNQtekG3WdZV9/uo/ypt5 79dIXw+5TpoXPCkbnrjJ+xnDPILn45B9lW1DVNvZrX9dvovJXuszy848a1ii /s2eM84cmj4Hz5sLds0zU+TtqfrcPcJ+dqS8Y6v8bff7eUv4+K2a7eFGGcdi Nh/3lKWTr5c1j4/Nzr+cMy/zLJtMF7Y+ofddpdhHhZm/Okw57fPAzbv2Z7J6 zqKA71oEgJIzZYoY+Vg/m1sikpCP23r2NNq565pryMdYVpvVMd9+ntL0edSG WVPTc13u8fp5O9k5brzmQtX3w70yzcfvb5d1gc7ZLM/R+ZmT1ase2TbfLhsK vkdwwq2f2teXrZv1GclM/wX47t5+jjWveqQJ1Fd2ZZasfwP0YU5fRd+/hW3J mwt2ezpzrStBPxdS3rFV/rYH+7wld/zm/q5yt4cbZRyLlnxs5MD3M+3Ma7/j szgh6hPlti96HxXmmBymHK4fAySe116T7DPIKm+Wgtjz8ebNcuDww402bvvb 38jHWG4BAACSTX4+3u69CABAVaLnrT76aDHy8cyZJSki7ny87dln0+076CDZ 8s475GMstwAAAMmGfAwA0M4554iRH6+5piSrz+Tc/fsPyHvvbZN/LFyf/dlf p79e8ny8a9w4o337v/hF4+/Jx1hmAQAAkg35GACgnbFjJfsMcgneg2zNx2vX bZOHH10lE3+/UG753QLDUufjtl69jPbtbWggH2McAgAAJBvyMQBAO2+9JdKh gxgZ+ZlnIl+9NefaZeRS5uOtr7ySbduO2bPJxxiHAAAAyWaP97xjAAA1RZ8+ YuTjCy4oelUbN+2QhS9vyObb+x94XV6avy77+/yMXMp8vPsXvzDatf/kk2Xr 1q3kY4xDAAAAAACoJB58UIx8fMghKuBuLGpV+flYZ+O1a7fm/E0mIz/9zDuy bPlm2/UUnY/V7w987GNGu3bdeiv5GOMSAAAAAAAqibY2kbo6MTLyzTeXpUid kT9o3a1yq/1NPMXm4x1TpxrtOdCpk2xtbiYfY1wCAAAAAEClcf31YuTjE04o yTxdQSk2H7f16GG0Z8+gQdLa2ko+xrgEAAAAAIBK4913RTp2FCMjP/FE3LUp Kh9v/cc/0u1Qbnv6afIxxikAAAAAAFQi554rRq7s3l2H0VirUkw+3nv22UY7 9n3pS8Z/k48xRgEAAAAAoBJZvjw9R5fOyBMmxFqVsPl4x7Rpkrl2vP3hh8nH GLcAAAAAAFCp/PrXYuTLI48UFURjq0aYfNy6fr3sP+44o/57f/jD7PVk8jHG KAAAAAAAVCo7doiYGTOK9yGHJUw+3j10qFHvAx/+sLS+8Qb5GJMgAAAAAABU Mg89JJl7lOWpp2KpQtB8vHXBguy94bvGjcuZy5p8jDEKAAAAAACVzjnniJGP TzxRZNeushcfKB+r/2874wzJzsnV0kI+xqQIAAAAAACVzttvi3TqJEZGHjBA ZP/+shYfJB/vvvzydD07dJBtTz6Zk43JxxizAAAAAABQDfz5z0bmNLLnz39e 1qJfmrfO0IpdPt41apRk7gXX/87PxuRjjFkAAAAAAKgWxo+X7LPI119ftmL9 5OPtv/99tm67VX63y8bkY4xZAAAAAACoJsx5oQ3vuKMsRXrdX/3BffeJdOxo 1GnPgAHGM8jkY0yao0ePFkRERERErCJHjZJXTzlFjHcndeggTwwaJK+88ooE pW3fftm+fa/Kq7sN9b/1z+xwy8ebp02TA4cdZtSnrVcv2bJxo/EcMvkYk2bs n11ERERERIzcG0eOlLc+/3nJXEferOfs0u9KzmPjph2y8OUN2Xz7j4XrZf2G 7ba/m/Nis7yzprVgHRrbfLx7t+waPlzkoIOMOuz7ylfkg7Vrs+94Ih9jAgUA AAAAgGpku8q5P/6xZO+17tZNZPHinD/Jz8AzHlphZGRNS8suef2NFnlt2T8N H350lbw0f51dSYX5ePVqOfCNb2TL3qOvG69alc3G5GNMqAAAAAAAUM3ce6/I EUeIkVU/9CGRSZNyfm3NyNOmv15wHTiD3TVi299NmybSuXO6vIMPltbf/EY2 bdzY/v5j8jEmVwAAAAAAqHbeeEPkK1+R7LXk009P59i9e6PJx3ObZfnv7hPp 27e9jOOPl61PP13w/mPyMSZYAAAAAACoBXbvFhkyRLL5VXvssSI33CCyPn1P te9rxBlUzpUJE2TXZz6Xu94LLpB9LS0F7z8mH2PCBQAAAACAWmLBgvRzyfpe 60yePeQQkQsvlLd/MVpe/6+pIq++KrJtW85i8557Uxb95TmRxx4Tuf12kUsu ab9vW8+VfVBHkXPPFXnqKePv899/TD7GChAAAAAAAGoRlVll7FiR446TnGu/ Vj/2MZEvf1nkU58S6dDB/m8+/nFp/slVsvChBTmrJx9jBQoAAAAAALWMyrEy c6bIoEGy9eSvyt6PHCOOeVl79NEi9fUi558vct99xn3bbu8/Jh9jBQkAAAAA AGCQzbkqn8qSJSIPPihy220i06fLkqmzZMGTy9yXs0A+xgoUAAAAAADAIPD8 XC6/Ix9jBQoAAAAAAGBAPsYaFwAAAAAAwIB8jDUuAAAAAACAAfkYa1wAAAAA AAAD8jHWuAAAAAAAAAbkY6xxAQAAAAAADMjHWOMCAAAAAAAYkI+xxgUAAAAA ADAgH2ONCwAAAAAAYEA+xhoXAAAAAADAgHyMNS4AAAAAAIAB+RhrXAAAAAAA AAPyMda4AAAAAAAABuRjrHEBAAAAAAAMyMdY4wIAAAAAABiQj7HGBQAAAAAA MCAfY40LAAAAAABgQD7GGhcAAAAAAMCAfIw1LgAAAAAAgAH5GGtcAAAAAAAA A/Ix1rgAAAAAAAAG5GOscQEAAAAAAAzIx1jjAgAAAAAAGJCPscYFAAAAAAAw IB9jjQsAAAAAAGBAPsYaFwAAAAAAwIB8jDUuAAAAAACAAfkYa1wAAAAAAAAD 8jHWuAAAAAAAAAbkY6xxAQAAAAAADMjHWOMCAAAAAAAYkI+xxgUAAAAAADAg H2ONCwAAAAAAYEA+xhoXAAAAAADAgHyMNS4AAAAAAIAB+RhrXAAAAAAAAAPy Mda4AAAAAAAABuRjrHEBAAAAAAAMyMdY4wIAAAAAABiQj7HGBQAAAAAAMCAf Y40LAAAAAABgQD7GGhcAAAAAAMCAfIw1LgAAAAAAgAH5GGtcAAAAAAAAA/Ix 1rgAAAAAAAAG5GOscQEAAAAAAAzIx1jjAgAAAAAAGJCPscYFAAAAAAAwIB9j jQsAAAAAAGBAPsYaFwAAAAAAwIB8jLXs/wdE0jXq "], {{0, 234}, {968, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], BaseStyle->"ImageGraphics", ImageSizeRaw->{968, 234}, PlotRange->{{0, 968}, {0, 234}}]]] }], "Text", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGCQAGIQvXIJ25fGM68dhZbOZG4C0gkhcoIgesWxciEQLfBy fjaIvjf5fTGIXv5iQf0cIH3MW7wJRJs09CssAtLty4VVQLRI4nFdEK18e6ch iN5+cQvDYiDd4FGuDqInnU7WBtEnNLt0QLTt7KkPVwHpC5tT34Ho6f93fALR 9scuiK8G0p2X5M6tAdLdf7deBNFNgVtvgOgyf7m7IFpIJP4FiFY79A5MZ/l+ z10LpE8ZSRSC6CWMOWtB9BVOgXUgmm+fwda1YPV520D0W96wXSD6x5OFYJpp T+IREN0fHHMcRH/bucp5PZBesNoxbD04PJhjQHRORVQKiH6j73z+EJC+49Z/ AUR7vfpxD0R/2nDiA4i+dvTmFxDN2fWZ4TCQZpgoxQqiZaJOCIFoAFoe4hU= "], FontSize->16], Cell[CellGroupData[{ Cell["\<\ Desplegar este grupo de celdas haciendo doble click en la flechita de la \ derecha de la pantalla.\ \>", "Subsection", CellChangeTimes->{{3.7311885233361998`*^9, 3.7311885725601997`*^9}, { 3.7311892079362*^9, 3.7311892268691998`*^9}, {3.7311892644542*^9, 3.7311892684502*^9}}], Cell[CellGroupData[{ Cell["\<\ Vayan par\[AAcute]ndose en cada celda y apretando el Enter del teclado num\ \[EAcute]rico o Shift + Enter. Tambi\[EAcute]n pueden seleccionar con el \ mouse el corchete que engloba a todo el grupo de celdas y ejecutarlas de una \ sola vez a todas.\ \>", "Subsubsection", CellChangeTimes->{{3.7312329321948*^9, 3.7312329877202*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Un", " ", "cl\[AAcute]sico", " ", "siempre", " ", "vigente"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"1", " ", "+", " ", "1", " "}]}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875369112*^9}, {3.7311926541182003`*^9, 3.7311926572872*^9}, { 3.7312344653336*^9, 3.7312344668206*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Esta", " ", "instrucci\[OAcute]n", " ", "va", " ", "a", " ", "asignarle", " ", "el", " ", "valor", " ", "1", " ", "a", " ", "la", " ", "variable", " ", "\"\\"", " ", "pero", " ", "no", " ", "va", " ", "generar", " ", "una", " ", "salida", " ", "en", " ", "pantalla"}], ",", " ", RowBox[{"debido", " ", "al", " ", "\"\<;\>\"", " ", "final"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"numero", " ", "=", " ", "1"}], ";"}], " "}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875595692*^9}, 3.731233383626*^9, {3.731234414584*^9, 3.7312344243796*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Aqu\[IAcute]", " ", "s\[IAcute]", " ", "muestra", " ", "el", " ", RowBox[{"resultado", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ SqrtBox["2"], "+", "numero", " "}]}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875595692*^9}, {3.731233383626*^9, 3.7312333864908*^9}, { 3.7312344388318*^9, 3.7312344442332*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "\[Pi]", " ", "entendido", " ", "como", " ", "n\[UAcute]mero", " ", "de", " ", "precisi\[OAcute]n", " ", RowBox[{"infinita", ".", " ", "Para"}], " ", "escribir", " ", "el", " ", "s\[IAcute]mbolo", " ", "\"\<\[Pi]\>\"", " ", RowBox[{"usen", " ", "[", RowBox[{"Tecla", " ", "Esc"}], "]"}], RowBox[{ RowBox[{"p", "[", RowBox[{"Tecla", " ", "Esc"}], "]"}], ".", " ", "\[CapitalIAcute]dem"}], " ", "para", " ", "\[Mu]"}], ",", " ", "\[Epsilon]", ",", " ", "\[Theta]", ",", " ", RowBox[{"y", " ", RowBox[{"as\[IAcute]", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", "\[Pi]", " "}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875425872*^9}, {3.7311881386542*^9, 3.7311881499182*^9}, { 3.7312345155726*^9, 3.7312345169046*^9}, {3.7312345544514*^9, 3.7312345551387997`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Las", " ", "primeras", " ", "50", " ", "cifras", " ", "de", " ", RowBox[{"\[Pi]", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"N", "[", RowBox[{"\[Pi]", ",", " ", "50"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875310382*^9}, 3.7311876287111998`*^9, {3.7311881572032003`*^9, 3.7311882079222*^9}, {3.7311926883852*^9, 3.7311926975282*^9}, { 3.7311927294231997`*^9, 3.7311927359052*^9}, 3.7312330700584*^9, 3.7312345514774*^9, {3.7314180891646*^9, 3.7314180914154*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "Tambi\[EAcute]n", " ", "pueden", " ", "escribir", " ", "la", " ", "palabra", " ", "Pi", " ", RowBox[{"directamente", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Pi", "\[IndentingNewLine]", RowBox[{"N", "[", RowBox[{"Pi", ",", " ", "50"}], "]"}], " "}]}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875310382*^9}, 3.7311876287111998`*^9, {3.7311881572032003`*^9, 3.7311882079222*^9}, {3.7311926883852*^9, 3.7311926975282*^9}, { 3.7311927294231997`*^9, 3.7311927359052*^9}, {3.7312330700584*^9, 3.7312330720768003`*^9}, {3.7312345736452*^9, 3.731234602282*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Al", " ", "multiplicar", " ", "\[Pi]", " ", "por", " ", "un", " ", "n\[UAcute]mero", " ", "real", " ", "de", " ", "precisi\[OAcute]n", " ", "finita"}], ",", " ", RowBox[{ "indicado", " ", "por", " ", "el", " ", "\"\<.\>\"", " ", "luego", " ", "del", " ", "\"\<1\>\""}], ",", " ", RowBox[{ "el", " ", "resultado", " ", "tambi\[EAcute]n", " ", "es", " ", "de", " ", "precisi\[OAcute]n", " ", "finita"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"1", " ", "\[Pi]"}], "\[IndentingNewLine]", RowBox[{"1.", " ", "\[Pi]", " "}]}]}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875310382*^9}, 3.7311876287111998`*^9, {3.7311881572032003`*^9, 3.7311882079222*^9}, {3.7311926883852*^9, 3.7311926975282*^9}, { 3.7311927294231997`*^9, 3.7311927359052*^9}, {3.7312330700584*^9, 3.7312330744574003`*^9}, {3.7312346088238*^9, 3.7312346392386*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Eso", " ", "es", " ", "v\[AAcute]lido", " ", "en", " ", RowBox[{"general", ".", " ", "Comparen", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ FractionBox["100371", "1000"], "\[IndentingNewLine]", "\[IndentingNewLine]", FractionBox["100371.", "1000"]}]}]], "Input", CellChangeTimes->{{3.7312359161372004`*^9, 3.7312359959639997`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "Esta", " ", "celda", " ", "contiene", " ", "varias", " ", "operaciones", " ", "y", " ", "muestra", " ", "un", " ", "resultado", " ", "tras", " ", "otro"}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"La", " ", "constante", " ", "de", " ", "Euler", " ", RowBox[{"es", " ", "[", "Esc", "]"}], RowBox[{"ee", "[", "Esc", "]"}], " ", "o", " ", "E", " ", RowBox[{"may\[UAcute]scula", "."}]}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"La", " ", "unidad", " ", "imaginaria", " ", RowBox[{"es", " ", "[", "Esc", "]"}], RowBox[{"ii", "[", "Esc", "]"}], " ", "o", " ", "I", " ", RowBox[{"may\[UAcute]scula", "."}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"10", "!"}], "\[IndentingNewLine]", SuperscriptBox["2", SuperscriptBox["2", SuperscriptBox["2", "2"]]], "\[IndentingNewLine]", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "\[ImaginaryI]"}], " ", "\[Pi]"}]], "+", "1"}], " ", "\[IndentingNewLine]", RowBox[{"Log", "[", "E", "]"}], "\[IndentingNewLine]", SuperscriptBox["I", "2"], "\[IndentingNewLine]", RowBox[{"ComplexExpand", "[", SqrtBox["\[ImaginaryI]"], "]"}]}]}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875499132*^9}, {3.7311876351702003`*^9, 3.7311878630542*^9}, { 3.7311880129572*^9, 3.7311880133302*^9}, {3.7311927188512*^9, 3.7311927198831997`*^9}, {3.7311927639351997`*^9, 3.7311927693882*^9}, { 3.7312331170438004`*^9, 3.7312332185218*^9}, {3.7312339305294*^9, 3.7312340041482*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Una", " ", "primitiva"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Integrate", "[", RowBox[{ FractionBox["1", "u"], ",", " ", "u"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.7311878727452*^9, 3.7311879515452003`*^9}, { 3.7311879817441998`*^9, 3.7311879974902*^9}, {3.7311882360302*^9, 3.7311882685692*^9}, {3.7311927872142*^9, 3.7311927900092*^9}, { 3.7312332523046*^9, 3.7312333374066*^9}, {3.7312340250638*^9, 3.7312340369063997`*^9}, {3.7312346788691998`*^9, 3.7312346847986*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Una", " ", "integral", " ", RowBox[{"definida", ".", " ", "El"}], " ", "s\[IAcute]mbolo", " ", "\[Infinity]", " ", RowBox[{"es", " ", "[", "Esc", "]"}], RowBox[{"inf", "[", "Esc", "]"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{"Sin", "[", "x", "]"}], "x"], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "Infinity"}], ",", " ", "Infinity"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.7311878727452*^9, 3.7311879515452003`*^9}, { 3.7311879817441998`*^9, 3.7311879974902*^9}, {3.7311882360302*^9, 3.7311882685692*^9}, {3.7311927872142*^9, 3.7311927900092*^9}, { 3.7312332523046*^9, 3.7312333429188004`*^9}, {3.731234044124*^9, 3.7312340548314*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Si", " ", "tienen", " ", "fijaci\[OAcute]n", " ", "con", " ", "los", " ", "s\[IAcute]mbolos"}], ",", " ", RowBox[{ "la", " ", "integral", " ", "definida", " ", "se", " ", "escribe", " ", RowBox[{"con", " ", "[", "Esc", "]"}], RowBox[{ RowBox[{"dintt", "[", "Esc", "]"}], ".", " ", "Raramente"}], " ", "se", " ", "usa", " ", RowBox[{"as\[IAcute]", ".", " ", "Es"}], " ", "m\[AAcute]s", " ", "pr\[AAcute]ctico", " ", "el", " ", "comando", " ", RowBox[{"Integrate", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "y"], RowBox[{ RowBox[{"Exp", "[", RowBox[{"-", "x"}], "]"}], RowBox[{"\[DifferentialD]", "x"}]}]}], "\[IndentingNewLine]", RowBox[{"Integrate", "[", RowBox[{ RowBox[{"Exp", "[", RowBox[{"-", "x"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "0", ",", " ", "y"}], "}"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7311878727452*^9, 3.7311879515452003`*^9}, { 3.7311879817441998`*^9, 3.7311879974902*^9}, {3.7311882360302*^9, 3.7311882685692*^9}, {3.7311927872142*^9, 3.7311927900092*^9}, { 3.7312332523046*^9, 3.7312333517592*^9}, {3.731234063462*^9, 3.7312342520022*^9}, {3.7312343149334*^9, 3.7312343159972*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{ "Los", " ", "c\[AAcute]lculos", " ", "se", " ", "simplifican", " ", "si", " ", "aclaran", " ", "las", " ", "cosas", " ", "que", " ", "el", " ", StyleBox["Mathematica", FontSlant->"Italic"], " ", "puede", " ", "asumir"}], ";", " ", RowBox[{"si", " ", "no"}]}], ",", " ", RowBox[{"procede", " ", "con", " ", "morosa", " ", RowBox[{"cautela", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", "Comparen", " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "x", " ", "\[Lambda]"}], "]"}], RowBox[{"1", " ", "+", " ", SuperscriptBox["x", "2"]}]], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "\[Infinity]"}], ",", " ", "\[Infinity]"}], "}"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Integrate", "[", RowBox[{ FractionBox[ RowBox[{"Exp", "[", RowBox[{"\[ImaginaryI]", " ", "x", " ", "\[Lambda]"}], "]"}], RowBox[{"1", " ", "+", " ", SuperscriptBox["x", "2"]}]], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "\[Infinity]"}], ",", " ", "\[Infinity]"}], "}"}], ",", " ", RowBox[{"Assumptions", " ", "\[Rule]", " ", RowBox[{"\[Lambda]", " ", ">", " ", "0"}]}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7311878727452*^9, 3.7311879515452003`*^9}, { 3.7311879817441998`*^9, 3.7311879974902*^9}, {3.7311882360302*^9, 3.7311882685692*^9}, {3.7311927872142*^9, 3.7311927900092*^9}, { 3.7312332523046*^9, 3.7312333517592*^9}, {3.731234063462*^9, 3.7312343632328*^9}}], Cell["\<\ Siguen varios ejemplos de c\[OAcute]mo simplificar expresiones [\[EAcute]sta \ es una celda de texto].\ \>", "Text", CellChangeTimes->{{3.7311928491962*^9, 3.7311928579611998`*^9}, { 3.7311930025202*^9, 3.7311930030702*^9}, {3.731233424125*^9, 3.7312334871832*^9}, {3.7312347357304*^9, 3.731234811828*^9}, { 3.7312348791046*^9, 3.731234879402*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Una", " ", "expresi\[OAcute]n", " ", "en", " ", "bruto"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "x", "]"}], "2"], "+", SuperscriptBox[ RowBox[{"Sin", "[", "x", "]"}], "2"], " "}]}]], "Input", CellChangeTimes->{{3.7311928491962*^9, 3.7311928579611998`*^9}, { 3.7311930025202*^9, 3.7311930030702*^9}, {3.731233424125*^9, 3.7312334871832*^9}, {3.7312347357304*^9, 3.7312347794035997`*^9}, { 3.7312348155684*^9, 3.731234822244*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "La", " ", "misma", " ", "expresi\[OAcute]n", " ", "pero", " ", "aplicando", " ", RowBox[{"Simplify", ".", " ", "Ver"}], " ", "tambi\[EAcute]n", " ", RowBox[{"FullSimplify", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "x", "]"}], "2"], "+", SuperscriptBox[ RowBox[{"Sin", "[", "x", "]"}], "2"]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.7311928491962*^9, 3.7311928579611998`*^9}, { 3.7311930025202*^9, 3.7311930030702*^9}, {3.731233424125*^9, 3.7312334871832*^9}, {3.7312347357304*^9, 3.7312347794035997`*^9}, { 3.7312348274723997`*^9, 3.7312348384812*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "Para", " ", "simplificar", " ", "funciones", " ", "trigonom\[EAcute]tricas"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"TrigReduce", "[", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], " ", RowBox[{"Cos", "[", "x", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"TrigExpand", "[", RowBox[{"Cos", "[", RowBox[{"2", "x"}], "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7311928227072*^9, 3.7311928449592*^9}, 3.7311929368272*^9, 3.7311930646032*^9, {3.731234846043*^9, 3.731234866557*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"El", " ", StyleBox["Mathematica", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], "es", " ", "por", " ", "dem\[AAcute]s", " ", RowBox[{"melindroso", ".", " ", "No"}], " ", "va", " ", "a", " ", "cancelar", " ", "la", " ", "ra\[IAcute]z", " ", "cuadrada", " ", "de", " ", "una", " ", "expresi\[OAcute]n", " ", "al", " ", "cuadrado", " ", "as\[IAcute]", " ", "porque", " ", RowBox[{"s\[IAcute]", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Pero", " ", "pueden", " ", "indicarle", " ", "qu\[EAcute]", " ", "cosas", " ", RowBox[{"asumir", ".", " ", "Comparen"}], " ", "los", " ", "tres", " ", "resultados", " ", "siguientes"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Simplify", "[", SqrtBox[ SuperscriptBox["x", "2"]], "]"}], " ", "\n", RowBox[{"Simplify", "[", RowBox[{ SqrtBox[ SuperscriptBox["x", "2"]], ",", " ", RowBox[{"x", " ", "\[Element]", " ", "Reals"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"El", " ", "s\[IAcute]mbolo", " ", "\"\\"", " ", RowBox[{"es", " ", "[", "Esc", "]"}], RowBox[{"elem", "[", "Esc", "]"}]}], " ", "*)"}], "\n", RowBox[{"Simplify", "[", RowBox[{ SqrtBox[ SuperscriptBox["x", "2"]], ",", " ", RowBox[{"x", " ", ">", " ", "0"}]}], "]"}], " "}]}]], "Input", CellChangeTimes->{{3.7311928662102003`*^9, 3.7311928753062*^9}, { 3.7311929752712*^9, 3.7311930422852*^9}, {3.7311930776652*^9, 3.7311931257142*^9}, {3.7312335098958*^9, 3.7312336044952*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Un", " ", "gr\[AAcute]fico", " ", "simple"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", SuperscriptBox["x", "2"], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "0", ",", " ", RowBox[{"2", "\[Pi]"}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.7311655838900003`*^9, 3.7311656809374*^9}, { 3.7311657121948*^9, 3.7311657264708*^9}, {3.7311675944846*^9, 3.7311676520116*^9}, {3.7311690340076*^9, 3.7311690469948*^9}, { 3.7311838145322*^9, 3.7311841299332*^9}, {3.7311874419172*^9, 3.7311875634941998`*^9}, {3.7311880628742*^9, 3.7311880866512003`*^9}, { 3.7312336186758003`*^9, 3.7312336296387997`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"RandomReal", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], "]"}], " ", RowBox[{"(*", RowBox[{ RowBox[{ "Un", " ", "n\[UAcute]mero", " ", "real", " ", "pseudo", " ", "random", " ", "entre"}], " ", "-", RowBox[{ "1", " ", "y", " ", "1.", " ", "A", " ", "ver", " ", "si", " ", "adivinan", " ", "cu\[AAcute]l", " ", RowBox[{"ser\[AAcute]", "."}]}]}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.7311880981702003`*^9, 3.7311881081892*^9}, { 3.7311882931802*^9, 3.7311883078642*^9}, {3.7311931774892*^9, 3.7311931811152*^9}, {3.7312336522702*^9, 3.7312336853302*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ "Una", " ", "lista", " ", "de", " ", "1000", " ", "n\[UAcute]meros", " ", "random", " ", "graficados", " ", "con", " ", "el", " ", "comando", " ", "ListPlot"}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"tabla", " ", "=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"RandomReal", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", " ", "1", ",", " ", "1000"}], "}"}]}], "]"}]}], ";"}], " ", RowBox[{"(*", RowBox[{ RowBox[{ RowBox[{"Alternativamente", " ", "tabla"}], " ", "=", " ", RowBox[{"RandomReal", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", "1000"}], "]"}]}], ";"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"ListPlot", "[", "tabla", "]"}]}]}]], "Input", CellChangeTimes->{{3.7312337052412*^9, 3.7312338314334*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "La", " ", "definici\[OAcute]n", " ", "de", " ", "una", " ", "funci\[OAcute]n", " ", "cuya", " ", "variable", " ", "no", " ", "tiene", " ", "atributos", " ", RowBox[{"determinados", ".", " ", "Es"}], " ", "lo", " ", "m\[AAcute]s", " ", RowBox[{"usual", ".", " ", "Notar"}], " ", "el", " ", "guion", " ", "bajo", " ", "y", " ", "el", " ", "s\[IAcute]mbolo"}], " ", StyleBox[":=", FontWeight->"Plain"]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"funcion", "[", "x_", "]"}], ":=", " ", SuperscriptBox["x", "2"]}], ";"}], " ", "\[IndentingNewLine]", RowBox[{"funcion", "[", "y", "]"}], "\[IndentingNewLine]", RowBox[{"funcion", "[", "4", "]"}]}]}]], "Input", CellChangeTimes->{{3.7311883124751997`*^9, 3.7311883565112*^9}, { 3.7311884712522*^9, 3.7311884737122*^9}, {3.7311932031842003`*^9, 3.7311932336602*^9}, {3.7312338452464*^9, 3.7312338579168*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "Una", " ", "funci\[OAcute]n", " ", "que", " ", "s\[OAcute]lo", " ", "se", " ", "eval\[UAcute]a", " ", "cuando", " ", StyleBox["x", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], "tiene", " ", "el", " ", "atributo", " ", "de", " ", "ser", " ", "un", " ", "entero"}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"uncion", "[", "x_Integer", "]"}], ":=", " ", RowBox[{"x", "+", SqrtBox["2"]}]}], " ", ";"}], "\n", RowBox[{"uncion", "[", "\[Pi]", "]"}], "\[IndentingNewLine]", RowBox[{"uncion", "[", "1", "]"}]}]}]], "Input", CellChangeTimes->{{3.7311883597422*^9, 3.7311884197892*^9}, { 3.7311932525632*^9, 3.7311932633262*^9}, {3.7312338726942*^9, 3.7312338983642*^9}}], Cell[TextData[{ StyleBox["\[FilledSquare] Algunos shortcuts \[UAcute]tiles: ra\[IAcute]z, \ potencia y cociente.\n\n", FontSize->18], "Ctrl + 2", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"\[Rule]", " "}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SqrtBox["\[Placeholder]"], TraditionalForm]]], StyleBox[" \n\nCtrl + 6 ", FontSize->18], Cell[BoxData[ FormBox[ RowBox[{"\[Rule]", " ", SuperscriptBox["\[Placeholder]", "\[Placeholder]"]}], TraditionalForm]]], "\n", StyleBox["\nCtrl + 7 ", FontSize->18], Cell[BoxData[ FormBox[ RowBox[{"\[Rule]", " ", FractionBox["\[Placeholder]", "\[Placeholder]"]}], TraditionalForm]]], "\n\nLa flechita es ->[espacio]." }], "Text", CellChangeTimes->{{3.7311841723702*^9, 3.7311843956612*^9}, { 3.7311932843082*^9, 3.7311933314681997`*^9}, 3.7312458765396*^9}, FontSize->18] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Primer ejemplo: problema 5.\n\n", Cell[BoxData[ GraphicsBox[{ {RGBColor[0.87, 0.94, 1], AbsoluteThickness[2], ArrowBox[{{-1.5, 0}, {1.5, 0}}], ArrowBox[{{0, 1.5}, {0, -1.5}}]}, {RGBColor[0, 0, 1], AbsoluteThickness[2], LineBox[NCache[{{Rational[-1, 2], 0}, { 0, Rational[-1, 2] 3^Rational[1, 2]}}, {{-0.5, 0}, { 0, -0.8660254037844386}}]]}, { {RGBColor[0, 1, 0], DiskBox[NCache[{Rational[-1, 2], 0}, {-0.5, 0}], 0.05]}, {RGBColor[1, 0, 0], DiskBox[NCache[{0, Rational[-1, 2] 3^Rational[1, 2]}, { 0, -0.8660254037844386}], 0.05]}}, { {GrayLevel[1], InsetBox[ StyleBox["\<\"\[ScriptL]\"\>", StripOnInput->False, FontSize->40], {-0.42320508075688773`, -0.5330127018922193}]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\\!\\(\\*SubscriptBox[\\(m\\), \\(1\\)]\\)\"\>", StripOnInput->False, FontSize->30], NCache[{Rational[-1, 2], 0.25}, {-0.5, 0.25}]]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\\!\\(\\*SubscriptBox[\\(m\\), \\(2\\)]\\)\"\>", StripOnInput->False, FontSize->30], NCache[{0.33, Rational[-1, 2] 3^Rational[1, 2]}, { 0.33, -0.8660254037844386}]]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\[ScriptX]\"\>", StripOnInput->False, FontSize->30], {0.2, -1.35}]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\[ScriptY]\"\>", StripOnInput->False, FontSize->30], {1.35, 0.2}]}}}, AspectRatio->Automatic, Axes->False, Background->GrayLevel[0], ImageSize->400, PlotRange->{{-1.6500000000000001`, 1.6500000000000001`}, {-1.6500000000000001`, 1.6500000000000001`}}, Ticks->False]], "Output", CellChangeTimes->{3.7310996831682*^9}], "\n", StyleBox["\nDesplegar este grupo de celdas desde la flecha de la derecha, \ como muestra la figura\nsiguiente:", FontSize->30], "\n\n", Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzs3Q2cJFddL/wOiRBJQOVFojASXuRNVHCDIqICoozAxYAI3FlERFbxWbOg goZnQ7I8ooiLa1TkeleSmzUqgqwkBta34JqLygausAEUYRU2gEZuXF0NmkCA 85ya7pmprqmuruqu7lM9+/1+Pj/CTlVXnaqu7q4+/z5V93vBi5/xw3fo9Xo/ eWb8n2f84E8//id+4gdf/r1nxH884eUv3fGC0+P/eehpvd4v3bHXy/4YJpA9 TKR7+aWY7Pg8nKwNL75wT/joybD63/T7Q6R7AQAAAOr5hV+8vrVpt912W/af nohIgzSWuu9RtmaWlneFPXv2hD3bt024jKfG3BLzsZiHJNkGtSOR6gAAAAD1 qB2JSOI0lrrvUbZesrrRZUdO9A+wYwfC8tIkyzkj5ndjsmP0DYN/z3c71I5E qgMAAADUo3YkIonTWOq+R9l6yWpH1xxbO8JOhCvPn3RZ58XcFPrjjx439+1Q OxKpDgAAAFCP2pGIJE5jqfseZWvmhw+tHWHT1I6yvComO07/KObsuW6D2pFI dQAAAIB61I5EJHEaS933KFsx28K+9XFHx8Klj5lmWfeJ+UDM7THPn+t2qB2J VAcAAACoR+1IRBKnsdR9j7IFs7Q7bHyqXR92T3S/o3yeH/q1o6yGdM7ctkPt SKQ6AAAAQD1qRyKSOI2l7nuUrZelHYfCibUD7Ni+sG3qZWbXqrs2Jjte98xt O9SORKoDAAAA1KN2JCKJ01jqvkfZalkKOw6tV47CsX3bWlruE2Nuibkp5hFz 2Ra1I5HqAAAAAPWoHYlI4jSWuu9Rtli2XRiOrJWOThwJF25ra9lnxLwhJjtm f2vw79lui9qRSHUAAACAetSORCRxGkvd9yhbK9v2HVs/tk4c2hGWWl3+Q2I+ Efrjj5Znvi1qRyLVAQAAAOpROxKRxGksdd+jbKVsCxuloxPh0I6lGazjopjs uD0c86Uz3R61I5HqAAAAAPWoHYlI4jSWuu9RtlC27QvrpaNjB8Ly0izWc07M e2Juj3nRTLdH7UikOgAAAEA9akcikjiNpe57lK2SpbB8YON6dcf2bZvhup4T +rWjD8WcO7P1qB2JVAcAAACoR+1IRBKnsdR9j7JFsrQjHDqxdlQdC/u2zXJ9 Z8e8LSY7fn9+ZutROxKpDgAAAFCP2pGIJE5jqfseZWtkacehsFY6OnHkwrBt 5ut8bMzNMTfFPHom61A7EqkOAAAAUI/akYgkTmOp+x5lK2RbuPDIeuUoHNqx NId1nhHzupjsGH5L6I9Fancdakci1QEAAADqUTsSkcRpLHXfo2yBbNsX1u90 dOxAWF6a17ofEnMs5paYZ7a+fLUjkeoAAAAA9agdiUjiNJa671EWPUth+cB6 5Sgc27dtzut/acztMe+MOafVZasdiVQHAAAAqEftSEQSp7HUfY+y4FnaEQ6t Xa4uHAv7ts27DVm9KKsbZfWjl7a6bLUjkeoAAAAA9agdiUjiNJa671EWO0s7 DoX1Ox0duTBsS9KO80P/unXZ9ese0tpy1Y5EqgMAAADUo3YkIonTWOq+R1nk bAsXHlmvHIVDO5YStePMmLfEZMfz62LOaGW5akci1QEAAADqUTsSkcRpLHXf oyxwtu0L63c6OnEo7FhK2Z5Hx9wUc3PMY1tZptqRSHUAAACAetSORCRxGkvd 9yiLmqWwfGC9chROHNoRlpK36edjsmP6bTFnT708tSOR6gAAAAD1qB2JSOI0 lrrvURY0SzvCobXL1SW9Xl0+58Z8KOb2mOdMvTy1I5HqAAAAAPWoHYlI4jSW uu9RFjNLOw6FjdJR6uvV5fOi0K8dvSfmnKmWpXYkUh0AAACgHrUjEUmcxlL3 PcoiZins2Bh01JHr1a3lS2MOx2TH9kVTLUvtSKQ6AAAAQD1qRyKSOI2l7nuU RcxjwqUbtzoK1+/uwvXq8lmOuSXmEzEPmXg5akci1QEAAADqUTsSkcRpLHXf oyxihscdHTuw3KFxR1nOiPmtmOz4fsPg382Xo3YkUh0AAACgHrUjEUmcxlL3 PcpiZvh+R0fC3uWujT16RMxNoT/+6IkTLUPtSKQ6AAAAQD1qRyKSOI2l7nuU Bc3Scth7JKsenQhHLtsTdnWudpRlT0x2jF8bc3bjx6sdiVQHAAAAqEftSEQS p7HUfY+yuFla3hX27NkVlpfSt6U858S8L+b2mOc3frzakUh1AAAAgHrUjkQk cRpL3fcoMts8N/RrRx+IuU+jx6odiVQHAAAAqEftSEQSp7HUfY8is012rbo/ ismO9Vc1eqzakUh1AAAAgHrUjkQkcRpL3fcoMvs8LubfYm6KeUTtx6kdiVQH AAAAqEftSEQSp7HUfY8is88ZMb8ekx3vvzX49/jHqR2JVAcAAACoR+1IRBKn sdR9jyLzyUNiPhZzS8xTaz1G7UikOgAAAEA9akcikjiNpe57FJlfLozJjvl3 xnzp2PnVjkSqAwAAANSjdiQiidNY6r5HkfnlnJh3xdwe86Kx86sdiVQHAAAA qEftSEQSp7HUfY8i880zY26N+VDMAyvnVTsSqQ4AAABQj9qRiCROY6n7HkXm m7NjrorJjv3XVs6rdiRSHQAAAKAetSMRSZzGUvc9isw/j425aZBHj5xP7Uik OgAAAEA9akcikjiNpe57FJl/zoi5NCY7/t8S+mORNs+ndiRSHQAAAKAetSMR SZzGUvc9iqTJQ0L/nke3hP49kDbPo3YkUh0AAACgHrUjEUmcxlL3PYqky4/F 3B7zzphzNk1XOxKpDgAAAFCP2pGIJE5jqfseRdLlHqFfN8rqRy/dNF3tSKQ6 AAAAQD1qRyKSOI2l7nsUSZunhv51646F/nXsNqapHYlUBwAAAKhH7UhEEqex 1H2PImlzRszvxmSvhdcN/t2fpnYkUh0AAACgHrUjEUmcxlL3PYqkz3kxN8Xc HPPY9b+rHYlUBwAAAKhH7UhEEqex1H2PIt3Iq2Ky18PbYs5e/ZvakUh1AAAA gHrUjkQkcRpL3fco0o3cJ+ZDMbfHPGf1b2pHItUBAAAA6lE7EpHEaSx136NI d/LC0K8dvSfmnHDJzp8I/3j4PeGVP/qSma1zaXlX2LNnT8yusLxU/Nsgu5bD UuFx27bvGZ5nz/awLfn+k1MtAAAAQD1qRyKSOI2l7nsU6U6ya9VdGx7be2e4 qHfv8K6v/4bwn096arj+ax8R9sTpT4w5o9X1LYUdh070X4gnDoUdS9vC9j17 wmVHTgy/SE8cCZet1Y+WlsOuOM81x4qv5GPhmlz9SWQeAQAAAOpROxKRxGks dd+jSJfy/N7jw7W9J4bbe6dnL46hvDPmJTFntra+88OVa2Wi668MF+65Jmwq Ca05cSRcuLwcdl12JJwYNU+ccmjHUvJ9KKdOAAAAgHrUjkQkcRpL3fco0pU8 LuZjvdOyF8XI3BzznLbWubQ7rH/6Hzs2qBtl44c2rkWXH1907MigbpSNQ8rN kx+ndOLK85PvRzl1AgAAANSjdiQiidNY6r5HkS4kG0t0bUyokffF3KOFdS7t ODQ0hujYNZvvW7TtwiOb5ync/yi/HLUjmWcAAACAetSORCRxGkvd9yjShTwk 5vaYUDNPbGGd51+ZGy905LKwfVvJfOdfuVE7OnZN+TyPuXT9WnfX73bNOplf AAAAgHrUjkQkcRpL3fco0oU8NyY0yEVTr3Mp7N64YF04sFxe81lan+lEOHLh tvJlrdeXToQrzy88fnnX+rXt1rJrxLpEmgYAAACoR+1IRBKnsdR9jyJdyAtj QoO8atp1Lu0Ih9YGFB3bN3Sduo0shR1rM504FHYslS9r275jpfNs2z58L6Q1 2Rin4nXvRCYJAAAAUI/akYgkTmOp+x5FupDzYkKDPHPadeauRXds34jxRL3z w5XrpaMdI2o9ufFL1+/emGfb9nDNaknpRDhyWW7c0WWD+yedOBL2Gn8kUwYA AACoR+1IRBKnsdR9jyJdyNkxH4gJNfKJmPtMub78teiK15lbz7Z94+9jlBu/ dOLK8wd/XwrLB46Fft1oV1jOj1daWg67BvWj0fUokXoBAAAA6lE7EpHEaSx1 36NIV/L8mFtiQkVu77Vxr6NeOP/K8dei26gvHQv7to1YVu5eR4d2rNWX1sYr XR92ly17aXdYXfKxS8NjOrDfZXEDAAAA1KN2JCKJ01jqvkeRriQbe7Qn5lhM KEk23ujXY86Zel3bwr6NAUUjr0W3ca+jK8P5I5a1bWNBuTrRtrB99Rp128vv o7RWOxq5bpF6AQAAAOpROxKRxGksdd+jSJeS1Y+eG/OGmA+fe//w2Yc9PBz7 qnPDFfHfL+xNf6261azVbkL+OnPFbNzrqKq+tDE0aV95najkMct7s2vWHQvX bB91nyWRegEAAADqUTsSkcRpLHXfo0gXk9WIfvm5Lwif+s23hF9deX44t83l l15nrpBcfenYvhE1ntJ7HVVn2/bLwpH4mGPXjBiTJNIgAAAAQD1qRyKSOI2l 7nsU6WpefOGe8NGTYfW/bS63/Dpzw1nacWi9vnTl+SOWtV6DyoYmjahBrWcp LO/aE645ltWN9oTto+6fJNIgAAAAQD1qRyKSOI2l7nsU6WpmUzvKXWeu4j5G 529csG50fWnjgnVhX1UtaGk57NqTjTc6EY5cpm4k7QUAAACoR+1IRBKnsdR9 jyJdzaxqR9n4nz17YnYtj7iPUXZtucE8e0ZfW25pedfYeXrbtoc9lx0JJ04c CZft2RWWR9ShRCYJAAAAUI/akYgkTmOp+x5FuppZXbNuXlmtLfUvUldZpxKZ NAAAAEA9akcikjiNpe57FOlqFrp2tG17uOzIiX7daPu29O2RLRkAAACgHrUj EUmcxlL3PYp0NYtbO9oWtmfjjTJZ7WjPnvIYiyRTBgAAAKhH7UhEEqex1H2P Il3NwtaOtu0Lx+q8+I9dGh6Tuq2y0AEAAADqUTsSkcRpLHXfo0hXs7i1o+2j xxoZdyQtBgAAAKhH7UhEEqex1H2PIl3NwtaOROYUAAAAoB61IxFJnMZS9z2K dDVqRyLVAQAAAOpROxKRxGksdd+jSFejdiRSHQAAAKAetSMRSZzGUvc9inQ1 akci1QEAAADqUTsSkcRpLHXfo0hXo3YkUh0AAACgHrUjEUmcxlL3PYp0NWpH ItUBAAAA6lE7EpHEaSx136NIV6N2JFIdAAAAoB61IxFJnMZS9z2KdDVqRyLV AQAAAOpROxKRxGksdd+jSFejdiRSHQAAAKAetSMRSZzGUvc9inQ1akci1QEA AADqUTsSkcRpLHXfo0hXo3YkUh0AAACgHrUjEUmcxlL3PYp0NWpHItUBAAAA 6lE7EpHEaSx136NIV6N2JFIdAAAAoB61IxFJnMZS9z2KdDVqRyLVAQAAAOpR OxKRxGksdd+jSFejdiRSHQAAAKAetSMRSZzGUvc9inQ1akci1QEAAADqUTsS kcRpLHXfo0hXo3YkUh0AAACgHrUjEUmcxlL3PYp0NWpHItUBAAAA6lE7EpHE aSx136NIV6N2JFIdAAAAoB61IxFJnMZS9z2KdDVqRyLVAQAAAOpROxKRxGks dd+jSFejdiRSHQAAAKAetSMRSZzGUvc9inQ1akci1QEAAADqUTsSkcRpLHXf o0hXo3YkUh0AAACgHrUjEUmcxlL3PYp0NWpHItUBAAAA6lE7EpHEaSx136NI V6N2JFIdAAAAoB61IxFJnMZS9z2KdDVqRyLVAQAAAOpROxKRxGksdd+jSFej diRSHQAAAKAetSMRSZzGUvc9inQ1akci1QEAAADqUTsSkcRpLHXfo0hXo3Yk Uh0AAACgHrUjEUmcxlL3PYp0NWpHItUBAAAA6lE7EpHEaSx136NIV6N2JFId AAAAoB61IxFJnMZS9z2KdDVqRyLVAQAAAOpROxKRxGksdd+jSFejdiRSHQAA AKAetSMRSZzGUvc9inQ1akci1QEAAADqUTsSkcRpLHXfo0hXo3YkUh0AAACg HrUjEUmcxlL3PYp0NWpHItUBAAAA6lE7EpHEaSx136NIVzOb2tFSWN61PWzr wPaJTBsAAACgHrUjEUmcxlL3PYp0NTOpHS3tCIdOHAvX7FoOS6XzZLWlPWHP nj1h1/JS8n0gUhUAAACgHrUjEUmcxlL3PYp0NbOoHS3tOBROZC+8E0fC3tLa 0PnhyhP91+axSx+TfB+IVAUAAACoR+1IRBKnsdR9jyJdzUxqR8u7wjXH+q+9 E4d2FMYeZWOOrgn9ySfCoR3GHUm3AwAAANSjdiQiidNY6r5Hka5mNvc76oVt +9aLR2HH0trfs7rRZeHIYMzRiSN7w/JSe+sUmUUAAACAetSORCRxGkvd9yjS 1cyqdtTbtm99bNGV58d/Ly2HXXs26kbh2DXudSQLEQAAAKAetSMRSZzGUvc9 inQ1M6sd5e9pdM2esOeyI2GtbHTiyGVhz/ZtybddpE4AAACAetSORCRxGkvd 9yjS1cyudrQUdhw6UXglnghHLttjvJEsVAAAAIB61I5EJHEaS933KNLVzKp2 tLS8K1x2JFc7OnZN2LNnl/sbycIFAAAAqEftSEQSp7HUfY8iXcp9vurccMYZ Z6z+/2LtKPt7Nn3y5S+F5V17hutG0Ykrz0++3SKTBAAAAKhH7UhEEqex1H2P Il3Ks5/3wvBjL7soPGH5qeGS1/zKau3oop/bFx792MeFF/34hWH7C1404bK3 he17LgtHNm5sFK4Z/OPEoR1hqQPbLtI0AAAAQD1qRyKSOI2l7nsU6VKy2tGH PnVruPrP3xMO/eX7V2tH1/zv94Y3vu3Pwwf/8ZbwvB/+sQmWm9WNrgnHBq+5 E0cuC3t2LYdtOw6F1erRiUNhh+vVyQIGAAAAqEftSEQSp7HUfY8iXcq5939g OHbi9tWaUTHZ3x/woIc0XOZw3ejYNXvCruWl/rSlHeFQv3gUDu1YSr7tIk0D AAAA1KN2JCKJ01jqvkeRLiW7p9G17/lQae3o8PuOrd8LqV7ydaMT4chle8L2 bfnpS2HHocF1645cGLZ1YPtFmgQAAACoR+1IRBKnsdR9jyJdyyv3vq60drT3 9Vc0Ws627fm60a6wXHJduqW169bFOQ8sG3skixUAAACgHrUjEUmcxlL3PYp0 Ld/1lPNLa0fnP/u5DZazLewbXKjuxJG9pXWj1SwthwPr8122cT07kQUIAAAA UI/akYgkTmOp+x5FupZ73uuccPTGfxuqG33oU7eGr7j3fRos5/xwZX9AUbh+ d3U9aGN8UgjHLn1M8u0XqRsAAACgHrUjEUmcxlL3PYp0MW982+Gh2tHb3/m+ hsvo38vo2DXFexyVZe2+SCfCoR3GHcniBAAAAKhH7UhEEqex1H2PIl3Miy/c M1Q7esWrL228jKXlXTXqRmvJ6kfl90QS6WoAAACAetSORCRxGkvd9yjSxTzy UY8eqh09Yfmpydsk0rUAAAAA9agdiUjiNJa671Gki7nTmWeG6z9y02rdKLv3 0d3ufo/kbRLpWgAAAIB61I5EJHEaS933KNLV/Npv/t5q7Si791Hqtoh0MQAA AEA9akcikjiNpe57FOlqnv28F67WjrJ7H6Vui0gXAwAAANSjdiQiidNY6r5H ka7m3Ps/MBw7cXs475sfm7wtIl0MAAAAUI/akYgkTmOp+x5FupozzjgjvPHt h8NZZ5+dvC0iXQwAAABQj9qRiCROY6n7HkW6nCcsPzV5G0S6GgAAAKAetSMR SZzGUvc9inQ5dzrzzORtEOlqAAAAgHrUjkQkcRpL3fcoIiKLGQAAAKAetSMR SZzGUvc9iojIYgYAAACoR+1IRBKnsdR9jyIispgBAAAA6lE7EpHEaSx136OI iCxmAAAAgHrUjkQkcRpL3fcoIiKLGQAAAKAetSMRSZzGUvc9iojIYgYAAACo R+1IRBKnsdR9jyIispgBAAAA6lE7EpHEaSx136OIiCxmAAAAgHrUjkQkcRpL 3fcoIiKLGQAAAKAetSMRSZzGUvc9iojIYgYAAACoR+1IRBKnsdR9jyIispgB AAAA6lE7EpHEaSx136OIiCxmAAAAgHrUjkQkcRpL3fcoIiKLGQAAAKAetSMR SZzGUvc9iojIYgYAAACoR+1IRBKnsdR9jyIispgBAAAA6lE7EpHEaSx136OI iCxmAAAAgHrUjkQkcRrbs2ePiIhI4wAAAAD1qB2JSOIAAAAAANAhakcikjgA AAAAAHSI2pGIJA4AAAAAAB2idiQiiQMAAAAAQIeoHYlI4gAAAAAA0CFqRyKS OAAAAAAAdIjakYgkDgAAAAAAHaJ2JCKJAwAAAABAh6gdiUjiAAAAAADQIWpH IpI4AAAAAAB0iNqRiCQOAAAAAAAdonYkIokDAAAAAECHqB2JSOIAAAAAANAh akcikjgAAAAAAHSI2pGIJA4AAAAAAB2idiQiiQMAAAAAQIeoHYlI4gAAAAAA 0CFqRyKSOAAAAAAAdIjakYgkDgAAAAAAHaJ2JCKJAwAAAABAh6gdiUjiAAAA AADQIWpHIpI4AAAAAAB0iNqRiCQOAAAAAAAdonYkIokDAAAAAECHqB2JSOIA AAAAANAhakcikjgAAAAAAHSI2pGIJA4AAAAAAB2idiQiiQMAAAAAQIeoHYlI 4gAAAAAA0CFqRyKSOAAAAAAAdIjakYgkDgAAAAAAHaJ2JCKJAwAAAABAh6gd icg8E50dc8/c3wAAAAAA6BC1IxGZV6KHxXwi5t9j7jb4OwAAAAAAHaJ2JCLz SnRtTPZ/bo25++DvAAAAAAB0iNqRiMwj0SMHdaMsP52bBgAAAABAh6gdicg8 Ev3poG70dzGn56YBAAAAANAhakciMutE9475QsxtMV9fmA4AAAAAQIeoHYnI rBM9azDm6BUl0wEAAAAA6BC1IxGZdaIXxFwbc8eS6QAAAAAAdEjXa0fRF8V8 ScxdY7bFPD7mSTHLMd8V820x5w2uiXV62+ufd6LTUrdhHtsRnRXzfTEPaXm9 9xkcJ+ek3gcj2veEmC+f0bLvFPPYmJ+KuTDme2Pu3ODx2WvszAbz3znmwYP9 /dXZa7XQluxvX5E9F4Pn5SGD12yWx8Q8clBLAgAAAACgQ7pcO8pqQTHHB9e6 +sLgv1X5l5hrYq6I+blBnemL2mxTSRtPi/mWmB8frPPFTWpYg2387zEHYz4Z c3vMO2MeOst2z2A/PDTmNTHvj7kl5v/GvLRim981eM4+F/PoltqQ1RE/nzse Ph6zL6tbpN4/g/a9ZtCuE1lNpcXlPiDmrTG3lrwm/mNQSxr5Ooi+LOYDucf8 a8xfx/xhzA0xO2NeH3OHwfxZferI4FjNr+u2wXNwRszhGq/XLNcFAAAAAAA6 pcu1o9Dvp76sZh/0qPxTzL3bbleuff+jZJ1Pa/D4swd1luIyLppVm2e0H15V sg3/NmLeby7Mt7ulNmRj0cpqjJ+OOTfx/jlnUFtZa9Ovt7DMbGzPxYWa0T/E /E7M/4n5bO7vWT3yrBHL+aEar6OsJnfmYP6fqZjvBwfz/HXN1+drAgAAAAAA nbIAtaM7xNw35qUj+p5/O+arYr4x5ukxz4/5kZg35ebJxriU3Weljfa9r6RN r2u4jPvFHC0s4+WzaO+s0uuPv7qosA3/MmLeBxTm+94W25HVj/64UDfJ8pTE +2dnoT3ZmJ2Jx5b1+tf8+4vc8j6VbXthnofHvDc3z5UjlvWtMTeNeH1l+/HX Y7bn5s/GFR0ozJeN8Xp1zD0G82TXmvyOmF8dsdzrYx4zWCYAAAAAAB3S9dpR 2Oivzq5z9veF/ucP9gbX0RrxmA/l5t07o3Y9qNe/t0y+VvG/J1jOowrbtmNW +3KWybY9tw1/N2KerB741sE8R2dR1+v1762T35/fnXi/XFVSP3njhMs6M+bP css5FrM0Yt679Tau+5jlJSPmy+pBP1XSxmdXtOPK3Hw/XDHfrxWWmY0D++Lc PAAAAAAAdMii1I5Cvw+6eH24146Z/7/l5s2uZfa4Gbbtrbl1fXzCZeT7+JPW OqbYD7ty23BtxXxf3Ovf5+leM2rHHQvHyrcl3i+fLKnLnJxgOdn4rrfnlvHv WZ1szGNemJs/u/bcyGsq9vr3YlqbN7vG3sh7d0XfkJv3SRXzFWtSf1GYBwAA AACADlmw2tHLCn3Qz6vxmD/Mzf832fiKGbXtxbn1fC7miyZYRr6tXzfLfTnD 5+iJuW04kLgtn861ZVvCdnxJSd1oLXdvuKxnFx7/6hqP+crCY26JeeCIefM1 ri9UHcfRPXPzflPFfC8srP8dhXkAAAAAAOiQBasdfV+hD3rk9bRyj3lW4TE/ Nmb+r425OOZPYv5gMGbi6eNqTtG3FdbzoMGyXj1Y1u/E3HfMMn499/gH1Ni2 bOxOdn+f/y/mJTHfE/PlM9jvd415WszPxLwh5jsq5v3q3Da8vmT6OYOaw9fE PLVX854/WY0l5pkxPxfzG1W1itxj/jXXlocWpn1pr3+fqa8Y99y2sP+KtZN8 HtFgOdlYqo8WHl855ij32OJ9ua4YMd+thfm+smKZZ+WP94r5nl5Y5jsL8wAA AAAA0CELVjt6cqEP+uU1HnNu4TH/FHNmyXzZtcCycU2f7ZX38f95Vm+oWM95 hfnfH/OPg9rRjw3+djLmtVktZsQyXp17/NdUrCu73012/b7bStqZ1Uu2t7S/ nxHzppJ6QpZnjHjMPXLzXF6YltU+/r2wnP+IudOIZWX3uNre69fwis9LNibm W8e0/59z89938Lfsvj5HCsvKxollY9Iqr4E44T7M6is3DfZh9vz+V2Hd5zdY 1osLj729V3FNucJj31x47Gd6hTFFvf69kYrP83kVy/yi3HxfXTFfsa56Q2Ee AAAAAAA6ZMFqR08q9EH/do3HnFnSH76zZL59uenZNb2eEvPQXn+czdrf3x1z 5xHr+aHCOn43368f/Wxu2mtGLOPC3DzfWLFNB3Lz7e31x/rs7A3XeEbe06bm vn5OblnZct8V88Hc37J6yFklj8vXE95UMj27pl2xDnTvEW14SeE5+cuYY7m/ /V1V7ST6eG7ecwd/y+pXf1NyTGT3AfqNGRyzPzpY/i8M/r27sN5LGizrPYXH 3tjgsb9dss1fV5jnq0vmeWbFMu+Qm2/k+Kno4YVl/l1hHgAAAAAAOmTBakff WeiD/vOajyuOz3lzYXr+unZZDWG5MH1vbvqfjaiZ7C2s49GF6fnxOH8zop0v zc3z+BHz5Mc3ZbWs03LT8rWn92Z9+xPu5zv1Nq6N9uGYu+WmvS23jk3jm3r9 8Vtr0982Yvk/XthXm66B1+tfU+5fBtOv7w1qdoPlvzv32CdWbEe+zvTg3N+z 6+bdNPh7dmxkY8FK7//TwjH7gcF6HhDK6yjH8s9hxXLO7vXHWuUf+4EG7bii 8Ngs31JxbK3lxyuWeUZuvsdWzHf/wjLfVZgHAAAAAIAOWbDaUXHc0V/UfFzx Omnvzk27c8ynctNeUfL47Fpen8/Ns69knlcW1nF2yTz5a6gtlUzPj7N5yoht edOo2k2vf/+jfH1hx4T7OX/fpRcVpuXHYb15xOPX2vBnI6ZnNYf/yC1nV8k8 b81Nf3ph2jW5ab9asR0fyM33mMK0P+7160aPHvX4Fo7Xbxms++bC328pHCvf WWNZDyqp6zSpHb2h5PHFcUePK5lnd8Uy75ib77sq5junsMzfKswDAAAAAECH LFjt6PsKfdB/XPNx/1J43J/npv1k7u9ZjWnUvXfel5vvEyXTfyA3/eYRyziY m+eFJdPz43E2XSssWur1782zNs/DC9OzOlj+fjq/P8E+fnxhX31FYXp+zM+f jFjGWu2oOL4kP09+TND3FKblr5eX1Xe+uDD9xtz0KyvW8X9y8/1Q7u8P7PXv 9zNyTE1Lx+tvDNb9G7m/PbLXv09RVS2lbFnFsTtZPtigLcXa0YleYVxar389 weI6NtVSC8fb2nxPrZjvSwrL/PnCPAAAAAAAdMiC1Y5eUOiD/sWaj7u58Lif y037i9zf31OxjD8oLOMuhelPyE37+xHLeHFunt8rmf6y3PTnlUwvjm36kcHf s3E83xzzxsL0ayfYx/8r9/hPlUy/ITf9wpLp+Xvg3FCxnvy9e4rXCPzD3LT3 lzw2P07sByvW8Ze5+d42+Ft27cCjMSd7JdcebPFYzcaArY13y66Pl13T8Fdi /q2kPpO15Y5jlpft1/8sPO6mBu0p1o7eUjLPd5W07acqlpm/DuPTK+a7a2GZ Ly7MM7mTR8PB/fvD/kY5HI5PtdJFdzIcPbi2Lw6GoydTtweYm+OHN94LD5/a 74QbWnpPPFX37Vbc7s5sU0c/r/P7Z+jcqqPtLXH88Ebbt8phO8QxPLnO7Ltu 2vKvHWBGOvp5sAXOaWAeFqx29JJCH/RFNR/3r4XHPSU37frc399RsYyDhWU8 tjD923LT3jdiGV+fmydrU3Hcx0/npr+o5PF/X9K3/w8xny75e5b/McE+fkfu 8ZuuiRZ9e8wf9fp1rNNLpufvgXOsYj3Xjqr/RB/MTdtU/4r+W8yfxvxkg23J 8vu9jTFLR2Z8rK6MeE5GZeQ133LLPFJ4THYdxTNqtqd4P65N1zPslY87qrrf 0Vfl5ts0Ti4335cVlvncwjyTu/FA2H7eeeG8RlkJu0/p7zw3hgPb1/bF9nDg xtTtoS3Z9/mDTmypct0lG++Fl1yXujUd0dJ74qm6b7fidndmm7r3eb3ab7x7 ZfO51fKl4YYOtneU6y7ZaPtWOWyHOIYn15l9101b/rUDC2Lxvvd17/Ngq5zT wDwsWO3oVYU+6F+q8ZjTesPXecvuOXSn3PT8fXUOVSznTwrr/v7C9G/OTRtV Ozq9N3zfpOL9Znbnpr205PE3lfTtZ2NWsjrU8V5/bMnrY94SczjmURPs4/z4 qo9O8Pgzc4//p4r5DuXme0JhWv66eCOve1ejLfl1FPOJ3pixPlMeq3+aW1d2 /H2y1x/jdnnM22NuLbRn5H2bcst8Xsl23K9me/LXIsyuV7dpzFV0XsnyR9Zn o6/JzfetFfPdvbDMJxfmmdxEtaONrOxetHOuNjgP2nIGv5favXJe2O4JpYo+ sRJqR1PZitvdmW3q2Od1/KzJPmdKz6kuuDqc7Fp7K2z5/m/H8OQ6s++6acu/ dqDrFvZ7X8c+D7bQOQ3Mw4LVjn6j0Af9mzUe85WFx7yiMP1XctPeWLGcvygs 5/GF6fkxRZuus5ab759H9bcP6gpr03625LGvKbThj2awj/PXvcvGM53W8PH5 esKtFfPla3YPLkw7nJvWuH6VW06+3vcPveEaYml9rqV9mN2Xaq1G+De9wj2j BvNk91w6mWvLjTWWm9UejxW24Vk125QfU/SqEfM8sKR29NqKZeavcXefivm+ urDMJxXmmVy+drS8M+wde726vWHn8vC5wfLOvafY+CPnQVvNyasvWD+eF+s7 BHOnT6yE2tFUtuJ2d2abuvV5nf+syY/h3ri2V7faW2XL9387hifXmX3XTVv+ tQMdt7jf+7r1eTD+nMY16yBvwWpH1xT6oN9e4zE/kZv/v2LuUZievw7e3orl /O/cfJ+N+bLC9Aflpn+sYjlr9YL/iPmiwrT8NdY2XW8u+u7C9n9nxXruFHPv Cfbxiwvr+PoR8z1s0N5HFf7+5MLj7zzi8Vfl5nl6YdrPFZZxzohlPKbXH9/z 4BHT35lbxiNj7huzP/e327K/z+A4zY+P+/aK+YrXQfzGGssuHgO/lJv27TF7 ev3xZxfFfFNu2tq9tLLXwL1GLPvOveFxcVn+rKItPzSY5zO9wvUXC/M9obDM pxXmmVy+drT9QBh/CjI4B9i7MywPjT86lc4HunXexvRuPLB9Qb9DMHf6xEqo HU1lK253Z7apW5/XN1y6nPvdzVUl947sVnurbPn+b8fw5Dqz77ppy792oOMW 93tftz4Pxp/TAHkLVjv6YKEPuvJ6Zr3+fVbyYzteVjLPN+WmH65Y1odz811Z Mv0B+ZrEiGXcNzfPW0um/15uetl9frLt+cK4ukSvf52+d8XcPME+vsegFjBy bM5g+e8ZTH91YdqjC8/R1414/M25eV5ZmP6gwjK+v2QZd+z1xxJl/3jxiG35 m9wy7pX7+/+b+/vHRtVSJjxGz+oN31/riyrm/dXCdta6P1U2X+4x/xLz4Jij vf79sF4U8/Be/55U2fSsjnTX3L9/Ycyyi+OaslrTF4+Y938O5vndMct8YWGZ P1aYZ3KNa0cDJ4+Gg0P1o+Ww86rjUzVlcXTrvI3pLe53COZOn1gJtaOpbMXt 7sw2devzOt9nfMHVZT+46VZ7q2z5/m/H8OQ6s++6acu/dqDjFvd7X7c+D8af 0wB5i1I76vWv13VboQ/6I2Mek78eXXbNudKxEb2N69HdHnPPkukPyNVssnm+ dsQ8+bZtGq8TPSM3/b+XTM9fL660Pz56b26elRHz/PfB9ImuaZerCWTJ7rF0 l8L0H8hNXy5MK94zZ9PYp16/PhWqtrU3PC7p74o1mOinctMfUfL4cwvr+MrC 9Dflpl3fGzE+aoJ997OF9T60Yt4rS+o0I6/9lntcNj7obwuP29fL3b8p+t7C srNkY93uPmbZv1PyuLIxcN/R26jLfsOYZV5bWN4fFOaZ3KS1o8zJo+HynRu/ Nzlv5YpT5Pcm3TpvY3qL+x2CudMnVkLtaCpbcbs7s03d+rwe32fcrfZW2fL9 347hyXVm33XTln/tQMct7ve+bn0eeC+DZhaodvT8kj7t7PpaXzNi/h8u1D8e ULHsbKzMfw7mfX3J9Etzyxp1r5iHFdr2P0vm+a3BtI/EnF6YltU6Pl3o5//y kmU8KzdP2dilrIb1qV7/3j6VffoV+yMb05O/Rt8f9Pr3jbpLzPfE/Nvg7+/P b0evP57oQGE/XFSy/O8pzJPdA+quhXnuOlj+2jy/GXO3mC+N2T6ol2R/f0fJ 8rM64xsL63hoYZ7vLEz/w17FGKEa+yzbP9m19r7Q27xtz87Nd69e/1p723P7 MZ9s3M8vDpJdd25pxPq+NubW3OMO93L1sd7w/bfW8oYa2/Gokset1vdiVmKe 3uvfl2vt3lF/UrGs7P5JP16yT7LXbVZHPSukrB2FrHy0L6zkrnN7xdjiUf66 t7kcPBrq/1rleDg84n5M1fddyj/u8Hqd6+TRg5uWc7DyAnxNztva2N7sVpTF ZeSuETi43+fGtYUrFzT6XlZj2rTehrX5srFno9rUWJrjYm2b9u7Mj7nfW+M4 aOd5rbVVUzz3TZ6z/HrG3r+scr1tvc6KRuzz3DraUrnPG/WJze84GXX8t7KP x7a3/ntia/t2ivey3EIm/CxprlvHVDvb3a1tGqXB53Urx1SJ3Ptu/p7SK7vL lj//84tJ3zvK+4zK29To9TSr56F0VY7hSXVh301+Xuq1Mx8zPC8Ze+5Xfi5a to9K93Hd+Vptcx3z3a72vhsUdf9738TvLzN9HXbg86DROU39+x218n23heO0 m887W0XXa0eDvur3lPQ/5+tCe2JeHvONvf59bfJjUm4Y1f9eWM+TehvjmrI+ +2+IeVrMK3ob94DJxifdccTjv6vQrmx80r3CRi3mwty0Hyg89ltj/m/Jtm26 n1OvX59562B6tk929/r3ucnuM/SaXD3i56fc71md5k9ybbk1Vy9YG+vyzbn5 s7Ewby7ZhuzeUF+bm69YE6qqgy3F/FVunv8sHAfZteGKNaFzesN1r7U8vzDf N5bMk9WbRt63Z8R+unevf124z5QsL59sHNVdRmx7VX6tYt3P6A2PxcuuX/d9 YaO2VFzW99TcpuweYf9eo23/1BsxTqrXf/1UPTY7lv4lJK4dhXBDuHQ5d95Q UTzqfxbvDTtz869neWfYO/Zcc+38Y3euXjWctfOW8sVcFy5Zn/eScN2gTfnz x+HzyFHn+PXO26bf3rB+HpI/N1u/RuDewTlune/oa+czu1dK91udNq1/182O k03XLJz8908pj4v89/eylP0WrZXntd6Omfq5b/KcNfrtWOV623qd5XdFxT5f uzdrR/Z5rTa3eZyMOf5b2cdj21vjPbFD72XTf5Y00KljqqXt7tQ2jVP/2Jzu mKpqQu48a1TWz7/meH4x5XvHps+M1f1Y3qb+cTXmfWjWz0PJuhzDE+jQvpvk vNRrp43zjnHmcF4y9tyv/Fy0bB9l+3ijZjBo+4j56v3Obpbnq/Pdrva+G2xY lO99jd9f5vI67MDnQdvnNG1+323hOO3m885WsQC1o//WG75/zKhk9Z2sjpTV mbLrbl3c64/RObvBur68169B3VSy/KwGdbeKx2Z1jrfEbIv5yV7/3kxZu7P7 Dq3Vc7JxRXtizig89rd7w3WZtXxoxLqy+tGFg3WUPS6r+Uw8hqawru+L+URu 2dm2ZOOwHlSY73Ejnqfb8zWLwfaXPX9/XLGtP9Qbvj9SVmd7dcxXlcz/HSOW /5nBcXH3mJf2ysf8ZNeuG1tnLKzvcGEZWU0tq6n87eC5z8ZsZfcnuqTXH+uU XafufTEnYn4/5sW9fv0vG6/1V4O//+3geHtnzP3GrP/xvf44tuLzf3XJ9j2r wXZ9ScxLBu0o25/Z/b8eVvH47LqE2fik6wb7Ye/guc+2MxvDlNXpfjAkrx2d DNddnOsTvuDqknPP/jne5nOCkqzsHvG5mi1j1Ll42Tlj2W9Chs+3rz56cEyb snP8su82486D2tje0P+dS9V5yHn971+Hrxhzrp6d04xZTrFNZU3aOJe6NFx1 efE8KnvcvobjjtIfF81qRy09r7V2TTvPfZPnbFa1o8lfZ6s7otk+n/Z3dG28 3uZ5nNQ+/lvcx6XtHfOe2Kn3sjY+S2rq1DHV0nZ3apvqGHNstvT5WN2ENvtZ 2tpv0793DH1mXFFnP8ZljXqPnsfzsL7pjuGtsu+anZd67Uz9/NeS4LykdN80 Oxdd3nn56rFy/PCYti/vDJeXfzjO6Xx1vtvVbu0o/bnMZLWjGu8vc3sdduDz oM1zmra/77ZwnHbzeWer6HrtKGz0RWfjO7LxOU8Y9MFn+aaYh8Z8Ra9Qj5ly Xdm1trLry/10r3/trwf3Row3GrOcrxssJ7seWjZ+adM16HLz3nGw3kcPtjEb NzK2/hOdGfMtMT8/qGP8bJv7YrCOL475kUENp/K+QL1+/e0Rg5pGti33KkzP rjn3gl6/JvXVg32UPa9fNma5dx3UWVbGPRe9/v2eshreXQbHRnYdv+xacacN pt8p5tt6/euzZfv83Jj7rk1vuG/uNDg+7tfbuAbb3NPrXzNv22C7sv2Ur/et 5TcmXHb2PL0o5vt7/bFOz+8VrjE4ZSY3de1o+JrBZcs4efTy4XO87HywMJZ3 aExC2efq8auGljE0LnotQ78LWQ47ryouJH++vRJ2rq2z0J7hc9r43WZTY6rP g1rZ3uzcuHC+MrzNG+fNKyu5c5ZN50BxOftWhrenbCx14Tc1ZePH1s+lllfC Srbutd/Q1L4+wQz205THRZNrF7TzvNbaMy09982es9nUjqZ5nZXs82L7i79b HPmdftp9vtHecft8fsdJjXXtHn7t7yvZN+20t+o9sa1929J7WSufJbWenW4d U61sd8e2qZYxx2ZLn4+VWrxmXVv7rY33jvxnxvrzvek9uvC78OWLw3WbFjWn 52FtXY7hLbPvmpzjeO1M+/zXM5PzkonO/crORfvjGMr3cX9a/xguzFfYf8sX X7ep7jO/89X5blebffJdOJdp8r2v/vvLPF+HHfg8aO2cZgbfd1s4Trv5vLNV LErtSETqp9cfB3dbSf3of/XGjGVKkMm1UDs6efUFuXPhS8MNQ1OPh6vWz88G 56ylfcQHhz5bV/YN/x4rv47V359sXkT/XCb/+bxpDFT+fDt3jlJY2Kax/5vG 01SdB7WzvcPnxv3lDC9mxG+3iudAJ68LF68tZ/XcZ8Rv/Mbuu+JvtSp+A1lL l46LOvdMbel5raOt5z40e85mUzua5nWW3+eD73ab2t/fF8PfAa9qfj35sft8 xDVBNm37HI+TwpjP1f2zudFDbd68b9pqb8V7Ylv7tqX3srbeM8bq2DHVynZ3 bJvqqTg2W/x8rGv8++wczi9aee/Y/Pvt8vfo7H4Uw/2YFxc7wOf5PDiGt9S+ q3+O47XT1ntYtbbPS6Y59yuei2b9usXxtHEfbxozUDJfcWxE5XfdGZ+vznW7 2vxu0KFzmVDne1+D95e5vg478nkwMNU5zYy+77ZWO+rU885WoXYksjXT618T b9Q1Hj8+bgzZHDO5FmpHQ5/Bg2svrzt+xfr3nrVx8yPFz9Z9KxufwfmfPOfP 8ZYvLZ7ZDi/j4KbfvKw3dPh8u+o3oIXzmeHva1XnQe1s7/Er8t+rRi/n+OHC 7xI39bdW7Y9ie64OF1QcC0PnUitXTPB9J9/wLh0XNb5DtNTeOlp77kOz52xm taNJX2e5fT7u95nDv2VcCU1/0lV3n2d9HkPfYYrbPsfjZOg7y4jf7vbblN/H F4Sr87O11t7R74mt7duW3svaes8Yp2vHVBvb3bVtqqeqn6W9z8e6putnaWm/ tfHeUdiW6vfoQr9q8fib4/PgGN5a+672OY7XTmvvYZVaPi+Z7txv+Fx0ZJ0m v77a882qzXXMcbtCi98NOnQuU1xOrdpR1fvLXF+H3fg8WDPNOc2svu+2Wjvq zPPOVqF2JLI1M7geYNl9nbK8P3X7cpncjGtH+fOCC4onpCVGzT80tmltjHvj /rzh8+3qMcPD45CHz0/rnQdNvr0nw9UXrC2/5HeGw0sY+l5ZdW/SynYcLlzj YEztqPJ8vc76OnVcjP8O0VZ7x2v3uW/ynM2qdjTp62zot4Ul19EoLif/W9dm 1wNoss+Hn9vits/vOIluuHTjN2wrVXXBtXsKZxn+zWl77R31ntjevq1r3HtZ W+8Z1bp3TE2/3d3bpnpq3Kd53PprfD7W1VY/y1T7rYX3juK2jHuPHjr+Znie Us0xvNX2Xd1zHK+d9t7DKrWwf9o79xs+Fx39vOfnqzq28/MNvw7nd7463+1a ndrSd4PunMv0Na0dTfu9N9PO67ALnwcbJj+nmd333TZrR9153tkq1I5Etm4G 93p6b65m9OmYN8U8KHXbcpncTK9Zlz8vGHFN4mJ2j6jXFK5v3F9X0/vtDF8j euzpev470ND44jrnQdNs7w3h0orfK1bu/zEnTKvXAihpx6bx4GNqR837PoZa 0bHjYtx3iBbbO1a7z32T52xW9zua9HV2w6XLjY63yftXmu3zofYOrWeex0mh r2Ki7zZttnfUe2Jb+7ak9ZO+l7X0njFmQ7p3TE293R3cplrq97NM8/lYVzv9 LNPtt+nfOzZvy6i+t9zMczlPqeYY3mr7rt45jtdOm+9hletvYf+0d+43XDsZ 3Zzq2kmuZbnXwPB88ztfne92rS6lle8GXTqX6WtaO2ryvXe2r8MufB5smPyc Znbfd9usHXXneWerUDsS2dqJ7hTzDdl9jmJOS92ekkyuhdpR/px5dI1lghTO O48eLIxZLiZ3z82DpQOf8+fRhWvrNdo3o86DWtre/LjmkmtPb96s8d8ry+4N WpkxtaMJf/rY7n5a1cZxMe47RJvtHaPl577Jczab2tGkr7Ph36PV6oIYWesd o+k+z88/tO1zPE5C876Kzdps74j3xNb27Ybp38vaec+ovR2dOaam3O5OblMd 4/tZ2vh8rGvyfpb29tv07x2bt2XscuZ0nlLJMbzl9l298xavnTbfw6pMv3/a PPerWzvJz1fVh11v/MRMz1fnul2DpbTy3aBD5zJrLWpYO6rztj2f12EHPg9y Jj6nSfh9t9Vlhe68/7IY1I5q978/JeYHUrdDZAtmclPXjoZ/SzT6t+jTnitm svvG7h+6h+aolN6fdVFqR02fkzHnQNk5zabfvAxlcM/SBtes607taHULpzwu OlQ7avm5X9za0QTXZJj0vazx43Lbl7A/cfrX4xxqR63t27623svaeM+o3rVd Paam2O7ObtM41e8l7R1T9XShdtTWZ3mbfTVzeR4cw1tu3827dnTKvnZmsV2l 2jz3q1s7qXvOWue9ecbnq3PdrsFSOlc7ysz6e1/zbZ/f67ADnwc5E5/TJPy+ 2+ayuvT+y2JQOxqfaFfM52IuT90WkS2YyU1bOxq6L2rVPTDWPjsbZFR/Xf7e hPv3bh67vpaV3WF4EfOsHU2xvW3+Fqd4z+LiuP7VrF1rPLd/5lY76sJx0aR2 1GJ7S7fBuKP+3l/0cUczPk5C2+OOpm3vHMYdtfhetrG+yd8zam9HF4+pSba7 69s0UkU/yyyOqTHaqR1Nt99mMXZiqs+MeT0PjuEtt++a1468dmbZd9ndcUdV 56LT1lhSjjua5XYNltJ67WgRvvc13Pa5vg478HmQc0qPO+rY+y+LQe1odKLT Y34t5gsxF6duj8gWzeSmrB0dvyr/m5/ivU0mOJ9uLH+/1f2bfoM0+r6pNa6t e93FU9zvaJrtbe8awPl7ja7+9qrqfqL587iZ1o66dlw0ud/RrNq7Jt31n5tc Q6V6ve28zhb7fkezPk6Gj9mx9xWI3282fwdvs72zv99Rm+9lIx7U6D1jzIYs 0DFVd7sXaZvyRvezzP6Y2qyt+0pPs9+mf++ouy1DM8/lPKWaY3ir7bt6x6DX zqqW3sOqtLF/ZnO/o9nWWNLd72i+taPJvxt06Vymr83a0Xxfh134PNjQxfsd TfcdtqvPO1uF2lF5orvG/FHMrTHPSd0ekS2cyU1TOzp5NOzL/d5ieedV4Xhh lvz59Mq+is/U6hXlzgfXfr8xuk0Hc/fXrLpv6sXXVS4oXHdxru1D55z1vj9M s735c+wm7aw6V69eTqEPv+Q3QO3Vjrp2XIz/DtFOe+to67nva/Kc5bex+t7N 49bbzuts6J7LJe8tdZczXrPXSf67QnHb53ecFF6vK1dU7p+h7ze5L+3ttbde f/fk+7at97L23jOqde2YamO7u7ZNdbVzbI77fKxr8n6W9vZbG+8d9bZlaOa5 nKdUcwxvtX1X9xj02mnvPaz2OibcP+2d+82vxjK/89X5blemre8G3TmX6Wuv djTv12EXPg82tPV7mGm/77b3HbarzztbhdrR5kTnxnww5lMx35y6PSJbPJOb tHaUnZPtHR5ztK/kJG74s3JnuHzMfcfX7jeYZeP+lsOfzzuvOl7dtPw5/ND3 tfx5dHZ+f/nI886TRy/PjX0vjqeq+P7QyvYWf8tSt51TfK88frhwj8fN3zPa rB1167gY/x2iree1jnae+74mz9nQ70UrvvuOX29Lr7PjV+TuhbsSdldc56f6 9Tpe3X2+6foEhW2f53EyfL3Q8vff/nz5Gv/w+0B77a37O8hJ921b72XtvWeM 061jqp3t7tY21dVCv0WNz8e6pqkdtbbfWnjvqLctQzPP5TxlHMfw1tp3dY9B r5323sMqtbF/Wjv3m2ONZY7nq/OuHbX13aBL5zLF7Zpb7aiV12E3Pg/WTHNO 0+b33fa+w3b1eWerUDsaTvToQc0oqx2dm7o9IqdAJte4dlR+f8qV3YfLP6c3 jU0aNaa3v9yN75vD3zmGro23snvkuODsXHPj87n4WT7cp712zeXiKf7wMsrO QSrOs1va3rjBuXObeu0sOwfK/w4na8vmrzOD33Dlf5s14rymzdpRt46Lwjnn xddtXkZbz2sdLT33mSbP2dD3uuy77/7ia7p/rIxfb1uvs+Phqvy6Sp/fzW0a /5vPEiX7vNhHdfLowc2vk+K2z/M4ybZ9X+F6CcWdXKzxr+wb/m7WWnur+lTa 2bdtvZe19Z4xVseOqVa2u2PbVE+9cTzTfj7WNU0/S3v7rYX3jlrbMjTzXM5T xnIMb6l9V/sY9Npp7T2sWhv7p61zv3nWWOZ4vjrn2lFr3w26dC4TanzvC5ON a5z967AjnwcDU53TtPh9t73vsF193tkq1I42Ej17cI267Fp1d03dHpFTJJPL 145K7/FXzO7cb6uqPi83HD9cuH9l6XqGl7upFlU45xzV1qHfOm76PjJ8La3l 5bV1VSyj9LdR1efZrWxvdr5xebE+V9HOEedAJ4/uK6yn2I5R9xbdfO3hVmtH be2nVo6L4d+m5ZcxNBaslee1jnae+0yj52zo96Jr595lx0o8v99ZdS2vtl5n Jb8P27TPNz8n437LOHKfHxx+7la/1+bWtfZ9Y3llZeQ9eTLzO07C5t93Z9+j 8+spjA0t+y1sO+2tek9sZ9+29l7W0nvGeB07plrZ7o5tUy0V43ha/Hysa6p+ ltDifmvhvaOt/u/5Pg+O4a2075ocg147Y57/rKaz/rgx1wOr0sL+aefcb841 lrmdr853u9r7btClc5l63/tqj2uc6/twVz4P+qY7p2nv+26bx2k3n3e2CrWj fqJXxHwh5vUxp6duj8gplMnla0eNU/67w82y75ujPj83J/vsLatFZb89KT2H KEv2XWXTQvLn0ReEfQc318GGkp1Llv6eacx5dkvbu/m6gCOeg8pzoOPh8Kbf u4xqR77Nm+9l2nbtqDvHRShcb2Ijw9c4aOl5rbVr2njumz5nm/uFyl/zB8PR q0f3ZbT3OltrU83nt+L3h/Vkv3Uc09a4joNXXbrxnlm6U+d4nIS6x3/Zb/Da bO+498Q29m1772WtvGfUe3Y6dUy1s93d2qbxqn8r3tYxVde0taM299u07x1t 9X/P/3lwDG+Vfdf8HMdrZ+TzP/R9sOy9p752zkumPfebc41lbuerKbarje8G a8vqyLlMje999V+n83wf7tLnQQvnNC19323zOO3m885WcarXjqI7xlwZ8/mY F6duj8gpmMlNUjta/31Ok9+lrd3fcvQ549rvF6vHMFUvY/33baULKZ5HF8fF r2XwW5WR5/XjzrPb29613wGWniOvbmuNc/XjhyvbsfY7mcPHq+/f2n7tqL39 NN1xsbqEeCyUPP6Cq0uvOTH181pr10z/3Dd/zgbXpCw7F15b58kwpi+jrddZ rlVVz+/a+9HUO7x6+1dfJ9k68u+ZI3fqHI+TsLZ/yr8z9dcz7rfu07a3znti C/u2pfeyjX02zXtGXd06ptrZ7m5tU7Uxx2aLx1Qd09eOMu3tt2neO9rr/w5z fx4cw1tj3zU/x/HaGfn8t1g76jdr2vOSac/95l1jaaPNdaTYrja+G6zpyrnM +O99jV6nc3sf7tbnQSvnNG30dfQ3vpXjtJvPO1vFqVw7iu4R886YW2Kekro9 IqdoJjd0jYKameq3/WvnjJtT//7Bo5dRfT5efh6dvx9nP3X7WevUz9rY3rI2 bmxrvfu5j25HfpetXqt+xL7Mt6GVbvqa7Zv9cVHx+DHjYaZ9XuuY5rmf+Dlb PReu2If56ZXj+6Z5nRWN2OdTvR+NULL965uZf88cu1Pnd5ysf2eaaj2TtrfB e+LU+3b697I6y2qnFpnTqWOqpe3u1DbVWfaoY7PNY6ra+Pfk+Z9fTPre0ejz pfIzY/z2tP08lLbLMdxMB/bd5OelXjul9yHav3bdwelrR4PGz+68ZOy5X37d Veebdedr4b25lfPVhNs11XeDqvVOenzM9ntf8/eXebwPd+vzoM1zmun7OtYX NNVx2s3nna3iVK0dRQ+J+fuYj8d8fer2iJzCoZa6v8FKr8l5S/5ejRe4gO7C W/znfnFeZwAA9PXPK9uqHQGUW/zvu9DcqVg7ir4j5t9i3hPzFanbI3KKh1oW p087P166clzz0D1DXT93K1j8535xXmcAAGROhqsvyK6tdnG4TvcsMEOL/30X mjvVakfRjpjbY94S88Wp2yMi1LM4fdrHr8hdq3fUPVWzMdf5+0uu7Gtw/ym6 avGf+8V5nQEAkHXRXr56346VfTO4FjBAzuJ/34XmTpXaUXSHmL0x2T9eHXNa 6jaJyGqoZYH6tI9fNXzfxbX7q+YzdC/IlbDbtXO3hoV/7hfodQYAcMrr38t9 eefeGdzTFKBg4b/vQnOnQu0oOivmqpjPxvxg6vaIyFCoZZH6tLP7Lu4dPqca mXguVXkfVBbLoj/3i/Q6AwA41fXv937Qj/qBuVj077vQ3FavHUVfGfPXMSdi Hpe6PSKyKdSyaH3a/e9xQ2O1h7Icdu7NfpfjXGrrWeTnftFeZwAAAMzPIn/f hea2cu0oemTMJ2M+EvOg1O0RkdJQy/FweH0c9AKdg5w8Gg4Wx3Cv5qBr/m51 C/ncL+jrDAAAgPlZyO+70NxWrR1F3xPz6Zg/j7lb6vaIyMgAAAAAANAhW7F2 FP1kzOdj/lfMHVO3R0QqAwAAAABAh2yl2lF0Rsz+mC/EvDxlW0SkdgAAAAAA 6JCtUjuKvjTmT2P+K+aZqdohIo0DAAAAAECHbIXaUXT/mA/F3BTzqBRtEJGJ AwAAAABAhyx67Sj6lpibY26IWZr3+kVk6gAAAAAA0CGLXDuKtsfcFvP2mLvM c90i0loAAAAAAOiQ1RpQVvO59toQ9u8P4ad/OoRnPWs1R57wvP7fsmm33rr5 cQXzqh1Fp8W8Mib7xy/HnD6P9YrITAIAAAAAQFfceGN413f8QAj3vGcIvV4I d75zCA9/eAhPe9pqbj7nAf2/ZdPufvcQXvayED760dWHpqodRWfGvDHmczH/ z6zXJyIzDwAAAAAAqf3zP4fwzGeGcPrp4dN3uVsIF18cwvHjm2Zbrw9l0y65 JIRzzgnhDncI4RnPCL+25w83zT/r2lH05TF/FfPvMcuzXJeIzC0AAAAAAKSU XX/uXvcKYWkphCuvDL/4C38xctZNY4s++9kQ3vjGEO53v37NKVtWTpu1o+z+ RTHflfv318R8LOZ4zMPbWo+IJE9CJ8PRg/vD/v1ZDoajJ9O2ZrxFay8AAAAA 0HmveEV/3ND554fwr/+6+qeya8+tGTnt5Mnw4a97fH9ZF10Uwhe+sPrnlmtH b455xuD/P2kw1uhd2dijttYhIp1IQjeGA9vPC+edl2V7OHBj2taMt2jt3XD8 8P5wULELAAAAALrl5S8P4YwzQnjd64b+PFHtaG1atqzTTw/hx3989W9t1Y6y 69HF3BbzrTE/Ori30e9m9zpqY/ki0qkktGi1mEVrb8iKRqvjpHavnBe2L0SD AQAAAOAU8au/GkKvF8Lll2+aNFXtKPOGN/SX/drXtlI7Glyr7qMxn4r5zZjs jz8Tc9q0yxaRTiahRbsG3KK1N7b46gsGtS61IwAAAADojN/7vf615X72Z0sn T107yrzqVSGcdlr47BVXZP+aqh85+r2Yz8fcGnN7zIUx94y5w7TLFpFOhi3s xgPb1Y4AAAAAoEs+/vEQzjorhB/90ZGztFI7ymTryNbV6y2FCfuQoyfH/Otg rFGWTw/+/YWYa2MeOemyRaSzYQtTOwIAAACAjnn600O43/1C+K//GjlLa7Wj uI4v3Pe+IfR6vx8m6D8eXKvuH3N1o6xe9MmYP4z5ukmWKSILEbYwtSMAAAAA 6JC3vz2s3ofobW+rnK212lH02d///f46e70nh4b9x9GRXM3oEzFviLl30+WI yMKlBcfD4f1r9wEazuHjVY+rf/+g44eLy87Nf/zwxt9LVph/bHV7xi2ryf2O 8vPmcvBoGHebpPL2lu/jgyMasbaMvTuX12tHyzv3jn3cNO0OJ4+Gg2XPDwAA AAAQwmc+E8L97x/C+eePnbXN2tFtt90WQq/31ph/iDkz1Ow7Hlyr7rMx/x7z ymwMUt3HisjCZwprdYbdYWVQnyhmZXdVzebGcGD72rzbQ+mwmEEtZ/dKcdnL YefeQQ3kuks2/n7JdZsWcd0lG48rmVycuWJZNdob1uo2e8PO5ZJ9srwz7B1R 4xrV3v7yyvdxvx60uU6TX0ZZysYgTdvucOOBsP288fsHAAAAAE5Jb3pTCHe4 Qwgf/ejYWWdQO7pfzOdjnhdq9BtHd415e8zzY86o8xgR2VKZUFY3GlFnKK0h lY1DGVOLycax7F6pXG5WOzl8RVdqR/1a2uY6V0lWdo+sw+Tbm9Xexi9vOey8 fHhcULPaUTvtVjsCAAAAgApPeEIIT35yrVlbrx1lfcG93ttj/jLU6DeO7lRn PhHZspnM8auG6kZr44uGsndnWM7XN646XlhIVS3mZDh6ef7xxXVs1K1WVnL1 pYS1o5NHLx+upWV1lsI+yV9DLpteVoYprfusjftZT2EcUpye371NrlnXVrtd sw4AAAAARvjIR0I47bQQrr661uwzqh09bXDfo4eH9P3SItLtTOTk1RcMj/05 XjbT0XAwXz+64OrCPXMqajFDtan+9emG1zFirEyy2tHxcNV6faXf3rLxOSeP Hhyqqa3s23wfoWLtaLXms+l+Q9n9j4bHfV1w9eZizY0Htldep67NdgMAAAAA I7zsZSHc5z4hfO5ztWafUe3o9JhPxPxKSN8vLSLdzkTyNYnlS28YPWN+LMqm +sfoWszxK1ZytZPLR45hOX64MP4mVe3o+BXr7ahq76q4T/at1bwK44WK7c2m Xz5yYSfDdRcvVz4PY2tHLbYbAAAAABhhaSmE3btrzz6T2lGWXu9VMf8c0vVH i8hiZCL5cUfr1zjbVBsaZ1Qt5mS4+oKNMUcXX1e11Py4mXS1o3ytq2z8z6ZW V8yfb+/yxddV7tOh56Fk48bVjtpsNwAAAABQ4uMfD6vXirv22toPmWHt6AmD 69Y9IMy/L1pEFieTKdzvqPSePKUXsssbVYu5IVy6vuwLwrgSxbj6yexrR/la 14h7PxWzOzeuqjBeKN/e8uvM1W3vuNpRu+0GAAAAAEr8zu+EcPrpIdxyS+2H zLB2dFbM52K+P6TrkxaR7mdC2f2GCteLK2ZtPFLMwdKLoY2oxZy8OlywXo+6 NIytUIypn8y+dpT/+wQprCff3rFje6aqHbXbbgAAAACgxM6dITzykY0eMrPa UZZe769jXh/S9EeLyGJkCsfD4aw2tHdnWB5TZ1jeubdkHNKIWsyNB8L2tcdu PxDGjLzZ0rWj6dqrdgQAAAAAyT3iEf36UQMzrh39SszRkKY/WkQWI9M7eTQc XL+u2d7N17LLjUMaLh9txXFHy2Hn3hrXfqu4tl+a2tH07QYAAAAAStzlLiG8 /vWNHjLj2tEPxXwmpOmPFpHFSMuya9nl6guFMUkrVxzPzTv/+x2NuwZc9bLq 3O9oOUx7G6D51Y7abTcAAAAAUPCZz4TQ64Xw5jc3etiMa0dPW21Tr3d2mG9f tIgsTiaQrw8dDKW3Mlqf9Wg4uHtlRH2jXi3m4usqVxCuu3i5sn5yw6XLFfWT Jssa1d7hdazsOxrGlLsqza921G67AQAAAICCT34yrNZp3vGORg+bce3oWwa1 o/uG+fZFi8jiZALDtZ2dVx2vnjs3lmd5aHDL6FrM8Ss26k3LOy8fWZ86efTy 4WvkjamfLO+8Koxq7fhljW7vyesu3hhftbwzXF5ZUIvbd3hjbNbBwrzzrB21 2W4AAAAAoODo0bBap7mh2UV/Zlw7evCgdvTIMN++aBFZnEzk+FW5a9Gt7A77 D5aPWclqDXt3Lq/XmYbHEI2uxcQV5Oo4/XvxFG+vM7zs0fWTofrIeSth9/7D hfpRfxzV+GVVtPfk0bBv5bxcjWrviH1yPBzevz/sXtloz7451Y6WL75uc3ta bDcAAAAAUHDttWG1TpONP2pgxrWjuw9qR08M8+2LFpHFyWQKNYdszMre/bn7 HA2yOz/Pyr7C+KGKWkxWz7m8cK+k3RXLrqifhJPXhYvz44lW60f5Ze0d1KmW w86do66vN669WS1r7/C4pdJ9sjusDG1TsY7Vbu1o6P5Nufbkxwy11e7V6xPu r3ktQwAAAAA4FRw6FFbrNJ/6VKOHzbh2dNdB7ei7w3z7okVkcTKxbNxPaf2m LNnYpOLAoTG1mNVaxN7h+tHmjKv3rC4oHD04XPsoXc7eg+Ho1VW1mDHtXV1P oQ5TkawWtmmXhHZrR+H4FaXbPXztwHbaHW48ELafV7V/AAAAAOAU8+53h9U6 zd/+baOHzbh2dL9B7eibwnz7okVkcTKV/v1vKuoOWc0oG4dSXmwYU4sJ62NZ SmtUq8seV+9Zb+nqNdf2714ZvZxsnExlLaZGewfXv6vaJ6vXhds/ov4SWq4d rW53SVsuuLpwXbrp2612BAAAAAAFH/tYWK3TvPOdjR4249rRowa1oweG+fZF i8jipAVrdYeSjCw0FB9XfY2zfo2qfNn567INj6cpXVB1G/PTN7W9fnur9snB MRdzy29r5e4b296Ktoy4P9U07XbNOgAAAAAouOWWsFqneetbGz1sxrWj7x7U jr4szLcvWkQWJ53UpH5yw6XL67WjC65WsAAAAAAAOuTMM0N4wxsaPWTGtaPn xnwu5rSQpk9aRLqfTspft23liuOjZzx5NOxbv57dchg37AgAAAAAYK7uf/8Q du9u9JAZ145eGXM8pOmPFpHFSCcdvyJ3X6LsfkRl11jLrte2d2dYXp9vn+uk AQAAAADdsrISwuMf3+ghazWgT3/6s+HIu/8p/OVffXI9LdSOro15Y0jTHy0i i5FuOn5V2Lm8MfbovOWdYW/xHjy7c/Wl81bC7rE3BwIAAAAAmLPXvz6Es84K 4fbbaz8kXzt6x+Ebw75ffs/q39Yy7nF5Q7WjXu/0mFtiLgjp+qRFpPvpqJPh 6MG9w/WjkVkJu/cfDsdTNxkAAAAAoOiGG0Lo9UL467+u/ZB8DaisflTncWsK taNvWG1Lr7ctzL8vWkQWJx2W1Y/2D1+XbijLYefebAySuhEAAAAA0FGf/3wI X/IlIfzyL4+c5YN/c/PQdemyOlFesX40So3a0a6Y/4w5I6TrkxaR7qf7Th4N B4vXq1vNQfc3AgAAAAC67znPCeFRjxo5uVg7yu5xVLRWP7r8wPtHLqdG7ejd MW8N6fqjRWQxAgAAAADALP3Zn4XVa8W9971TLSarH334IydGTq+sHfV6jxxc r+67Q/p+aRHpdgAAAAAAmLUHPSiEH/mRma5iTO3o12OOx9whpO+XFpFuBwAA AACAWXvta0O4y11CuOWWma1iZO2o1zs75j9idof0fdIi0v0AAAAAADBrN98c wllnhXDRRTNbRUXt6GdiPh1zTkjfJy0i3Q8AAAAAAPPwmteEcKc7hfDhD89k 8WW1o8984AMh9HqfiXlZSN8fLSKLEQAAAAAA5uGznw3hYQ8L4YlPnMniy2pH n3/848P/394drEQVhQEc/xbFEC2CNkLPUG7cpBCufY6ibOcj5NLEdS4VIhfS 4BPMuFFoE7YQe4JCd7oQy+j4OXMXRhAmtzlM/P7wg5nLzHfv2Z4DMyViP90u 9fejgfEgSZIkSZIkSRpV29ulRJSyvt766N/OjtbWhveKmC3196KB8SFJkiRJ kiRJGmULC8Pfruv3Wx37y9lRrze4x4/5+ct3tfehgfHyCgAAAACAETo/XyxT U/ul0zkrW1tv2pr7euXD8HW3uzqYPTn5+ezkZLHU34cGAAAAAADgTyI6qZ++ pEctzn2YvqaddKf6OgEAAAAAALieiHvNGc9petHCvGfNrN10v/r6AAAAAAAA +DsRt9JS+pk204MbzJhI75oZS4OZtdcFAAAAAADAzUXMpcP0PW2k6Wt853F6 m76lo8GM2usAAAAAAACgHZf/TxTxNH1MeSUO0vu0nF42lptrB81nPqXn6W71 5wcAAAAAAODfiJi5ck60l44be1fOk55Uf07gv3MBBG0sVg== "], {{0, 297}, {1679, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], BaseStyle->"ImageGraphics", ImageSizeRaw->{1679, 297}, PlotRange->{{0, 1679}, {0, 297}}]], "Input"] }], "Chapter", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.7310896562732*^9, 3.7310896723052*^9}, { 3.7310929577434*^9, 3.7310929620646*^9}, {3.7311558834132*^9, 3.7311559004296*^9}, {3.7311900163182*^9, 3.7311900290462*^9}, { 3.7311901735931997`*^9, 3.7311903425242*^9}, {3.7311904268732*^9, 3.7311904345931997`*^9}, {3.7311904773782*^9, 3.7311905335432*^9}, { 3.7311906196252003`*^9, 3.7311906348262*^9}, {3.7311933841191998`*^9, 3.7311933852622004`*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Se trata del sistema de dos masas que se mueven cada una sobre un eje, \ unidas por una barra de longitud \[ScriptL]", StyleBox[".", FontSlant->"Italic"], " Elegimos como coordenada el \[AAcute]ngulo \[Phi] que forma el vector ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox[ OverscriptBox[ StyleBox["r", FontWeight->"Bold"], " "], FontWeight->"Bold"], "21"], "=", " ", RowBox[{ SubscriptBox[ StyleBox["r", FontWeight->"Bold"], "2"], "-", SubscriptBox[ StyleBox["r", FontWeight->"Bold"], "1"]}]}], TraditionalForm]]], " con el eje x." }], "Subchapter", CellChangeTimes->{{3.7310896794072*^9, 3.7310897208803997`*^9}, { 3.7310898236387997`*^9, 3.7310898255418*^9}, 3.7310904816296*^9, { 3.731155890258*^9, 3.7311558936784*^9}, {3.7311844263512*^9, 3.7311844726692*^9}, {3.7312350673622*^9, 3.7312350760492*^9}}, FontSize->24], Cell[BoxData[ GraphicsBox[{ {RGBColor[0.87, 0.94, 1], AbsoluteThickness[1], ArrowBox[{{-1.5, 0}, {1.5, 0}}], ArrowBox[{{0, 1.5}, {0, -1.5}}]}, { {RGBColor[0, 1, 0], Opacity[0.6], DiskBox[NCache[{Rational[-1, 2], 0}, {-0.5, 0}], 0.05]}, {RGBColor[1, 0, 0], Opacity[0.6], DiskBox[NCache[{0, Rational[-1, 2] 3^Rational[1, 2]}, { 0, -0.8660254037844386}], 0.05]}}, {GrayLevel[1], AbsoluteThickness[4], ArrowBox[NCache[{{Rational[-1, 2], 0}, { 0, Rational[-1, 2] 3^Rational[1, 2]}}, {{-0.5, 0}, { 0, -0.8660254037844386}}]]}, {GrayLevel[1], AbsoluteThickness[1], Dashing[{Small, Small}], LineBox[NCache[{{Rational[-1, 2], 0}, { Rational[-1, 2], -1.5}}, {{-0.5, 0}, {-0.5, -1.5}}]]}, {RGBColor[1, 1, 0], AbsoluteThickness[2], ArrowBox[{{-0.5000000000000002, -1.}, {-0.45002083072932203`, \ -0.9987502603949663}, {-0.4001665833531724, -0.9950041652780258}, \ {-0.3505618675264006, -0.9887710779360422}, {-0.3013306692049388, \ -0.9800665778412416}, {-0.25259604074547726`, -0.9689124217106448}, \ {-0.2044797933386608, -0.9553364891256061}, {-0.15710219254454916`, \ -0.9393727128473791}, {-0.11058165769134937`, -0.921060994002885}, \ {-0.06503446588876982, -0.9004471023526769}, {-0.02057446139579716, \ -0.8775825618903728}}]}, { {GrayLevel[1], InsetBox[ StyleBox["\<\"\\!\\(\\*SubscriptBox[\\(r\\), \\(1\\)]\\)\"\>", StripOnInput->False, FontSize->30], NCache[{Rational[-1, 2], 0.25}, {-0.5, 0.25}]]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\\!\\(\\*SubscriptBox[\\(r\\), \\(2\\)]\\)\"\>", StripOnInput->False, FontSize->30], NCache[{0.33, Rational[-1, 2] 3^Rational[1, 2]}, { 0.33, -0.8660254037844386}]]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\[ScriptX]\"\>", StripOnInput->False, FontSize->30], {0.2, -1.35}]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\[ScriptY]\"\>", StripOnInput->False, FontSize->30], {1.35, 0.2}]}, {RGBColor[1, 1, 0], InsetBox[ StyleBox["\<\"\[Phi]\"\>", StripOnInput->False, FontSize->25], NCache[{Rational[-1, 2] + (Rational[1, 2] 2^Rational[-1, 2]) (-1 + 3^Rational[ 1, 2]), -1.1159258262890681`}, {-0.24118095489747932`, \ -1.1159258262890681`}]]}}}, AspectRatio->Automatic, Axes->False, Background->GrayLevel[0], ImageSize->400, PlotRange->{{-1.6500000000000001`, 1.6500000000000001`}, {-1.6500000000000001`, 1.6500000000000001`}}, Ticks->False]], "Output", CellChangeTimes->{3.731099700148*^9}] }, Open ]], Cell[TextData[{ "El lagrangiano es (\[DownExclamation]revisen!):\n\n", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{"\[ScriptCapitalL]", "(", RowBox[{"\[Phi]", ",", " ", FractionBox["d\[Phi]", "dt"]}], ")"}], " ", "=", " ", RowBox[{ RowBox[{ FractionBox["1", "2"], SuperscriptBox[ RowBox[{ SuperscriptBox["\[ScriptL]", "2"], "(", FractionBox["d\[Phi]", "dt"], ")"}], "2"], " ", RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["m", RowBox[{"1", " "}]], SuperscriptBox["cos", "2"], " ", "\[Phi]"}], " ", "+", " ", RowBox[{ SubscriptBox["m", "2"], SuperscriptBox["sin", "2"], "\[Phi]"}]}], ")"}]}], " ", "+", " ", RowBox[{ SubscriptBox["m", RowBox[{"2", " "}]], "g", " ", "\[ScriptL]", " ", "cos", " ", "\[Phi]"}]}]}], ","}], FontColor->GrayLevel[0], Background->RGBColor[0.87, 0.94, 1]], TraditionalForm]], FontColor->GrayLevel[0.5]], "\n\ny la ecuaci\[OAcute]n de movimiento:\n\n", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ StyleBox[ RowBox[{ RowBox[{"[", RowBox[{ SubscriptBox["m", "1"], "+", RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["m", "2"], "-", SubscriptBox["m", "1"]}], ")"}], " ", SuperscriptBox["sin", "2"], "\[Phi]"}]}], "]"}], FractionBox[ RowBox[{ SuperscriptBox[ StyleBox["d", FontSlant->"Plain"], "2"], "\[Phi]"}], SuperscriptBox["dt", "2"]], RowBox[{"+", RowBox[{"(", RowBox[{ SubscriptBox["m", "2"], "-", SubscriptBox["m", "1"]}], ")"}]}], " ", "sin", " ", RowBox[{"(", RowBox[{"2", "\[Phi]"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", FractionBox["d\[Phi]", "dt"], ")"}], "2"]}], FontColor->GrayLevel[0]], StyleBox[ RowBox[{"+", " ", RowBox[{"(", RowBox[{"g", "/", "\[ScriptL]"}], ")"}]}], FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], SubscriptBox[ StyleBox["m", FontColor->GrayLevel[0], Background->RGBColor[0.87, 0.94, 1]], "2"], StyleBox["sin", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["\[Phi]", FontColor->GrayLevel[0]]}], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox["=", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ RowBox[{ StyleBox["0", FontColor->GrayLevel[0]], "."}]]}], Background->RGBColor[0.87, 0.94, 1]], TraditionalForm]]], "\n\nIntroduciendo la relaci\[OAcute]n \[Mu] = ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["m", "1"], "/", SubscriptBox["m", "2"]}], TraditionalForm]]], " y el tiempo adimensional \[Tau] = ", Cell[BoxData[ FormBox[ RowBox[{ SqrtBox[ RowBox[{"g", "/", "\[ScriptL]"}]], "t"}], TraditionalForm]]], ", queda una ecuaci\[OAcute]n que depende de un \[UAcute]nico \ par\[AAcute]metro:\n\n", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"\[Mu]", "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Mu]"}], ")"}], " ", SuperscriptBox["sin", "2"], "\[Phi]"}]}], "]"}], RowBox[{"\[Phi]", "''"}]}], " ", "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Mu]"}], ")"}], " ", "sin", " ", RowBox[{"(", RowBox[{"2", "\[Phi]"}], ")"}], " ", SuperscriptBox[ RowBox[{"\[Phi]", "'"}], "2"]}], "+", " ", RowBox[{"sin", " ", "\[Phi]"}]}], " ", "=", " ", "0."}], Background->RGBColor[0.87, 0.94, 1]], TraditionalForm]], FontColor->GrayLevel[0]], "\n\nAqu\[IAcute] las derivadas de \[Phi] son respecto a la variable \[Tau]. \ Si las masas son iguales, ", Cell[BoxData[ FormBox[ RowBox[{"\[Mu]", " ", "=", " ", "1"}], TraditionalForm]]], ", resulta la ecuaci\[OAcute]n del p\[EAcute]ndulo. \n\nAl haber \ introducido el tiempo \[Tau], la escala de tiempo caracter\[IAcute]stica del \ problema es 2\[Pi], salvo casos extremos.\n\nPara explorar las soluciones \ posibles ya no es necesario variar por separado ", Cell[BoxData[ FormBox[ SubscriptBox["m", "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["m", "2"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox["g", TraditionalForm]]], " y \[ScriptL]. Todo lo que importa es \[Mu], la relaci\[OAcute]n entre las \ masas." }], "Subchapter", CellChangeTimes->{{3.7310916222022*^9, 3.7310920927525997`*^9}, { 3.7310921693796*^9, 3.7310924018894*^9}, {3.7310926136842003`*^9, 3.7310926724852*^9}, {3.7310927678814*^9, 3.7310928171521997`*^9}, { 3.7310928479664*^9, 3.7310929406096*^9}, {3.731092986311*^9, 3.7310929871066*^9}, {3.7310935713887997`*^9, 3.7310936214944*^9}, { 3.7310939831546*^9, 3.731093984075*^9}, {3.731099821882*^9, 3.7310998729926*^9}, {3.7311539669482*^9, 3.7311539691976*^9}, { 3.7311540012452*^9, 3.7311540430256*^9}, {3.7311845060652*^9, 3.7311845275872*^9}, {3.7311893885212*^9, 3.7311894178602*^9}}, FontSize->24], Cell[CellGroupData[{ Cell[TextData[{ "La soluci\[OAcute]n num\[EAcute]rica de la ecuaci\[OAcute]n de movimiento: \ \n\nEl comando esencial es ", StyleBox["NDSolve", FontWeight->"Bold"], ". Si buscan en la ayuda, van a encontrar numerosos ejemplos de aplicaci\ \[OAcute]n a problemas f\[IAcute]sicos, incluidos problemas de \ mec\[AAcute]nica.\n\nLo m\[AAcute]s concreto es proponer condiciones \ iniciales y resolver directamente. Va un ejemplo de prueba (pr). " }], "Subchapter", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.7310929888538*^9, 3.73109309509*^9}, { 3.7310931357047997`*^9, 3.7310933533808002`*^9}, {3.7310999806837997`*^9, 3.731099982825*^9}, {3.7311000313042*^9, 3.731100036656*^9}, { 3.7311002104252*^9, 3.7311002359351997`*^9}, {3.7311894545452003`*^9, 3.7311894549632*^9}}, FontSize->24], Cell["\<\ La celda de abajo es la primera celda que tienen que ejecutar s\[IAcute] o s\ \[IAcute] para poder seguir adelante. Recuerden: con el cursor dentro de la celda, tecleen Enter del teclado num\ \[EAcute]rico o Shift + Enter para ejecutar.\ \>", "Text", CellChangeTimes->{{3.7311002389782*^9, 3.7311002442042*^9}, { 3.7311540687139997`*^9, 3.7311540764323997`*^9}, {3.7311657701774*^9, 3.7311658054706*^9}, 3.7311845700552*^9, {3.7312351335568*^9, 3.731235166867*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "\[CapitalAAcute]ngulo", " ", "inicial", " ", "de", " ", "prueba"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Phi]0pr", " ", "=", " ", FractionBox[ RowBox[{"5", "\[Pi]"}], "6"]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Velocidad", " ", "angular", " ", "inicial", " ", "de", " ", RowBox[{"prueba", ":", " ", RowBox[{ "lo", " ", "leo", " ", "as\[IAcute]", " ", "\"\\""}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Phi]p0pr", " ", "=", " ", "0"}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Un", " ", "valor", " ", "de", " ", "\[Mu]", " ", "de", " ", "prueba"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Mu]pr", " ", "=", " ", "2"}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "Un", " ", "tiempo", " ", "m\[AAcute]ximo", " ", "de", " ", "prueba", " ", "hasta", " ", "el", " ", "cual", " ", "integrar", " ", "las", " ", "ecuaciones"}], ";", " ", RowBox[{ "elegido", " ", "en", " ", "realidad", " ", "despu\[EAcute]s", " ", "de", " ", "algunos", " ", "tanteos", " ", RowBox[{"preliminares", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"tmaxpr", " ", "=", " ", "20"}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"La", " ", "cosa", " ", "en", " ", RowBox[{"s\[IAcute]", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Notar", " ", "el", " ", "uso", " ", "del", " ", "doble", " ", "\"\<=\>\"", " ", "para", " ", "significar", " ", "\"\\"", " ", "entre", " ", RowBox[{"expresiones", ".", " ", "Un"}], " ", "solo", " ", "\"\<=\>\"", " ", RowBox[{"asigna", "."}]}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"Por", " ", "ejemplo"}], ",", " ", RowBox[{ "\"\\"", " ", "le", " ", "asigna", " ", "a", " ", "\"\\"", " ", "el", " ", "valor", " ", RowBox[{"\"\<1\>\"", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{ "Un", " ", "doble", " ", "\"\<=\>\"", " ", "significa", " ", "\"\\"", " ", "sin", " ", RowBox[{"asignaci\[OAcute]n", ".", " ", "\"\\""}], " ", "puede", " ", "ser", " ", "cierta", " ", "o", " ", "falsa"}], ",", " ", RowBox[{ "pero", " ", "no", " ", "le", " ", "asigna", " ", "a", " ", "\"\\"", " ", "el", " ", "valor", " ", RowBox[{"\"\<1\>\"", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"sol", " ", "=", " ", "\[IndentingNewLine]", RowBox[{"NDSolve", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Mu]pr", " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Mu]pr"}], ")"}], " ", SuperscriptBox[ RowBox[{"Sin", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}], "2"]}]}], ")"}], " ", RowBox[{ RowBox[{"\[Phi]", "''"}], "[", "t", "]"}]}], " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Mu]pr"}], ")"}], " ", RowBox[{"Sin", "[", RowBox[{"2", RowBox[{"\[Phi]", "[", "t", "]"}]}], "]"}], " ", SuperscriptBox[ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}], "2"]}], "+", " ", RowBox[{"Sin", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}]}], " ", "\[Equal]", " ", "0"}], ",", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"Ec", ".", " ", "de"}], " ", "movimiento"}], ",", " ", RowBox[{"copiada", " ", "de", " ", "m\[AAcute]s", " ", RowBox[{"arriba", " ", "[", "revisen", "]"}]}]}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Phi]", "[", "0", "]"}], " ", "\[Equal]", " ", "\[Phi]0pr"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "0", "]"}], " ", "\[Equal]", " ", "\[Phi]p0pr"}]}], " ", RowBox[{"(*", " ", RowBox[{"Condiciones", " ", "iniciales"}], " ", "*)"}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", "\[Phi]", ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "tmaxpr"}], "}"}]}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "La", " ", "funci\[OAcute]n", " ", "para", " ", "la", " ", "que", " ", "va", " ", "a", " ", "resolver", " ", "y", " ", "en", " ", "qu\[EAcute]", " ", "variable", " ", "e", " ", RowBox[{"intervalo", ".", " ", "Aqu\[IAcute]"}], " ", "es", " ", "la", " ", "funci\[OAcute]n", " ", "\[Phi]"}], ",", " ", RowBox[{ "de", " ", "la", " ", "variable", " ", "t", " ", "en", " ", "el", " ", "intervalo", " ", "entre", " ", "0", " ", "y", " ", "tmaxpr"}]}], " ", "*)"}], "\[IndentingNewLine]", "]"}]}]}]}]], "Input", CellChangeTimes->{{3.7310933168054*^9, 3.7310935639944*^9}, { 3.7310936512194*^9, 3.7310937854728003`*^9}, {3.7310939391446*^9, 3.7310939392538*^9}, {3.7310939949084*^9, 3.7310940111822*^9}, { 3.731094060884*^9, 3.7310940836348*^9}, {3.7310942751556*^9, 3.7310942882460003`*^9}, {3.7310963787258*^9, 3.7310964225601997`*^9}, { 3.7310965286104*^9, 3.7310965311084003`*^9}, {3.7311000600444*^9, 3.731100187144*^9}, {3.7311540826043997`*^9, 3.7311541610278*^9}, { 3.7311548737584*^9, 3.7311549014158*^9}, {3.731167110516*^9, 3.7311671110299997`*^9}, {3.7311849160792*^9, 3.7311849255572*^9}, { 3.7311849652662*^9, 3.7311849741242*^9}, {3.7311897327382*^9, 3.7311897376401997`*^9}, {3.7312351793442*^9, 3.7312352346636*^9}, { 3.7312352840078*^9, 3.7312352922318*^9}, {3.7314181076672*^9, 3.7314184099966*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Si todo sali\[OAcute] como es debido, deber\[IAcute]a aparecer justo arriba \ el siguiente cuadrito\n\n", Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"\[Phi]", "\[Rule]", TagBox[ TemplateBox[{RowBox[{ StyleBox[ TagBox["InterpolatingFunction", "SummaryHead"], "NonInterpretableSummary"], StyleBox["[", "NonInterpretableSummary"], DynamicModuleBox[{Typeset`open$$ = False}, PanelBox[ PaneSelectorBox[{False -> GridBox[{{ PaneBox[ ButtonBox[ DynamicBox[ FEPrivate`FrontEndResource[ "FEBitmaps", "SquarePlusIconMedium"]], ButtonFunction :> (Typeset`open$$ = True), Appearance -> None, Evaluator -> Automatic, Method -> "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], GraphicsBox[{{{}, {}, { Directive[ Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1]], LineBox[CompressedData[" 1:eJwB0QMu/CFib1JlAgAAADwAAAACAAAAStFeOa+/lj4a0DjGpvEEQFp+433D 3dQ/ggAXf0vSBEA+lvq2a77lP8wbZEeMZQRAT/QI4wkn8D+i/p0LZ60DQDc0 cm9ZVPU/PXETDnmSAkD2vGVz/fH6P6u35xLxsQBArkFwQdgXAECTvOGKCMn7 P03J8gTc7gJAv7u5+V5d8z9IHOSKnbgFQDpHAnxYL+c/Fw6ZlmZSCEBG/kre HYXVP1EkE95ZJAtAjycy9BQXrL9f2VCrVMYNQEhfwRC2H9u/5Kx+nYYtEEAj RbOH1T/qv05/NwP4kxFAby/GxA029b8iIdIrbeISQGwcv7kawPy/LFVPcvdM FEAS9d1KiSoBwOTuA5rgsBVAPzM7RlzaAsAGWJqEzfwWQHtaxi6a3APAXlMT jc9kGECWkQnT/YkEwCAebljVtBlAwCRAbujdBMAYe6tB8CAbQE8NMcfs7gTA vT0gDGqGHEDNXzqAqbYEwM3Pdpnn0x1AyRUVsCo7BMAT9K9Eej0fQPfxQoF2 VgPA4nNlWYhHIEAbc0tO2wUCwJCgDgED7SBAopMp6xcGAMBZFqk3iKAhQNRU gTqBL/m/13O0Tw9IIkAb5g7ri0Lxv3Aasfag/SJA32ciENpq47/gc0kO4q8j QK8H3bQpIsq/BbVSByVWJEDP+NtyY7PDP0U/TY9yCiVAbfaj4VbR4T86sbj4 wbIlQAwDdOFkcO8/Bta/0sBXJkDJPb8NVHT3P+1DuDvKCidAE3dqXcHc/j+J mSGG1bEnQG1fLSV3qQFAQDh8X+tmKEDccNr/ui8DQKy+RxoDEClAwNbE4BoX BEDv965FyrUpQM54B/kDogRATXoHAJxpKkB9LaoFqOkEQGDk0JtvEStAbA3H /uPpBECOl4vGTccrQCrkw0mFoQRAkv3hYdt5LEDw04UbMggEQEtLqd5qIC1A 8PdXZMAcA0Af4mHqBNUtQErHkYCJjAFAqGCL16B9LkCzZIEQgHP+PwiSUDXs Ii9AI9zqgwOI9z+DDAciQtYvQAStxJ7DU+4/WjcX+Mw+MEDrCdPZVvDgPwCN oyZ+mTBAgR5gbQ8nwD8RvH0Nh/IwQLWXIWuagdC//F4Q5ZBFMUCFJP1ejl/k v3WmGwSgnzFAIqcpj77D8b/IYd8TsPMxQJFx+XLrtvm/qcEba8VOMkD9fAdl 5TsAwPX6pXoyqDJAG/6yYmZHAsAcqOh6oPsyQMtxUZcdfgPA0PmjwhNWM0C6 /ADfO1MEwF+/F/uHqjNAN6Hye2fEBMAyFFD6//8zQMCwWOGx8ATAS6K5zg== "]]}}}, { DisplayFunction -> Identity, AspectRatio -> 1, Axes -> {False, False}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{True, True}, {True, True}}, FrameLabel -> {{None, None}, {None, None}}, FrameStyle -> Directive[ Opacity[0.5], Thickness[Tiny], RGBColor[0.368417, 0.506779, 0.709798]], FrameTicks -> {{None, None}, {None, None}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}], Method -> { "DefaultBoundaryStyle" -> Automatic, "ScalingFunctions" -> None}, PlotRange -> {{0., 20.}, {-2.6175267796698733`, 2.617993877991478}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.1], Scaled[0.1]}, { Scaled[0.1], Scaled[0.1]}}, Ticks -> {Automatic, Automatic}}], GridBox[{{ RowBox[{ TagBox["\"Domain: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "20.`"}], "}"}], "}"}], "SummaryItem"]}]}, { RowBox[{ TagBox["\"Output: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"scalar\"", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> { "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, BaseStyle -> { ShowStringCharacters -> False, NumberMarks -> False, PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}], True -> GridBox[{{ PaneBox[ ButtonBox[ DynamicBox[ FEPrivate`FrontEndResource[ "FEBitmaps", "SquareMinusIconMedium"]], ButtonFunction :> (Typeset`open$$ = False), Appearance -> None, Evaluator -> Automatic, Method -> "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], GraphicsBox[{{{}, {}, { Directive[ Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1]], LineBox[CompressedData[" 1:eJwB0QMu/CFib1JlAgAAADwAAAACAAAAStFeOa+/lj4a0DjGpvEEQFp+433D 3dQ/ggAXf0vSBEA+lvq2a77lP8wbZEeMZQRAT/QI4wkn8D+i/p0LZ60DQDc0 cm9ZVPU/PXETDnmSAkD2vGVz/fH6P6u35xLxsQBArkFwQdgXAECTvOGKCMn7 P03J8gTc7gJAv7u5+V5d8z9IHOSKnbgFQDpHAnxYL+c/Fw6ZlmZSCEBG/kre HYXVP1EkE95ZJAtAjycy9BQXrL9f2VCrVMYNQEhfwRC2H9u/5Kx+nYYtEEAj RbOH1T/qv05/NwP4kxFAby/GxA029b8iIdIrbeISQGwcv7kawPy/LFVPcvdM FEAS9d1KiSoBwOTuA5rgsBVAPzM7RlzaAsAGWJqEzfwWQHtaxi6a3APAXlMT jc9kGECWkQnT/YkEwCAebljVtBlAwCRAbujdBMAYe6tB8CAbQE8NMcfs7gTA vT0gDGqGHEDNXzqAqbYEwM3Pdpnn0x1AyRUVsCo7BMAT9K9Eej0fQPfxQoF2 VgPA4nNlWYhHIEAbc0tO2wUCwJCgDgED7SBAopMp6xcGAMBZFqk3iKAhQNRU gTqBL/m/13O0Tw9IIkAb5g7ri0Lxv3Aasfag/SJA32ciENpq47/gc0kO4q8j QK8H3bQpIsq/BbVSByVWJEDP+NtyY7PDP0U/TY9yCiVAbfaj4VbR4T86sbj4 wbIlQAwDdOFkcO8/Bta/0sBXJkDJPb8NVHT3P+1DuDvKCidAE3dqXcHc/j+J mSGG1bEnQG1fLSV3qQFAQDh8X+tmKEDccNr/ui8DQKy+RxoDEClAwNbE4BoX BEDv965FyrUpQM54B/kDogRATXoHAJxpKkB9LaoFqOkEQGDk0JtvEStAbA3H /uPpBECOl4vGTccrQCrkw0mFoQRAkv3hYdt5LEDw04UbMggEQEtLqd5qIC1A 8PdXZMAcA0Af4mHqBNUtQErHkYCJjAFAqGCL16B9LkCzZIEQgHP+PwiSUDXs Ii9AI9zqgwOI9z+DDAciQtYvQAStxJ7DU+4/WjcX+Mw+MEDrCdPZVvDgPwCN oyZ+mTBAgR5gbQ8nwD8RvH0Nh/IwQLWXIWuagdC//F4Q5ZBFMUCFJP1ejl/k v3WmGwSgnzFAIqcpj77D8b/IYd8TsPMxQJFx+XLrtvm/qcEba8VOMkD9fAdl 5TsAwPX6pXoyqDJAG/6yYmZHAsAcqOh6oPsyQMtxUZcdfgPA0PmjwhNWM0C6 /ADfO1MEwF+/F/uHqjNAN6Hye2fEBMAyFFD6//8zQMCwWOGx8ATAS6K5zg== "]]}}}, { DisplayFunction -> Identity, AspectRatio -> 1, Axes -> {False, False}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{True, True}, {True, True}}, FrameLabel -> {{None, None}, {None, None}}, FrameStyle -> Directive[ Opacity[0.5], Thickness[Tiny], RGBColor[0.368417, 0.506779, 0.709798]], FrameTicks -> {{None, None}, {None, None}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}], Method -> { "DefaultBoundaryStyle" -> Automatic, "ScalingFunctions" -> None}, PlotRange -> {{0., 20.}, {-2.6175267796698733`, 2.617993877991478}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.1], Scaled[0.1]}, { Scaled[0.1], Scaled[0.1]}}, Ticks -> {Automatic, Automatic}}], GridBox[{{ RowBox[{ TagBox["\"Domain: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "20.`"}], "}"}], "}"}], "SummaryItem"]}]}, { RowBox[{ TagBox["\"Output: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"scalar\"", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Order: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["3", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Method: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Hermite\"", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Periodic: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> { "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, BaseStyle -> { ShowStringCharacters -> False, NumberMarks -> False, PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}]}, Dynamic[Typeset`open$$], ImageSize -> Automatic], BaselinePosition -> Baseline], DynamicModuleValues :> {}], StyleBox["]", "NonInterpretableSummary"]}]}, "CopyTag", DisplayFunction->(#& ), InterpretationFunction->("InterpolatingFunction[{{0., 20.}}, <>]"& )], False, Editable->False, SelectWithContents->True, Selectable->False]}], "}"}], "}"}]], CellChangeTimes->{{3.7312352714446*^9, 3.7312352924814*^9}}], "\n\nEs la soluci\[OAcute]n de NDSolve, y la hemos asignado a la variable \ \"sol\".\n\nLa soluci\[OAcute]n est\[AAcute] dada en t\[EAcute]rminos de una \ funci\[OAcute]n num\[EAcute]rica, de la que no se dan otros detalles que su \ intervalo de definici\[OAcute]n y el tipo de salida que arroja, en este caso, \ un escalar.\n\nPara extraer de aqu\[IAcute] la funci\[OAcute]n \[Phi](t) hay \ que seleccionar la parte adecuada de \"sol\". Paso por paso nos vamos \ abriendo camino en la lista {{...}} hasta llegar a la funci\[OAcute]n \ propiamente dicha:" }], "Subchapter", CellChangeTimes->{{3.7311002522548*^9, 3.731100439532*^9}, { 3.7311542353084*^9, 3.7311542413536*^9}, {3.7311658184568*^9, 3.7311659397748003`*^9}, {3.731165996451*^9, 3.7311659981836*^9}, { 3.7311846110402*^9, 3.7311846253702*^9}, {3.7312352997986*^9, 3.7312353044458*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Con", " ", "este", " ", "nos", " ", RowBox[{"quedamos", "."}]}], " ", "*)"}]}]}], "Input", CellChangeTimes->{{3.731093866955*^9, 3.7310939183448*^9}, { 3.731154227901*^9, 3.7311542321502*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ La operaci\[OAcute]n de seleccionar partes de una lista se hace mediante los \ dobles corchetes. Lo \[UAcute]nico que nos interesa es el \[UAcute]ltimo \ resultado. Es decir, en general nos bastar\[IAcute]a con escribir:\ \>", "Subchapter", CellChangeTimes->{{3.7311004623744*^9, 3.7311004854351997`*^9}, { 3.7311005880892*^9, 3.7311006115108*^9}, {3.7311542677756*^9, 3.7311542681344*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"Phi", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{3.7311006213564*^9}] }, Open ]], Cell[CellGroupData[{ Cell["El gr\[AAcute]fico de la soluci\[OAcute]n \[Phi](t):", "Subchapter", CellChangeTimes->{{3.7310941227967997`*^9, 3.7310941284918003`*^9}, { 3.7311006377394*^9, 3.731100646727*^9}, {3.7311659893062*^9, 3.7311659935338*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"Phi", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "Repito", " ", "la", " ", "definici\[OAcute]n", " ", "por", " ", "las", " ", "dudas", " ", "de", " ", "que", " ", "se", " ", "olviden", " ", "de", " ", "ejecutar", " ", "la", " ", "celda", " ", "anterior"}], " ", "*)"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Plot", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Phi", "[", "t", "]"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "tmaxpr"}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"AbsoluteThickness", "[", "3", "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"AxesStyle", "\[Rule]", "White"}], ",", "\[IndentingNewLine]", RowBox[{"TicksStyle", "\[Rule]", RowBox[{"Directive", "[", RowBox[{"White", ",", " ", "16"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", "600"}], ",", " ", "\[IndentingNewLine]", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", " ", "\"\<\[Phi]\>\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"LabelStyle", "\[Rule]", "20"}], ",", "\[IndentingNewLine]", RowBox[{"ImageMargins", "\[Rule]", "40"}]}], "\[IndentingNewLine]", "]"}]}], "Input", CellChangeTimes->{{3.7310940268368*^9, 3.7310940333596*^9}, { 3.7310941312530003`*^9, 3.7310942708606*^9}, {3.7311006527964*^9, 3.7311007923724003`*^9}, {3.7311542876605997`*^9, 3.731154309432*^9}, { 3.7312353688182*^9, 3.7312353972176*^9}}], Cell["\<\ Lo \[UAcute]nico importante en la sintaxis del Plot son las comas. Los cortes \ de l\[IAcute]nea no tienen ning\[UAcute]n efecto, salvo facilitar la lectura. \ Todas las opciones despu\[EAcute]s de \"{t, 0, tmaxpr}\" pueden omitirse y \ permutarse. \ \>", "Text", CellChangeTimes->{{3.7311543334316*^9, 3.7311544249779997`*^9}, { 3.7311660182346*^9, 3.731166023135*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Phi", "[", "t", "]"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "tmaxpr"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7311895807562*^9, 3.7311895834572*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "La animaci\[OAcute]n. \n\nPrimero probamos graficar las cosas en un \ instante ", Cell[BoxData[ FormBox["t", TraditionalForm]]], " fijo. Elegido un ", Cell[BoxData[ FormBox["t", TraditionalForm]]], " de prueba, la celda de abajo combina varios gr\[AAcute]ficos: los ejes, \ las masas y la barra entre las masas." }], "Subchapter", CellChangeTimes->{{3.731094365837*^9, 3.7310943771012*^9}, { 3.7310944214172*^9, 3.7310944777256002`*^9}, {3.7310947778822002`*^9, 3.7310948661484003`*^9}, {3.7311008153542*^9, 3.731100825388*^9}, { 3.7311544383234*^9, 3.7311544423571997`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Un", " ", "valor", " ", "de", " ", "tiempo", " ", "de", " ", RowBox[{"prueba", ".", " ", "Tiene"}], " ", "que", " ", "estar", " ", "entre", " ", "0", " ", "y", " ", "tmaxpr"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Cambien", " ", "luego", " ", "este", " ", "valor", " ", "para", " ", "obtener", " ", "gr\[AAcute]ficos", " ", "a", " ", "distintos", " ", "tiempos"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"tpr", " ", "=", " ", "3.75"}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"Para", " ", "ese", " ", "valor", " ", "del", " ", "tiempo"}], ",", " ", RowBox[{ "la", " ", "configuraci\[OAcute]n", " ", "del", " ", "sistema", " ", "queda", " ", "dada", " ", "por", " ", "el", " ", "valor", " ", "de", " ", "\[Phi]", " ", "que", " ", "arroja", " ", "la", " ", "funci\[OAcute]n", " ", RowBox[{"Phi", "[", "tpr", "]"}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"Conocido", " ", "\[Phi]pr"}], " ", "=", " ", RowBox[{ RowBox[{"Phi", "[", "tpr", "]"}], " ", "deber\[IAcute]a", " ", "poder", " ", "graficar", " ", "la", " ", "configuraci\[OAcute]n", " ", "del", " ", "sistema"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Phi]pr", " ", "=", " ", RowBox[{"Phi", "[", "tpr", "]"}]}], ";"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "El", " ", "comando", " ", "Show", " ", "combina", " ", "varios", " ", "gr\[AAcute]ficos", " ", "que", " ", "comparten", " ", "la", " ", "misma", " ", "regi\[OAcute]n", " ", "del", " ", "plano"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{ "Empiezo", " ", "por", " ", "graficar", " ", "un", " ", "par", " ", "de", " ", "ejes"}], ",", " ", RowBox[{ "y", " ", "luego", " ", "agrego", " ", "las", " ", "masas", " ", "y", " ", "la", " ", "barra"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Show", "[", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", "Ejes", " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"ejes", " ", "=", " ", RowBox[{"Graphics", "[", RowBox[{"{", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "2", "]"}], ",", " ", "LightBlue", ",", " ", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.5"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"1.5", ",", " ", "0"}], "}"}]}], "}"}], "]"}], ",", " ", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", "1.5"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1.5"}]}], "}"}]}], "}"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"White", ",", RowBox[{"Text", "[", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\<\[ScriptX]\>\"", ",", " ", "30"}], "]"}], ",", RowBox[{"{", RowBox[{"0.2", ",", RowBox[{ RowBox[{"-", "0.95"}], "\[Times]", "1.5"}]}], "}"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"White", ",", RowBox[{"Text", "[", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\<\[ScriptY]\>\"", ",", " ", "30"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"0.99", " ", "1.5"}], ",", "0.22"}], "}"}]}], "]"}]}], "}"}]}], "}"}], "\[IndentingNewLine]", "]"}]}], ",", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Masas", " ", "y", " ", "barra"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Graphics", "[", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "2", "]"}], ",", " ", "Blue", ",", " ", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]pr", "]"}]}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]pr", "]"}]}]}], "}"}]}], "}"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"Green", ",", " ", RowBox[{"Disk", "[", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]pr", "]"}]}], ",", " ", "0"}], "}"}], ",", " ", "0.05"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Disk", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]pr", "]"}]}]}], "}"}], ",", " ", "0.05"}], "]"}]}], "}"}]}], "}"}], "\[IndentingNewLine]", "]"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Retoques", " ", RowBox[{"opcionales", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", RowBox[{"1.6", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "400"}]}], "\[IndentingNewLine]", "]"}]}]}]], "Input", CellChangeTimes->{{3.7310944144742002`*^9, 3.7310944512181997`*^9}, { 3.7310944962946*^9, 3.7310947727654*^9}, {3.731094844165*^9, 3.731094926943*^9}, {3.7311008450802*^9, 3.7311009024392*^9}, { 3.7311544504792*^9, 3.7311546003554*^9}, {3.731166049974*^9, 3.7311660595378*^9}, {3.7311660940861998`*^9, 3.7311661202866*^9}, { 3.7311846869402*^9, 3.7311846903022003`*^9}, 3.7311848837172003`*^9, { 3.7311896686422*^9, 3.7311896695052*^9}, {3.7312354302082*^9, 3.7312356830846*^9}, {3.731235729815*^9, 3.731235750811*^9}}], Cell[TextData[{ "OK. Si result\[OAcute] para un ", StyleBox["t", FontSlant->"Italic"], " dado, tiene que funcionar para un ", StyleBox["t", FontSlant->"Italic"], " cualquiera. M\[AAcute]s que nada, lo de arriba sirvi\[OAcute] para ajustar \ los tama\[NTilde]os de los discos, la extensi\[OAcute]n de los ejes, etc." }], "Text", CellChangeTimes->{{3.7311677133996*^9, 3.7311677687416*^9}, { 3.7311847081212*^9, 3.7311847099042*^9}, 3.7311896175982*^9, { 3.7312357240366*^9, 3.731235725507*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "En lugar de andar cambiando el valor de ", Cell[BoxData[ FormBox[ RowBox[{"tpr", " "}], TraditionalForm]]], "una y otra vez, lo m\[AAcute]s pr\[AAcute]ctico es usar el comando ", StyleBox["Manipulate", FontWeight->"Bold"], ", que nos permite variar el tiempo y graficar la configuraci\[OAcute]n de \ las masas de manera interactiva." }], "Subchapter", CellChangeTimes->{{3.7310948088542*^9, 3.7310948088698*^9}, { 3.7310949324342003`*^9, 3.7310949823194*^9}, {3.7311009089630003`*^9, 3.7311009168878*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Phi]pr", " ", "=", " ", RowBox[{"Phi", "[", "t", "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Show", "[", "\[IndentingNewLine]", " ", RowBox[{"ejes", ",", " ", RowBox[{"(*", " ", RowBox[{ "El", " ", "gr\[AAcute]fico", " ", "de", " ", "los", " ", "ejes", " ", "se", " ", "defini\[OAcute]", " ", "m\[AAcute]s", " ", RowBox[{"arriba", ".", " ", "Ahora"}], " ", "lo", " ", "uso", " ", RowBox[{"directamente", "."}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Graphics", "[", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "2", "]"}], ",", " ", "Blue", ",", " ", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]pr", "]"}]}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]pr", "]"}]}]}], "}"}]}], "}"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"Green", ",", " ", RowBox[{"Disk", "[", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]pr", "]"}]}], ",", " ", "0"}], "}"}], ",", " ", "0.05"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Disk", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]pr", "]"}]}]}], "}"}], ",", " ", "0.05"}], "]"}]}], "}"}]}], "}"}], "\[IndentingNewLine]", "]"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"1.6", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "400"}]}], "\[IndentingNewLine]", "]"}]}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Abajo", " ", "est\[AAcute]", " ", "contenida", " ", "la", " ", "informaci\[OAcute]n", " ", "sobre", " ", "la", " ", "variable", " ", "que", " ", "se", " ", "contrala", " ", "con", " ", RowBox[{"Manipulate", ".", " ", "El"}], " ", "tiempo", " ", "t", " ", "va", " ", "desde", " ", "0", " ", "hasta", " ", "tmaxpr", " ", "y", " ", "el", " ", "paso", " ", "por", " ", "defecto", " ", "es", " ", RowBox[{"\"\\"", ".", " ", "El"}], " ", "\"\<.\>\"", " ", "luego", " ", "del", " ", "\"\<1000.\>\"", " ", "es", " ", "para", " ", "que", " ", "no", " ", "use", " ", "n\[UAcute]meros", " ", "de", " ", "precisi\[OAcute]n", " ", RowBox[{"infinita", "."}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "tmaxpr", ",", " ", FractionBox["tmaxpr", "1000."], ",", " ", RowBox[{"Appearance", "\[Rule]", " ", "\"\\""}], ",", " ", RowBox[{"AnimationRate", "\[Rule]", " ", "2"}]}], "}"}]}], " ", "\[IndentingNewLine]", "]"}]], "Input", CellChangeTimes->{{3.7310949963196*^9, 3.7310950465596*^9}, { 3.7311009225671997`*^9, 3.731100955302*^9}, {3.7311546421809998`*^9, 3.7311546914614*^9}, {3.7311547318216*^9, 3.7311547828682003`*^9}, { 3.7311572357860003`*^9, 3.7311572495818*^9}, 3.731166146094*^9, { 3.7311671475346003`*^9, 3.7311671506926003`*^9}, {3.7311677900506*^9, 3.7311678593456*^9}, {3.7311848494302*^9, 3.7311848538992*^9}, { 3.7311849815052*^9, 3.7311849839782*^9}, {3.7311896525982*^9, 3.7311896964342003`*^9}, {3.7312357661146*^9, 3.731235901383*^9}}], Cell[TextData[{ "\[DownExclamation]Cautivador! Pueden mover el tiempo usando el mouse, \ apretar el bot\[OAcute]n Play, o ingresar un tiempo cualquiera en el cuadrito \ debajo de la barra. En tal caso usen el ", StyleBox["Enter", FontWeight->"Bold"], " ", StyleBox["normal", FontWeight->"Bold"], " para forzar el cambio." }], "Text", CellChangeTimes->{{3.7311009590928*^9, 3.731100976326*^9}, { 3.7311546735880003`*^9, 3.731154687572*^9}, {3.7312360688156*^9, 3.7312361054890003`*^9}}], Cell["\<\ Si le dan Play, al llegar al final del recorrido temporal la \ animaci\[OAcute]n vuelve a t = 0. Ser\[IAcute]a raro que justo el tiempo \ tmaxpr fuera igual a un m\[UAcute]ltiplo entero del per\[IAcute]odo del \ movimiento. De manera que, en general, la animaci\[OAcute]n no se cerrar\ \[AAcute] sobre s\[IAcute] misma, sino que tendr\[AAcute] un salto desde la \ configuraci\[OAcute]n en tmaxpr hasta la configuaci\[OAcute]n en t = 0. Si quieren hacer un gif prolijo, esos saltos son indeseables. Es necesario \ que la animaci\[OAcute]n termine justo un paso antes de completarse un per\ \[IAcute]odo del movimiento.\ \>", "Text", CellChangeTimes->{{3.7310950899386*^9, 3.7310956269722*^9}, { 3.7310956851078*^9, 3.7310957701458*^9}, {3.7310959279906*^9, 3.7310960648316*^9}, {3.7310962597*^9, 3.731096345289*^9}, { 3.7311009918968*^9, 3.7311010511138*^9}, {3.7311547092592*^9, 3.7311547209606*^9}, {3.7311661924516*^9, 3.7311661957432003`*^9}, { 3.73116622762*^9, 3.731166236204*^9}, {3.7311850243442*^9, 3.7311850251482*^9}}, FontSize->18] }, Open ]], Cell[TextData[{ "Podr\[IAcute]amos calcular el per\[IAcute]odo del movimiento de manera anal\ \[IAcute]tica (lo veremos m\[AAcute]s adelante en la materia). En lugar de \ eso, vamos a incluir dentro de ", StyleBox["NDSolve", FontWeight->"Bold"], " un comando que vigile los cruces por cero del \[AAcute]ngulo \[Phi]. Ese \ comando, que sirve tambi\[EAcute]n para muchas otras cosas, es ", StyleBox["WhenEvent", FontWeight->"Bold"], ". Ya que \[Phi] s\[IAcute] o s\[IAcute] cruza por cero (\[DownQuestion]por \ qu\[EAcute]?), esto servir\[AAcute] para medir el intervalo entre pasos \ sucesivos.\n\nCada vez que sea cierto que Sin[\[Phi](t)/2] = 0, lo que \ implica que \[Phi](t) es un multiplo entero y par de 2\[Pi], ", StyleBox["WhenEvent", FontWeight->"Bold"], " va a \"sembrar\", v\[IAcute]a ", StyleBox["Sow", FontWeight->"Bold"], ", el valor de t correspondiente. Cuando el c\[AAcute]lculo est\[EAcute] \ completo, el comando ", StyleBox["Reap ", FontWeight->"Bold"], "va a \"cosechar\" la lista de esos valores. La diferencia temporal entre \ estos sucesivos cruces por 0 deber\[IAcute]a ser constante e igual al per\ \[IAcute]odo del movimiento.\n\n", StyleBox["En realidad hay que hacer una salvedad. La barra puede rotar \ (siempre en el mismo sentido) u oscilar como un p\[EAcute]ndulo. Si oscila, \ el per\[IAcute]odo es igual a dos veces el intervalo entre pasos sucesivos \ por \[Phi] = 0 (\[DownQuestion]por qu\[EAcute]?). Si rota, el per\[IAcute]odo \ es igual al intervalo entre dos pasos sucesivos. Para cubrirnos en ambos \ casos, definimos T = 2 veces el intervalo entre pasos sucesivos, lo que es \ redundante para el movimiento de rotaci\[OAcute]n. Hay maneras de corregir \ esto.", FontSize->18] }], "Subchapter", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.7310950899386*^9, 3.7310956269722*^9}, { 3.7310956851078*^9, 3.7310957701458*^9}, {3.7310959279906*^9, 3.7310960648316*^9}, {3.7310962597*^9, 3.731096345289*^9}, { 3.7311009918968*^9, 3.7311011767504*^9}, 3.731154848229*^9, { 3.7311549233532*^9, 3.7311551174764*^9}, {3.7311564500668*^9, 3.7311564589435997`*^9}, {3.7311662786616*^9, 3.7311663636236*^9}, { 3.7311850503132*^9, 3.7311851478772*^9}, {3.7312361959298*^9, 3.7312361987146*^9}, {3.7312369355228*^9, 3.7312370129038*^9}, { 3.731237055374*^9, 3.7312370754004*^9}}, FontSize->24], Cell[CellGroupData[{ Cell[TextData[{ "He aqu\[IAcute] entonces el ", StyleBox["NDSolve", FontWeight->"Bold"], " que nos devuelve la soluci\[OAcute]n de la ecuaci\[OAcute]n de movimiento \ y la lista de tiempos en los que \[Phi] cruza por cero.\n\n", StyleBox["(Notar que el tiempo tmax tiene que ser lo suficientemente largo \ para que haya varios pasos por la vertical. Si en la primera elecci\[OAcute]n \ no funciona, tendr\[AAcute]n que cambiarlo.)", FontSize->18] }], "Subchapter", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.7310950899386*^9, 3.7310956269722*^9}, { 3.7310956851078*^9, 3.7310957701458*^9}, {3.7310959279906*^9, 3.7310960648316*^9}, {3.7310962597*^9, 3.731096345289*^9}, { 3.7311009918968*^9, 3.7311011767504*^9}, 3.731154848229*^9, { 3.7311549233532*^9, 3.7311551386806*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{"\[Phi]0pr", " ", "=", " ", FractionBox[ RowBox[{"5", "\[Pi]"}], "6"]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Phi]p0pr", " ", "=", " ", "0"}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Mu]pr", " ", "=", " ", "2"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"tmaxpr", " ", "=", " ", "50"}], ";"}], " ", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"sol2", " ", "=", "\[IndentingNewLine]", RowBox[{"Reap", "[", " ", RowBox[{"(*", " ", RowBox[{ "El", " ", "cosechador", " ", "de", " ", "lo", " ", "que", " ", "siembra", " ", "Sow", " ", "m\[AAcute]s", " ", "abajo"}], "*)"}], "\[IndentingNewLine]", RowBox[{"NDSolve", "[", " ", RowBox[{"(*", " ", RowBox[{ "El", " ", "mismo", " ", "sistema", " ", "de", " ", "ecuaciones", " ", "y", " ", "condiciones", " ", "iniciales", " ", "que", " ", "antes"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Mu]pr", " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Mu]pr"}], ")"}], " ", SuperscriptBox[ RowBox[{"Sin", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}], "2"]}]}], ")"}], " ", RowBox[{ RowBox[{"\[Phi]", "''"}], "[", "t", "]"}]}], " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Mu]pr"}], ")"}], " ", RowBox[{"Sin", "[", RowBox[{"2", RowBox[{"\[Phi]", "[", "t", "]"}]}], "]"}], " ", SuperscriptBox[ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}], "2"]}], "+", " ", RowBox[{"Sin", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}]}], " ", "\[Equal]", " ", "0"}], ",", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Phi]", "[", "0", "]"}], " ", "\[Equal]", " ", "\[Phi]0pr"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "0", "]"}], " ", "\[Equal]", " ", "\[Phi]p0pr"}], ",", "\[IndentingNewLine]", RowBox[{"WhenEvent", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], "/", "2"}], "]"}], " ", "\[Equal]", " ", "0"}], ",", " ", RowBox[{"Sow", "[", "t", "]"}]}], "]"}]}], " ", RowBox[{"(*", " ", RowBox[{ "WhenEvent", " ", "para", " ", "detectar", " ", "los", " ", "pasos", " ", "por", " ", "la", " ", "vertical", " ", "en", " ", "combinaci\[OAcute]n", " ", "con", " ", "el", " ", "sembrador", " ", "Sow"}], " ", "*)"}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", "\[Phi]", ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "tmaxpr"}], "}"}]}], " ", "\[IndentingNewLine]", "]"}], "\[IndentingNewLine]", "]"}]}]}], "Input", CellChangeTimes->{{3.7310957991436*^9, 3.7310958959021997`*^9}, { 3.7310960055718*^9, 3.7310960158064003`*^9}, {3.7310960853964*^9, 3.7310960859424*^9}, {3.7310963548986*^9, 3.7310963607516003`*^9}, 3.7310965000408*^9, {3.7310965720966*^9, 3.7310965825038*^9}, { 3.731096630987*^9, 3.7310966310962*^9}, {3.7311551514779997`*^9, 3.7311551883529997`*^9}, {3.7311671718556004`*^9, 3.7311671725046*^9}, { 3.7311851968282003`*^9, 3.7311852011772003`*^9}, {3.7312371096214*^9, 3.7312371651879997`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Si todo sali\[OAcute] bien, el resultado de la celda de arriba deber\ \[IAcute]a tener esta forma:\n\n", Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"\[Phi]", "\[Rule]", TagBox[ TemplateBox[{RowBox[{ StyleBox[ TagBox["InterpolatingFunction", "SummaryHead"], "NonInterpretableSummary"], StyleBox["[", "NonInterpretableSummary"], DynamicModuleBox[{Typeset`open$$ = False}, PanelBox[ PaneSelectorBox[{False -> GridBox[{{ PaneBox[ ButtonBox[ DynamicBox[ FEPrivate`FrontEndResource[ "FEBitmaps", "SquarePlusIconMedium"]], ButtonFunction :> (Typeset`open$$ = True), Appearance -> None, Evaluator -> Automatic, Method -> "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], GraphicsBox[{{{}, {}, { Directive[ Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1]], LineBox[CompressedData[" 1:eJwB0QMu/CFib1JlAgAAADwAAAACAAAAnIW2B5tvrD5UzzjGpvEEQPBdXF00 Feo/W6YiFqUkBEDOO7mkBi77P9Gf6i9imABAZDHLW8wwBECwkw0nJl7vP0XB TstvqQpAPZuvZknDhj8alh9oPtcQQPBMmk4eyfC/GlLMUc4dFEAA3BkOfOAA wKF7LwaTqhdAJjyUnDk8BMBaI53txCYbQNZS1+2X7gTAnFE/PABnHkDuR9ci i+sDwLP2yyq49iBAx/OdQHfA/7/chxLr9JsiQJ/r4kyZMuu/H1jeROg4JEA3 EdCPNEO3PyVfBQT2+CVAi0pllQDr8j9vqcZ2CJsnQICLwPv7aQFAfSrjTjVg KUCwBsCc1mMEQKTqhMAYHStAxOCsX4/nBEAP7sDlALwsQDhtYEvqtgNAPihY cAN+LkArQzqbGXD+P9jSRFcFETBA0zVZmSAB6D/zLAsplvQwQGCQcRdgFdG/ miaURwLUMUC6y1l8Ab/2v+RB6r9wpDJANj/bygo2AsCP+O1qbIYzQOS7s68A nQTA3NC+b2pZNEDUdb/0f9gEwLZIUsFDKDVA8AapwdhpA8DyW5NFqgg2QMMz BUJUgfy/0JChIxPaNkClOssAGwDkvxBhXTQJvTdAB6SFLREk2D/c0NuR2ps4 QB4s7569Cvk/SmInSa5rOUBBDM/FJ54CQBqPIDMPTTpAWBrS06+3BECM3eZ2 ch87QIwcQzPnxgRAi8tvB7HtO0BIh9TjQSIDQOxUpsp8zTxAfuztBlK9+j/v /6nnSp49QP0W9MQ84OA/VEZbN6aAPkB9LMq3n/Pdv1uu2eADVD9AxGMpnu/T +b/4Wo1rnhFAQKrlY1qfvgLAc6wEgAGCQEDxjqaJI78EwL+OYsHl6kBA+ksR N6DABMC7PheckFxBQIOboKSXugLAfj4tHSnMQUD/18rEi0P4vxLPKctCNEJA REBynILw2b9XLX0SI6VCQM8NTGJS3OI/bRy3hoQOQ0Dx6pMcxAP8P0lbUqHT dUNAaOAIyUYbA0DWZ0RV6eVDQEi9hYbX0wRANAUdNoBOREDwx7yBT6cEQENw TLDdv0RAn5CsEANZAkAYK93QKC9FQIFNFla0HvY/vnZUHvWWRUAa4ppHo7TT PxWQIgWIB0ZAv5g/7wok5r89OtcYnHBGQO5F2UzVnP2/FrLixXbiRkCK+w0A ep4DwLV5Txk/UkdAHfRkpmXqBMAm0qKZiLpHQDzccM/EbgTAR/hMs5grSEDG W8HQ/ZEBwDmv3fkplUhAp1L4quhj878+GeT4//9IQKQb9h94HMS/HEa84A== "]]}}}, { DisplayFunction -> Identity, AspectRatio -> 1, Axes -> {False, False}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{True, True}, {True, True}}, FrameLabel -> {{None, None}, {None, None}}, FrameStyle -> Directive[ Opacity[0.5], Thickness[Tiny], RGBColor[0.368417, 0.506779, 0.709798]], FrameTicks -> {{None, None}, {None, None}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}], Method -> { "DefaultBoundaryStyle" -> Automatic, "ScalingFunctions" -> None}, PlotRange -> {{0., 50.}, {-2.616500719197897, 2.61799387799139}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.1], Scaled[0.1]}, { Scaled[0.1], Scaled[0.1]}}, Ticks -> {Automatic, Automatic}}], GridBox[{{ RowBox[{ TagBox["\"Domain: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "50.`"}], "}"}], "}"}], "SummaryItem"]}]}, { RowBox[{ TagBox["\"Output: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"scalar\"", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> { "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, BaseStyle -> { ShowStringCharacters -> False, NumberMarks -> False, PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}], True -> GridBox[{{ PaneBox[ ButtonBox[ DynamicBox[ FEPrivate`FrontEndResource[ "FEBitmaps", "SquareMinusIconMedium"]], ButtonFunction :> (Typeset`open$$ = False), Appearance -> None, Evaluator -> Automatic, Method -> "Preemptive"], Alignment -> {Center, Center}, ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}]], GraphicsBox[{{{}, {}, { Directive[ Opacity[1.], RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1]], LineBox[CompressedData[" 1:eJwB0QMu/CFib1JlAgAAADwAAAACAAAAnIW2B5tvrD5UzzjGpvEEQPBdXF00 Feo/W6YiFqUkBEDOO7mkBi77P9Gf6i9imABAZDHLW8wwBECwkw0nJl7vP0XB TstvqQpAPZuvZknDhj8alh9oPtcQQPBMmk4eyfC/GlLMUc4dFEAA3BkOfOAA wKF7LwaTqhdAJjyUnDk8BMBaI53txCYbQNZS1+2X7gTAnFE/PABnHkDuR9ci i+sDwLP2yyq49iBAx/OdQHfA/7/chxLr9JsiQJ/r4kyZMuu/H1jeROg4JEA3 EdCPNEO3PyVfBQT2+CVAi0pllQDr8j9vqcZ2CJsnQICLwPv7aQFAfSrjTjVg KUCwBsCc1mMEQKTqhMAYHStAxOCsX4/nBEAP7sDlALwsQDhtYEvqtgNAPihY cAN+LkArQzqbGXD+P9jSRFcFETBA0zVZmSAB6D/zLAsplvQwQGCQcRdgFdG/ miaURwLUMUC6y1l8Ab/2v+RB6r9wpDJANj/bygo2AsCP+O1qbIYzQOS7s68A nQTA3NC+b2pZNEDUdb/0f9gEwLZIUsFDKDVA8AapwdhpA8DyW5NFqgg2QMMz BUJUgfy/0JChIxPaNkClOssAGwDkvxBhXTQJvTdAB6SFLREk2D/c0NuR2ps4 QB4s7569Cvk/SmInSa5rOUBBDM/FJ54CQBqPIDMPTTpAWBrS06+3BECM3eZ2 ch87QIwcQzPnxgRAi8tvB7HtO0BIh9TjQSIDQOxUpsp8zTxAfuztBlK9+j/v /6nnSp49QP0W9MQ84OA/VEZbN6aAPkB9LMq3n/Pdv1uu2eADVD9AxGMpnu/T +b/4Wo1rnhFAQKrlY1qfvgLAc6wEgAGCQEDxjqaJI78EwL+OYsHl6kBA+ksR N6DABMC7PheckFxBQIOboKSXugLAfj4tHSnMQUD/18rEi0P4vxLPKctCNEJA REBynILw2b9XLX0SI6VCQM8NTGJS3OI/bRy3hoQOQ0Dx6pMcxAP8P0lbUqHT dUNAaOAIyUYbA0DWZ0RV6eVDQEi9hYbX0wRANAUdNoBOREDwx7yBT6cEQENw TLDdv0RAn5CsEANZAkAYK93QKC9FQIFNFla0HvY/vnZUHvWWRUAa4ppHo7TT PxWQIgWIB0ZAv5g/7wok5r89OtcYnHBGQO5F2UzVnP2/FrLixXbiRkCK+w0A ep4DwLV5Txk/UkdAHfRkpmXqBMAm0qKZiLpHQDzccM/EbgTAR/hMs5grSEDG W8HQ/ZEBwDmv3fkplUhAp1L4quhj878+GeT4//9IQKQb9h94HMS/HEa84A== "]]}}}, { DisplayFunction -> Identity, AspectRatio -> 1, Axes -> {False, False}, AxesLabel -> {None, None}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, Frame -> {{True, True}, {True, True}}, FrameLabel -> {{None, None}, {None, None}}, FrameStyle -> Directive[ Opacity[0.5], Thickness[Tiny], RGBColor[0.368417, 0.506779, 0.709798]], FrameTicks -> {{None, None}, {None, None}}, GridLines -> {None, None}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], ImageSize -> Dynamic[{ Automatic, 3.5 (CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[ Magnification])}], Method -> { "DefaultBoundaryStyle" -> Automatic, "ScalingFunctions" -> None}, PlotRange -> {{0., 50.}, {-2.616500719197897, 2.61799387799139}}, PlotRangeClipping -> True, PlotRangePadding -> {{ Scaled[0.1], Scaled[0.1]}, { Scaled[0.1], Scaled[0.1]}}, Ticks -> {Automatic, Automatic}}], GridBox[{{ RowBox[{ TagBox["\"Domain: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox[ RowBox[{"{", RowBox[{"{", RowBox[{"0.`", ",", "50.`"}], "}"}], "}"}], "SummaryItem"]}]}, { RowBox[{ TagBox["\"Output: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"scalar\"", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Order: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["3", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Method: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["\"Hermite\"", "SummaryItem"]}]}, { RowBox[{ TagBox["\"Periodic: \"", "SummaryItemAnnotation"], "\[InvisibleSpace]", TagBox["False", "SummaryItem"]}]}}, GridBoxAlignment -> { "Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> { "Columns" -> {{2}}, "Rows" -> {{Automatic}}}, BaseStyle -> { ShowStringCharacters -> False, NumberMarks -> False, PrintPrecision -> 3, ShowSyntaxStyles -> False}]}}, GridBoxAlignment -> {"Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, BaselinePosition -> {1, 1}]}, Dynamic[Typeset`open$$], ImageSize -> Automatic], BaselinePosition -> Baseline], DynamicModuleValues :> {}], StyleBox["]", "NonInterpretableSummary"]}]}, "CopyTag", DisplayFunction->(#& ), InterpretationFunction->("InterpolatingFunction[{{0., 50.}}, <>]"& )], False, Editable->False, SelectWithContents->True, Selectable->False]}], "}"}], "}"}], ",", RowBox[{"{", RowBox[{"{", RowBox[{ "3.3428455269521837`", ",", "10.028534627349119`", ",", "16.714221051476557`", ",", "23.399904800155852`", ",", "30.085585873357363`", ",", "36.77126427101624`", ",", "43.45693999306614`"}], "}"}], "}"}]}]}]], "Output", GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{3.7311551920708*^9}], "\n\nVen que el resultado contiene dos partes: la soluci\[OAcute]n \ propiamente dicha, \[Phi](t), y una serie de tiempos en los que \[Phi] cruza \ por cero. La celda de abajo extrae cada cosa por separado:" }], "Subchapter", CellChangeTimes->{{3.7311552048817997`*^9, 3.7311552590861998`*^9}, { 3.7311664492724*^9, 3.7311664513627996`*^9}, {3.7311852147502003`*^9, 3.7311852256482*^9}, 3.7311899168392*^9, {3.7312371739519997`*^9, 3.7312371869052*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{"Phi", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"sol2", "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Tiempos", " ", "=", " ", RowBox[{ RowBox[{"sol2", "[", RowBox[{"[", "2", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}], "Input", CellChangeTimes->{{3.7310961605108*^9, 3.7310961998182*^9}, { 3.7310965046906*^9, 3.7310965060478*^9}, 3.7312372024968*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Verifiquemos que los sucesivos pasos por cero est\[EAcute]n equiespaciados\ \>", "Subchapter", CellChangeTimes->{{3.7310962137812*^9, 3.731096228185*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"lista", " ", "=", " ", RowBox[{"Differences", "[", "Tiempos", "]"}]}]], "Input", CellChangeTimes->{{3.731237265131*^9, 3.731237275862*^9}}], Cell[BoxData[ RowBox[{"ListPlot", "[", RowBox[{"lista", ",", " ", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Min", "[", "lista", "]"}], "-", FractionBox[ RowBox[{ RowBox[{"Max", "[", "lista", "]"}], "-", RowBox[{"Min", "[", "lista", "]"}]}], "4"]}], ",", RowBox[{ RowBox[{"Max", "[", "lista", "]"}], "+", FractionBox[ RowBox[{ RowBox[{"Max", "[", "lista", "]"}], "-", RowBox[{"Min", "[", "lista", "]"}]}], "4"]}]}], "}"}]}], ",", " ", RowBox[{"Frame", "\[Rule]", " ", "True"}], ",", " ", RowBox[{"ImageSize", "\[Rule]", "500"}], ",", " ", RowBox[{"FrameTicksStyle", "\[Rule]", "16"}]}], "]"}]], "Input", CellChangeTimes->{{3.731166596818*^9, 3.7311667824848003`*^9}, { 3.7311899483571997`*^9, 3.7311899513962*^9}, {3.7312372950808*^9, 3.7312373045692*^9}}], Cell["\[DownExclamation]OK dentro del error! Miren la escala vertical.", \ "Text", CellChangeTimes->{{3.7310966177014*^9, 3.731096624743*^9}, { 3.7311899423052*^9, 3.7311899427502003`*^9}, {3.7312372874898*^9, 3.7312372918104*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Entonces para hacer la animaci\[OAcute]n alcanza con variar t entre 0 y el \ per\[IAcute]odo, obtenido como la diferencia de dos valores consecutivos \ dentro de la lista de tiempos Tiempos (y multiplicado por 2 para abarcar los \ casos de oscilaci\[OAcute]n y rotaci\[OAcute]n).\ \>", "Subchapter", CellChangeTimes->{{3.7310966462906*^9, 3.7310967088297997`*^9}, 3.7311012284406*^9, {3.7311552875418*^9, 3.7311553323526*^9}, { 3.7311852502432003`*^9, 3.7311852551832*^9}, 3.7312372207392*^9, { 3.731237340416*^9, 3.731237368888*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"Periodo", " ", "=", RowBox[{"2", " ", "\[Times]", " ", RowBox[{ RowBox[{"Differences", "[", "Tiempos", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "El", " ", "s\[IAcute]mbolo", " ", "\[Times]", " ", "es", " ", "redundante"}], ",", " ", RowBox[{ "pero", " ", "a", " ", "veces", " ", "ayuda", " ", "a", " ", "la", " ", RowBox[{"lectura", ".", " ", "Se"}], " ", "escribe", " ", RowBox[{"como", " ", "[", "Esc", "]"}], "*", RowBox[{"[", "Esc", "]"}]}]}], " ", "*)"}], "\[IndentingNewLine]"}], "\n", RowBox[{ RowBox[{"Ncuadros", " ", "=", " ", "100."}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]", " ", "=", " ", FractionBox["Periodo", "Ncuadros"]}], ";"}], " ", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"ejes", " ", "=", " ", RowBox[{"Graphics", "[", RowBox[{"{", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "2", "]"}], ",", " ", "LightBlue", ",", " ", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.5"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"1.5", ",", " ", "0"}], "}"}]}], "}"}], "]"}], ",", " ", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", "1.5"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1.5"}]}], "}"}]}], "}"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"White", ",", RowBox[{"Text", "[", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\<\[ScriptX]\>\"", ",", " ", "30"}], "]"}], ",", RowBox[{"{", RowBox[{"0.2", ",", RowBox[{ RowBox[{"-", "0.9"}], "\[Times]", "1.5"}]}], "}"}]}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"White", ",", RowBox[{"Text", "[", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\<\[ScriptY]\>\"", ",", " ", "30"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"0.9", " ", "1.5"}], ",", "0.2"}], "}"}]}], "]"}]}], "}"}]}], "}"}], "\[IndentingNewLine]", "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Manipulate", "[", " ", RowBox[{"(*", " ", RowBox[{ "Ahora", " ", "el", " ", "tiempo", " ", "m\[AAcute]ximo", " ", "est\[AAcute]", " ", "elegido", " ", "igual", " ", "al", " ", RowBox[{"Per\[IAcute]odo", ".", " ", "El"}], " ", "paso", " ", "temporal", " ", "por", " ", "defecto", " ", "entre", " ", "cuadros", " ", "sucesivos", " ", "es", " ", RowBox[{"\[CapitalDelta]", "."}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Phi]pr", " ", "=", " ", RowBox[{"Phi", "[", "t", "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Show", "[", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"ejes", ",", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Graphics", "[", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "2", "]"}], ",", " ", "Blue", ",", " ", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]pr", "]"}]}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]pr", "]"}]}]}], "}"}]}], "}"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"Green", ",", " ", RowBox[{"Disk", "[", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]pr", "]"}]}], ",", " ", "0"}], "}"}], ",", " ", "0.05"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Disk", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]pr", "]"}]}]}], "}"}], ",", " ", "0.05"}], "]"}]}], "}"}]}], "}"}], "\[IndentingNewLine]", "]"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"AspectRatio", "\[Rule]", " ", "Automatic"}], ",", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"1.1", "\[Times]", " ", "1.5", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "400"}]}], "\[IndentingNewLine]", "]"}]}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", RowBox[{"Periodo", "-", "\[CapitalDelta]"}], ",", "\[CapitalDelta]", ",", " ", RowBox[{"Appearance", "\[Rule]", " ", "\"\\""}], ",", " ", RowBox[{"AnimationRate", "\[Rule]", " ", "5"}]}], "}"}]}], " ", "\[IndentingNewLine]", "]"}]}], "Input", CellChangeTimes->{{3.7310967218724003`*^9, 3.7310967310608*^9}, { 3.7311553100690002`*^9, 3.731155339758*^9}, {3.7311668791654*^9, 3.7311668868445997`*^9}, 3.7311670325176*^9, 3.7311857748082*^9, 3.7312372225996*^9, {3.731237374059*^9, 3.7312374210158*^9}, { 3.7312375294188004`*^9, 3.7312375751696*^9}}], Cell["\<\ \[DownExclamation]Ahora la animaci\[OAcute]n es un loop cerrado! \ \[DownExclamation]Vaya que hay diferencia!\ \>", "Text", CellChangeTimes->{{3.731096770491*^9, 3.7310968072974*^9}, { 3.7311012948456*^9, 3.7311013009306*^9}, {3.7311554688512*^9, 3.7311554848515997`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Para exportar la animaci\[OAcute]n como un ", StyleBox["gif", FontWeight->"Bold"], " usando el propio ", StyleBox["Mathematica.", FontSlant->"Italic"], " \n\nPrimero creamos una tabla con todos los gr\[AAcute]ficos. En esencia, \ es sustituir en la celda de arriba ", StyleBox["Manipulate", FontWeight->"Bold"], " por ", StyleBox["Table", FontWeight->"Bold"], "." }], "Subchapter", CellChangeTimes->{{3.7311600013428*^9, 3.7311600681056004`*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{"Ncuadros", " ", "=", " ", "200."}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"T", " ", "=", " ", "Periodo"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "Si", " ", "el", " ", "movimiento", " ", "fuera", " ", "de", " ", "rotaci\[OAcute]n"}], ",", " ", RowBox[{ "habr\[IAcute]a", " ", "que", " ", "dividir", " ", "el", " ", "Periodo", " ", "por", " ", "2"}], ",", " ", RowBox[{ "para", " ", "no", " ", "hacer", " ", "un", " ", "gif", " ", "el", " ", "doble", " ", "de", " ", "largo", " ", "de", " ", "lo", " ", "necesario"}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[CapitalDelta]", " ", "=", " ", FractionBox["T", "Ncuadros"]}], ";"}], " ", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"tgraficos", " ", "=", " ", RowBox[{"Table", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[Phi]pr", " ", "=", " ", RowBox[{"Phi", "[", "t", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{"ejes", ",", " ", "\[IndentingNewLine]", RowBox[{"Graphics", "[", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "2", "]"}], ",", " ", "Blue", ",", " ", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]pr", "]"}]}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]pr", "]"}]}]}], "}"}]}], "}"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"Green", ",", " ", RowBox[{"Disk", "[", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]pr", "]"}]}], ",", " ", "0"}], "}"}], ",", " ", "0.05"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Disk", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]pr", "]"}]}]}], "}"}], ",", " ", "0.05"}], "]"}]}], "}"}]}], "}"}], "\[IndentingNewLine]", "]"}], ",", "\[IndentingNewLine]", RowBox[{"AspectRatio", "\[Rule]", " ", "Automatic"}], ",", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"1.1", "\[Times]", " ", "1.5", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "400"}]}], "\[IndentingNewLine]", "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", RowBox[{"T", "-", "\[CapitalDelta]"}], ",", "\[CapitalDelta]"}], "}"}]}], " ", "\[IndentingNewLine]", "]"}]}], ";"}]}], "Input", CellChangeTimes->{{3.7310967396594*^9, 3.7310967568984003`*^9}, { 3.7311012536748*^9, 3.7311012767044*^9}, {3.7311553204776*^9, 3.7311554542274*^9}, {3.7311572069248*^9, 3.7311572204105997`*^9}, { 3.7311599368458*^9, 3.7311599427664003`*^9}, {3.7311599864955997`*^9, 3.7311599868406*^9}, {3.7311600790998*^9, 3.731160118392*^9}, { 3.7311669096236*^9, 3.7311669875861998`*^9}, 3.731167042257*^9, 3.7311853538892*^9, {3.7312376195032*^9, 3.7312376197693996`*^9}, { 3.7312376650456*^9, 3.7312376891896*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "En efecto, todos los cuadros est\[AAcute]n en la lista ", StyleBox["tgraficos", FontWeight->"Bold"], ", y se pueden animar directamente:" }], "Subsection", CellChangeTimes->{{3.7311679310076*^9, 3.7311679363536*^9}, { 3.7311853857382*^9, 3.7311853998662*^9}, {3.7311856765022*^9, 3.7311856959862003`*^9}}], Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"tgraficos", "[", RowBox[{"[", "i", "]"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"i", ",", " ", "1", ",", " ", RowBox[{"Length", "[", "tgraficos", "]"}], ",", " ", "1", ",", " ", RowBox[{"Appearance", "\[Rule]", " ", "\"\\""}], ",", " ", RowBox[{"AnimationRate", "\[Rule]", " ", "50"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7311679414966*^9, 3.7311679573826*^9}, { 3.7311680086556*^9, 3.7311680200554*^9}, {3.7311853253332*^9, 3.7311853420882*^9}, {3.7312376498978*^9, 3.7312376746358*^9}, { 3.7312377073856*^9, 3.7312377075416*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Ahora exportamos el archivo gif, que aparecer\[AAcute] en el mismo directorio \ que el Notebook. Puede tomar hasta 1 minuto. \ \>", "Subchapter", CellDingbat->"\[WatchIcon]", CellChangeTimes->{{3.731160135792*^9, 3.7311601902052*^9}, { 3.7311602434168*^9, 3.7311602931492*^9}, {3.7312377273204*^9, 3.7312377315327997`*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{"SetDirectory", "[", RowBox[{"NotebookDirectory", "[", "]"}], "]"}], ";"}], "\n", RowBox[{ RowBox[{"Export", "[", RowBox[{ "\"\\"", ",", " ", "tgraficos", ",", " ", "\"\\"", ",", " ", RowBox[{"\"\\"", "\[Rule]", " ", RowBox[{"1", "/", "25"}]}]}], "]"}], ";"}]}], "Input", CellChangeTimes->{{3.731160194182*^9, 3.7311602022736*^9}, 3.7311603161518*^9, {3.7311604377896*^9, 3.7311604465727997`*^9}}], Cell["\<\ Mientras se hace el gif, aprovecho para decir que lo mejor suele ser \ exportar cada figura a un archivo, y luego hacer el gif en otro programa.\ \>", "Text", CellChangeTimes->{{3.7311602957644*^9, 3.7311603062406*^9}, { 3.7311854337511997`*^9, 3.7311854683782*^9}}], Cell["El tama\[NTilde]o en kB del gif generado: ", "Text", CellChangeTimes->{{3.7311855934272003`*^9, 3.7311856065102*^9}}, FontSize->18], Cell[BoxData[ RowBox[{"Round", "[", RowBox[{ RowBox[{"FileByteCount", "[", "\"\\"", "]"}], "/", "1000"}], "]"}]], "Input", CellChangeTimes->{{3.7311854994561996`*^9, 3.7311855522522*^9}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Segundo ejemplo: problema 17.\n\n", Cell[BoxData[ GraphicsBox[{ {RGBColor[0.87, 0.94, 1], AbsoluteThickness[1], ArrowBox[{{-1.5, 0}, {1.5, 0}}], ArrowBox[{{0, 1.5}, {0, -1.5}}]}, { {RGBColor[0, 1, 0], Opacity[1], DiskBox[{-0.95, 0}, 0.05]}, {RGBColor[1, 0, 0], Opacity[1], DiskBox[NCache[{-0.44999999999999996`, Rational[-1, 2] 3^Rational[1, 2]}, {-0.44999999999999996`, -0.8660254037844386}], 0.05]}}, {RGBColor[1, 0.5, 0], AbsoluteThickness[3], Arrowheads[0.03], ArrowBox[{{0.75, -0.25}, {0.75, -0.75}}]}, {GrayLevel[1], AbsoluteThickness[4], ArrowBox[NCache[{{-0.95, 0}, {-0.44999999999999996`, Rational[-1, 2] 3^Rational[1, 2]}}, {{-0.95, 0}, {-0.44999999999999996`, -0.8660254037844386}}]]}, {GrayLevel[1], AbsoluteThickness[1], Dashing[{Small, Small}], LineBox[{{-0.95, 0}, {-0.95, -1.5}}]}, {RGBColor[1, 1, 0], AbsoluteThickness[2], ArrowBox[{{-0.9500000000000002, -1.}, {-0.900020830729322, \ -0.9987502603949663}, {-0.8501665833531723, -0.9950041652780258}, \ {-0.8005618675264006, -0.9887710779360422}, {-0.7513306692049387, \ -0.9800665778412416}, {-0.7025960407454772, -0.9689124217106448}, \ {-0.6544797933386608, -0.9553364891256061}, {-0.6071021925445491, \ -0.9393727128473791}, {-0.5605816576913494, -0.921060994002885}, \ {-0.5150344658887698, -0.9004471023526769}, {-0.4705744613957971, \ -0.8775825618903728}}]}, { {GrayLevel[1], InsetBox[ StyleBox["\<\"\\!\\(\\*SubscriptBox[\\(m\\), \\(1\\)]\\)\"\>", StripOnInput->False, FontSize->25], {-0.95, 0.25}]}, {RGBColor[1, 0.5, 0], InsetBox[ StyleBox["\<\"\\!\\(\\*OverscriptBox[\\(g\\), \\(\\(\[Rule]\\)\\(\\\\ \ \\)\\)]\\)\"\>", StripOnInput->False, FontSize->24], {0.95, -0.45}]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\[ScriptL]\"\>", StripOnInput->False, FontSize->30], {-0.51, -0.4330127018922193}]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\\!\\(\\*SubscriptBox[\\(m\\), \\(2\\)]\\)\"\>", StripOnInput->False, FontSize->25], {-0.24999999999999994`, -0.7160254037844386}]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\[ScriptX]\"\>", StripOnInput->False, FontSize->30], {0.2, -1.35}]}, {GrayLevel[1], InsetBox[ StyleBox["\<\"\[ScriptY]\"\>", StripOnInput->False, FontSize->30], {1.35, 0.2}]}, {RGBColor[1, 1, 0], InsetBox[ StyleBox["\<\"y\"\>", StripOnInput->False, FontSize->30], {-1.15, -0.12}]}, {RGBColor[1, 1, 0], InsetBox[ StyleBox["\<\"\[Phi]\"\>", StripOnInput->False, FontSize->25], {-0.6911809548974792, -1.1459258262890681`}]}}}, AspectRatio->Automatic, Axes->False, Background->GrayLevel[0], ImageSize->400, PlotRange->{{-1.6500000000000001`, 1.6500000000000001`}, {-1.6500000000000001`, 1.6500000000000001`}}, Ticks->False]]], "\n", StyleBox["\nIncluye comandos apenas m\[AAcute]s avanzados, opciones 3D y un \ exportador de gr\[AAcute]ficos. Se da por conocido lo explicado en el \ problema anterior.\n\nExpandir las celdas haciendo doble click en la flechita \ (por dem\[AAcute]s peque\[NTilde]ita) de la derecha.", FontSize->24] }], "Chapter", CellFrame->{{0, 0}, {0, 3}}, CellChangeTimes->{{3.7310969040467997`*^9, 3.731096936454*^9}, { 3.7311560086476*^9, 3.7311560129592*^9}, {3.7311905453122*^9, 3.7311906439182*^9}, {3.7311908569751997`*^9, 3.7311908792202*^9}, { 3.7311909402622004`*^9, 3.7311909705512*^9}, {3.7312377900094*^9, 3.7312378044622*^9}}], Cell[CellGroupData[{ Cell["\<\ Lo que sigue: \[FilledSmallCircle] La soluci\[OAcute]n num\[EAcute]rica. \[FilledSmallCircle] La extracci\[OAcute]n del per\[IAcute]odo. \[FilledSmallCircle] El gr\[AAcute]fico en bruto de las funciones \[Phi][t] e \ y[t].\ \>", "Section", CellChangeTimes->{{3.7310980899834003`*^9, 3.7310981529414*^9}, { 3.7311014944354*^9, 3.731101518539*^9}, {3.7311560340376*^9, 3.7311561126162*^9}, {3.7311634121303997`*^9, 3.7311634128993998`*^9}, { 3.7311907820472*^9, 3.7311907832782*^9}, {3.7312378162218*^9, 3.7312378178326*^9}}, FontSize->18], Cell[TextData[{ StyleBox["Antes que nada un ", FontSize->18], StyleBox["Quit", FontSize->18, FontWeight->"Bold"], StyleBox[". Reinicia el kernel del ", FontSize->18], StyleBox["Mathematica", FontSize->18, FontSlant->"Italic"], StyleBox[", libera la memoria y borra las definiciones y resultados \ anteriores.", FontSize->18] }], "Text", CellChangeTimes->{{3.7311561483652*^9, 3.7311563159108*^9}, { 3.7311910224722*^9, 3.7311910565522003`*^9}, {3.7311911610882*^9, 3.7311912712502003`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Quit", "[", "]"}], ";"}]], "Input", CellChangeTimes->{{3.7311910993822002`*^9, 3.7311911338022003`*^9}, 3.7311911861772003`*^9, {3.7311912343022003`*^9, 3.7311912350341997`*^9}}], Cell[TextData[{ StyleBox["El comando ", FontSize->18], StyleBox["Module", FontSize->18, FontWeight->"Bold"], StyleBox[" aqu\[IAcute] abajo encierra todo el c\[AAcute]lculo de la soluci\ \[OAcute]n num\[EAcute]rica de la ecuaci\[OAcute]n de movimiento. \n\n", FontSize->18], StyleBox["Module", FontSize->18, FontWeight->"Bold"], StyleBox[" sirve para definir variables locales. As\[IAcute] se evita el \ peligro de definiciones m\[UAcute]ltiples para un mismo s\[IAcute]mbolo. Es \ pr\[AAcute]ctica recomendada. Todo lo que est\[AAcute] definido dentro de las \ primeras llaves tiene significado \[UAcute]nicamente dentro del bloque \ encerrado por ", FontSize->18], StyleBox["Module", FontSize->18, FontWeight->"Bold"], StyleBox[".", FontSize->18] }], "Text", CellChangeTimes->{{3.7311561483652*^9, 3.7311563159108*^9}, { 3.7311910224722*^9, 3.7311910565522003`*^9}, {3.7311912879742002`*^9, 3.7311913163812*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"Module", "[", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ "\[Phi]", ",", " ", "\[IndentingNewLine]", "y", ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{"Ccondiciones", " ", "iniciales"}], "*)"}], "\[IndentingNewLine]", RowBox[{"\[Phi]0", " ", "=", RowBox[{"\[Pi]", "/", "2"}]}], ",", "\[IndentingNewLine]", RowBox[{"\[Phi]p0", " ", "=", " ", RowBox[{"-", "1.42"}]}], ",", "\[IndentingNewLine]", RowBox[{"y0", " ", "=", " ", "0"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{ "Hay", " ", "una", " ", "condici\[OAcute]n", " ", "inicial", " ", "suplementaria", " ", "que", " ", "est\[AAcute]", " ", "impl\[IAcute]cita", " ", "en", " ", "las", " ", "ecuaciones", " ", "de", " ", "movimiento", " ", "dentro", " ", "del", " ", "NDSolve"}], ";", " ", RowBox[{"a", " ", "saber"}]}], ",", " ", RowBox[{ SubscriptBox["P", "y"], " ", "=", " ", "0."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"La", " ", "elecci\[OAcute]n", " ", SubscriptBox["P", "y"]}], " ", "=", " ", RowBox[{ "0", " ", "no", " ", "significa", " ", "ninguna", " ", "p\[EAcute]rdida", " ", "de", " ", RowBox[{"generalidad", ".", " ", "Corresponde"}], " ", "a", " ", "tomar", " ", "un", " ", "SR", " ", "que", " ", "se", " ", "mueve", " ", "con", " ", "la", " ", "velocidad", " ", "horizontal", " ", "del", " ", RowBox[{"CM", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"Estoy", " ", "tomando", " ", "m2"}], " ", "=", " ", RowBox[{ RowBox[{"1", " ", "y", " ", RowBox[{"g", "/", "l"}]}], " ", "=", " ", "1"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"m1", " ", "=", " ", "5"}], ",", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"tmax", " ", "=", " ", "30"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{ "Soluci\[OAcute]n", " ", "num\[EAcute]rica", " ", "y", " ", "cosecha", " ", "de", " ", "los", " ", "tiempos", " ", "de", " ", "paso", " ", "por", " ", "la", " ", "vertical"}], ",", " ", RowBox[{"para", " ", "medir", " ", "el", " ", "per\[IAcute]odo"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"sol", ",", " ", "cruces"}], "}"}], " ", "=", RowBox[{"Reap", "[", RowBox[{"NDSolve", "[", "\[IndentingNewLine]", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}], "+", FractionBox[ RowBox[{" ", RowBox[{ RowBox[{"Sin", "[", RowBox[{"2", RowBox[{"\[Phi]", "[", "t", "]"}]}], "]"}], " ", SuperscriptBox[ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}], "2"]}]}], RowBox[{"2", RowBox[{"(", RowBox[{"1", "+", "m1"}], ")"}]}]], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox[ RowBox[{"Cos", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}], "2"], RowBox[{"1", "+", "m1"}]]}], ")"}], " ", RowBox[{ RowBox[{"\[Phi]", "''"}], "[", "t", "]"}]}]}], " ", "\[Equal]", " ", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "t", "]"}], " ", "\[Equal]", RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"1", "+", "m1"}]]}], RowBox[{"Cos", "[", RowBox[{"\[Phi]", "[", "t", "]"}], "]"}], " ", RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}]}]}], ",", " ", RowBox[{ RowBox[{"\[Phi]", "[", "0", "]"}], " ", "\[Equal]", " ", "\[Phi]0"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "0", "]"}], " ", "\[Equal]", " ", "\[Phi]p0"}], ",", " ", RowBox[{ RowBox[{"y", "[", "0", "]"}], " ", "\[Equal]", " ", "y0"}], ",", "\[IndentingNewLine]", RowBox[{"WhenEvent", "[", RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], "/", "2"}], "]"}], " ", "\[Equal]", " ", "0"}], " ", ",", " ", RowBox[{"Sow", "[", "t", "]"}]}], "]"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Phi]", ",", " ", "y"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "tmax"}], "}"}]}], "]"}], "]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"Phi", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], ",", " ", "\"\<\\n\>\""}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Print", "[", RowBox[{"\"\\"", ",", " ", RowBox[{"Y", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], ",", " ", "\"\<\\n\>\""}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", " ", RowBox[{"cruces", "[", RowBox[{"[", "1", "]"}], "]"}], ",", " ", "\"\<\\n\>\""}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Print", "[", RowBox[{ "\"\<\[CapitalDelta]t entre cruces consecutivos \[Rule] \>\"", ",", " ", RowBox[{"Differences", "[", RowBox[{"cruces", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], ",", " ", "\"\<\\n\>\""}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", " ", RowBox[{"T", " ", "=", RowBox[{"2", " ", RowBox[{ RowBox[{"Differences", "[", RowBox[{"cruces", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}]}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Phi", "[", "t", "]"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", RowBox[{"2", "T"}]}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", "Thick"}], ",", " ", RowBox[{"ImageMargins", "\[Rule]", "30"}], ",", " ", RowBox[{"ImageSize", "\[Rule]", "400"}], ",", " ", RowBox[{"PlotLabel", "\[Rule]", " ", RowBox[{"Style", "[", RowBox[{"\"\<\[Phi](t) \\n\>\"", ",", " ", "18"}], "]"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Y", "[", "t", "]"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", RowBox[{"2", "T"}]}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", "Thick"}], ",", " ", RowBox[{"ImageMargins", "\[Rule]", "30"}], ",", RowBox[{"ImageSize", "\[Rule]", "400"}], ",", " ", RowBox[{"PlotLabel", "\[Rule]", " ", RowBox[{"Style", "[", RowBox[{"\"\\"", ",", " ", "18"}], "]"}]}]}], "]"}]}], "}"}]}], "Input", CellChangeTimes->{{3.7309299208972*^9, 3.7309299302468*^9}, { 3.7309300836935997`*^9, 3.7309301202862*^9}, {3.7309302411535997`*^9, 3.730930276576*^9}, {3.7309303131898003`*^9, 3.7309303625973997`*^9}, { 3.7309877890039997`*^9, 3.7309879280081997`*^9}, {3.7309879602634*^9, 3.7309884510758*^9}, {3.7309885755596*^9, 3.7309887892925997`*^9}, 3.7309888221022*^9, {3.7309896759948*^9, 3.730989712383*^9}, { 3.7309897723873997`*^9, 3.7309897942402*^9}, {3.7309902840242*^9, 3.7309904315088*^9}, {3.7309904869948*^9, 3.730990530172*^9}, { 3.7309912317482*^9, 3.7309913086118*^9}, {3.730991460476*^9, 3.7309915676963997`*^9}, {3.7309918236628*^9, 3.7309918245364*^9}, { 3.7309919542376003`*^9, 3.7309919669254*^9}, {3.7310048663722*^9, 3.7310048806436*^9}, {3.7310049240706*^9, 3.7310049382806*^9}, { 3.731005038652*^9, 3.7310050478803997`*^9}, 3.7310051689264*^9, { 3.7310052402282*^9, 3.7310052499528*^9}, {3.7310052999700003`*^9, 3.731005323466*^9}, {3.7310053652816*^9, 3.731005402766*^9}, { 3.7310054383266*^9, 3.7310054777248*^9}, {3.7310201669844*^9, 3.7310202546912003`*^9}, {3.7310208508118*^9, 3.7310208562292*^9}, { 3.7310210031142*^9, 3.7310210057124*^9}, {3.7310211439642*^9, 3.7310212229154*^9}, {3.7310212747418003`*^9, 3.7310212831232*^9}, 3.7310213572518*^9, 3.7310213998722*^9, {3.7310214398154*^9, 3.7310214537088003`*^9}, {3.7310214842342*^9, 3.731021529708*^9}, 3.7310215686324*^9, {3.731021608182*^9, 3.7310217440572*^9}, 3.7310217752282*^9, {3.7310218076730003`*^9, 3.7310218239502*^9}, { 3.731096973901*^9, 3.7310971095478*^9}, {3.7310972051238003`*^9, 3.7310972367676*^9}, {3.7310972732720003`*^9, 3.7310973049374*^9}, { 3.731097416513*^9, 3.7310974461102*^9}, {3.7310981634421997`*^9, 3.7310982263613997`*^9}, {3.7311015262752*^9, 3.7311016212054*^9}, { 3.7311016685987997`*^9, 3.7311016985492*^9}, {3.7311032200559998`*^9, 3.7311032228016*^9}, {3.7311561201308002`*^9, 3.7311561467398*^9}, { 3.7311563391012*^9, 3.7311563505208*^9}, {3.7311564179608*^9, 3.7311564405846*^9}, {3.7311564934756002`*^9, 3.7311565448803997`*^9}, { 3.731156578178*^9, 3.7311566194435997`*^9}, {3.7311568048188*^9, 3.7311568072720003`*^9}, {3.7311605676482*^9, 3.731160690524*^9}, { 3.7311913476362*^9, 3.7311914291912003`*^9}, {3.7311915106652*^9, 3.7311915372892*^9}, {3.731237870126*^9, 3.7312378723784*^9}, { 3.731237947115*^9, 3.7312379870534*^9}, {3.7312380312028*^9, 3.7312380528204*^9}, {3.7312380950298*^9, 3.7312381105362*^9}, { 3.731238142583*^9, 3.7312382682718*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Abajo, la animaci\[OAcute]n:", "Subchapter", CellChangeTimes->{{3.7311017159926*^9, 3.7311017198450003`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "El", " ", "bloque", " ", "DynamicModule", " ", "es", " ", "igual", " ", "que", " ", "el", " ", "Module", " ", "pero", " ", "para", " ", "celdas", " ", "que", " ", "se", " ", "eval\[UAcute]an", " ", RowBox[{"din\[AAcute]micamente", ".", " ", "Puede"}], " ", "omitirse", " ", "sin", " ", "peligro", " ", "si", " ", "no", " ", "repiten", " ", "definiciones", " ", "de", " ", "los", " ", RowBox[{"s\[IAcute]mbolos", "."}]}], " ", "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ "DynamicModule", "[", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"y", ",", " ", "\[Phi]", ",", " ", "t", ",", " ", RowBox[{"ran", " ", "=", " ", "2"}], ",", " ", RowBox[{"\[CapitalDelta]", " ", "=", " ", RowBox[{"T", "/", "100"}]}], ",", " ", "ejes", ",", " ", "orb"}], "}"}], ",", " ", RowBox[{"(*", " ", RowBox[{"Todas", " ", "estas", " ", "variables", " ", "son", " ", RowBox[{"locales", ".", " ", "Fijo"}], " ", "en", " ", RowBox[{"T", "/", "100"}], " ", "el", " ", "intervalo", " ", "entre", " ", "dos", " ", "cuadros", " ", RowBox[{"sucesivos", "."}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"ejes", " ", "=", " ", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "2", "]"}], ",", " ", "White", ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "ran"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"ran", ",", " ", "0"}], "}"}]}], "}"}], "]"}], ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", "ran"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "ran"}], "}"}]}], "}"}], "]"}]}], "}"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"orb", " ", "=", " ", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Y", "[", "\[Tau]", "]"}], "+", RowBox[{"Sin", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Tau]", ",", " ", "0", ",", " ", "T"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Dotted", ",", "Orange"}], "}"}]}]}], "]"}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ "La", " ", "\[OAcute]rbita", " ", "de", " ", "la", " ", "segunda", " ", "masa"}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Manipulate", "[", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"y", " ", "=", " ", RowBox[{"Y", "[", "t", "]"}]}], ";", " ", "\[IndentingNewLine]", RowBox[{"\[Phi]", " ", "=", " ", RowBox[{"Phi", "[", "t", "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{ "ejes", ",", "\[IndentingNewLine]", "orb", ",", "\[IndentingNewLine]", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "2", "]"}], ",", " ", "Blue", ",", " ", RowBox[{"Line", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"y", ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}]}], "}"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Green", ",", " ", RowBox[{"Disk", "[", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}], ",", " ", "0.1"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Disk", "[", RowBox[{ RowBox[{"{", RowBox[{"y", ",", " ", "0"}], "}"}], ",", " ", "0.1"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"AspectRatio", "\[Rule]", " ", "Automatic"}], ",", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"ran", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "600"}]}], "\[IndentingNewLine]", "]"}]}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", RowBox[{ RowBox[{"Abajo", ":", " ", RowBox[{ RowBox[{ "el", " ", "tiempo", " ", "m\[AAcute]ximo", " ", "llega", " ", "hasta", " ", "T"}], " ", "-", " ", "\[CapitalDelta]"}]}], ",", " ", RowBox[{ "porque", " ", "si", " ", "llegase", " ", "hasta", " ", "T", " ", "la", " ", "animaci\[OAcute]n", " ", "repetir\[IAcute]a", " ", "el", " ", "primer", " ", "cuadro"}], ",", " ", RowBox[{ RowBox[{"ya", " ", "que", " ", "en", " ", "t"}], " ", "=", " ", RowBox[{ RowBox[{"0", " ", "y", " ", "t"}], " ", "=", " ", RowBox[{ "T", " ", "la", " ", "configuraci\[OAcute]n", " ", "es", " ", "la", " ", RowBox[{"misma", "."}]}]}]}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", RowBox[{"T", "-", "\[CapitalDelta]"}], ",", " ", "\[CapitalDelta]", ",", " ", RowBox[{"Appearance", "\[Rule]", " ", "\"\\""}]}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"TrackedSymbols", "\[RuleDelayed]", " ", RowBox[{"{", "t", "}"}]}]}], " ", RowBox[{"(*", RowBox[{ "S\[OAcute]lo", " ", "recalcula", " ", "las", " ", "cosas", " ", "cuando", " ", "cambia", " ", RowBox[{"t", "."}]}], "*)"}], "\[IndentingNewLine]", "]"}]}]}], "\[IndentingNewLine]", "]"}]}]], "Input", CellChangeTimes->{{3.7309893676366*^9, 3.7309896446464*^9}, { 3.730989872724*^9, 3.7309898902634*^9}, {3.7309905906118*^9, 3.7309905946521997`*^9}, {3.7310199483114*^9, 3.7310201463822002`*^9}, { 3.7310202717964*^9, 3.7310203332836*^9}, {3.7310215566038*^9, 3.7310215570425997`*^9}, {3.731021598109*^9, 3.731021598299*^9}, { 3.7310973353156*^9, 3.7310974361085997`*^9}, {3.731097487684*^9, 3.731097510391*^9}, {3.7311017222096*^9, 3.7311018121378*^9}, { 3.7311019121818*^9, 3.7311019409941998`*^9}, {3.7311045633208*^9, 3.7311045904376*^9}, {3.7311046494316*^9, 3.7311046748344*^9}, { 3.7311566346464*^9, 3.7311566640984*^9}, {3.7311567267236*^9, 3.731156781646*^9}, {3.7311571660044003`*^9, 3.7311571892084*^9}, 3.7311634184634*^9, {3.7311691040856*^9, 3.7311691160256*^9}, { 3.7311915657972*^9, 3.7311915661452*^9}, {3.7312383944700003`*^9, 3.7312383952686*^9}}], Cell["\<\ Ya est\[AAcute] ajustado para que el loop sea cerrado. Lo podemos subir a \ Tumblr sin que nos averg\[UDoubleDot]ence (al menos no por esa raz\[OAcute]n).\ \>", "Text", CellChangeTimes->{{3.731238299458*^9, 3.7312383786608*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "En ", Cell[BoxData[ FormBox[ SuperscriptBox["glorioso", "*"], TraditionalForm]]], " 3D, pero nada \[OAcute]ptimo." }], "Subchapter", CellChangeTimes->{{3.7310980596842003`*^9, 3.7310980627604*^9}, { 3.7310982700844*^9, 3.7310982720811996`*^9}, {3.7311019742448*^9, 3.7311019820164003`*^9}, {3.7311568515366*^9, 3.731156868163*^9}, { 3.7311691441586*^9, 3.7311691459370003`*^9}, {3.7311915796102*^9, 3.7311915800692*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"DynamicModule", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"y", ",", " ", "\[Phi]", ",", " ", "t", ",", " ", RowBox[{"ran", " ", "=", " ", "2"}], ",", " ", RowBox[{"\[CapitalDelta]", " ", "=", " ", RowBox[{"T", "/", "200"}]}], ",", " ", "ejes", ",", " ", "orb"}], "}"}], ",", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{"ejes", " ", "=", " ", RowBox[{"Graphics3D", "[", RowBox[{"{", " ", RowBox[{"White", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "ran"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "ran", ",", " ", "0"}], "}"}]}], "}"}], ",", " ", "0.012"}], "]"}], ",", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", " ", RowBox[{"-", "ran"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "ran"}], "}"}]}], "}"}], ",", " ", "0.012"}], "]"}]}], "}"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"orb", " ", "=", " ", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{ RowBox[{"Y", "[", "\[Tau]", "]"}], "+", RowBox[{"Sin", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Tau]", ",", " ", "0", ",", " ", "T"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", " ", RowBox[{ RowBox[{"AbsoluteDashing", "[", RowBox[{"{", RowBox[{"5", ",", " ", "8"}], "}"}], "]"}], ",", "Orange"}], "}"}]}]}], "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"y", " ", "=", " ", RowBox[{"Y", "[", "t", "]"}]}], ";", " ", "\[IndentingNewLine]", RowBox[{"\[Phi]", " ", "=", " ", RowBox[{"Phi", "[", "t", "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{ "ejes", ",", "\[IndentingNewLine]", "orb", ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "1", "]"}], ",", " ", "Blue", ",", " ", RowBox[{"Specularity", "[", RowBox[{"White", ",", "20"}], "]"}], ",", " ", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "y", ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}]}], "}"}], ",", " ", "0.03"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Green", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Sphere", "[", " ", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}], ",", " ", "0.08"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Sphere", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "y", ",", " ", "0"}], "}"}], ",", " ", "0.08"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"AspectRatio", "\[Rule]", " ", "Automatic"}], ",", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"ran", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "0.5"}], ",", " ", "0.5"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "600"}], ",", "\[IndentingNewLine]", RowBox[{"ViewVector", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"8", RowBox[{"{", RowBox[{"2", ",", " ", RowBox[{"-", "0.9"}], ",", " ", "0.5"}], "}"}]}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"ViewAngle", "\[Rule]", " ", RowBox[{"20", "\[Degree]"}]}]}], "\[IndentingNewLine]", "]"}]}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", RowBox[{"T", "-", "\[CapitalDelta]"}], ",", " ", "\[CapitalDelta]", ",", " ", RowBox[{"Appearance", "\[Rule]", " ", "\"\\""}], ",", " ", RowBox[{"AnimationRate", "\[Rule]", " ", "2"}]}], "}"}], ",", " ", "\[IndentingNewLine]", RowBox[{"TrackedSymbols", "\[RuleDelayed]", " ", RowBox[{"{", "t", "}"}]}]}], "\[IndentingNewLine]", "]"}]}]}], "\[IndentingNewLine]", "]"}]], "Input", CellChangeTimes->{{3.7310223701512003`*^9, 3.7310223841015997`*^9}, { 3.731022427184*^9, 3.7310225225062*^9}, {3.7310974700290003`*^9, 3.7310974827212*^9}, {3.731097516943*^9, 3.7310975174578*^9}, { 3.7310975600528*^9, 3.7310976150987997`*^9}, {3.7310976647756*^9, 3.7310978284254*^9}, {3.7310978781702003`*^9, 3.7310980400408*^9}, { 3.731101868626*^9, 3.7311018966563997`*^9}, {3.7311019466940002`*^9, 3.7311019606664*^9}, {3.7311043348476*^9, 3.7311044298828*^9}, 3.731104557736*^9, 3.7311046376516*^9, {3.7311571460668*^9, 3.731157153755*^9}, 3.731157719769*^9, 3.7311634319723997`*^9, { 3.7311637128814*^9, 3.7311637347264*^9}, {3.7311638867344*^9, 3.731163891903*^9}, {3.7311691664952*^9, 3.7311693500922003`*^9}, 3.7311693855643997`*^9, {3.7312384105243998`*^9, 3.7312384136689997`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ FormBox[ SuperscriptBox["\[InvisiblePrefixScriptBase]", "*"], TraditionalForm]], CellDingbat->"\[NeutralSmiley]", FontSize->18], StyleBox["Al menos hasta la versi\[OAcute]n 10.0, que es la que estoy \ usando, los gr\[AAcute]ficos 3D del ", FontSize->18], StyleBox["Mathematica", FontSize->18, FontSlant->"Italic"], StyleBox[" no se caracterizan por un ", FontSize->18], StyleBox["rendering", FontSize->18, FontSlant->"Italic"], StyleBox[" muy delicado ni muy veloz. \n\nSi todo lo que interesa es la \ velocidad, \[EAcute]sta puede mejorarse si calculan antes todos los cuadros y \ los guardan como im\[AAcute]genes 2D. Eso es lo que hace la tabla de abajo. \ Las im\[AAcute]genes se guardan en la tabla llamada tgraf. Para un \ gr\[AAcute]fico con tan pocos elementos, la ganancia no es muy notable.", FontSize->18] }], "Subchapter", CellDingbat->"\[NeutralSmiley]", CellChangeTimes->{{3.7310982793352003`*^9, 3.7310984112855997`*^9}, { 3.7311019906099997`*^9, 3.7311020169184*^9}, {3.7311020687696*^9, 3.7311020977396*^9}, {3.731156888818*^9, 3.7311568910986*^9}, { 3.731156923365*^9, 3.731157003677*^9}, {3.7311570790514*^9, 3.7311571056456003`*^9}, {3.7311573157844*^9, 3.731157331505*^9}, { 3.7311575965822*^9, 3.7311576010956*^9}, {3.7311694766032*^9, 3.7311694886942*^9}, {3.731238506949*^9, 3.7312385297244*^9}, { 3.7312385645282*^9, 3.7312385703943996`*^9}, {3.7312386095290003`*^9, 3.7312386196436*^9}}, FontSize->36], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Puede", " ", "tardar", " ", "varios", " ", "segundos"}], "*)"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Module", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"y", ",", " ", "\[Phi]", ",", " ", "t", ",", "\[Tau]", ",", " ", RowBox[{"ran", " ", "=", " ", "2"}], ",", " ", RowBox[{"\[CapitalDelta]", " ", "=", " ", RowBox[{"T", "/", "100"}]}], ",", " ", "ejes", ",", " ", "orb", ",", " ", "imagentemp"}], "}"}], ",", " ", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"ejes", " ", "=", " ", RowBox[{"Graphics3D", "[", RowBox[{"{", " ", RowBox[{"White", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "ran"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "ran", ",", " ", "0"}], "}"}]}], "}"}], ",", " ", "0.012"}], "]"}], ",", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", " ", RowBox[{"-", "ran"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "ran"}], "}"}]}], "}"}], ",", " ", "0.012"}], "]"}]}], "}"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"orb", " ", "=", " ", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{ RowBox[{"Y", "[", "\[Tau]", "]"}], "+", RowBox[{"Sin", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Tau]", ",", " ", "0", ",", " ", "T"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", " ", RowBox[{ RowBox[{"AbsoluteDashing", "[", RowBox[{"{", RowBox[{"5", ",", " ", "8"}], "}"}], "]"}], ",", "Orange"}], "}"}]}]}], "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"tgraf", " ", "=", " ", "\[IndentingNewLine]", RowBox[{"Table", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"y", " ", "=", " ", RowBox[{"Y", "[", "t", "]"}]}], ";", " ", "\[IndentingNewLine]", RowBox[{"\[Phi]", " ", "=", " ", RowBox[{"Phi", "[", "t", "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Image", "[", RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{ "ejes", ",", "\[IndentingNewLine]", "orb", ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "1", "]"}], ",", " ", "Blue", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "20"}], "]"}], ",", " ", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "y", ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}]}], "}"}], ",", " ", "0.03"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Green", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Sphere", "[", " ", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}], ",", " ", "0.08"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Sphere", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "y", ",", " ", "0"}], "}"}], ",", " ", "0.08"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"AspectRatio", "\[Rule]", " ", "Automatic"}], ",", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"ran", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "0.5"}], ",", " ", "0.5"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "600"}], ",", "\[IndentingNewLine]", RowBox[{"ViewVector", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"8", RowBox[{"{", RowBox[{"2", ",", " ", RowBox[{"-", "0.9"}], ",", " ", "0.5"}], "}"}]}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"ViewAngle", "\[Rule]", " ", RowBox[{"20", "\[Degree]"}]}]}], "\[IndentingNewLine]", "]"}], "]"}]}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", RowBox[{"T", "-", "\[CapitalDelta]"}], ",", " ", "\[CapitalDelta]"}], "}"}]}], "\[IndentingNewLine]", "]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}]}]], "Input", CellChangeTimes->{{3.731157008755*^9, 3.7311570761448*^9}, { 3.731157110474*^9, 3.7311571377254*^9}, {3.7311573446924*^9, 3.7311573882866*^9}, {3.7311574443176003`*^9, 3.7311574862068*^9}, { 3.7311575227386*^9, 3.7311575277854*^9}, {3.7311576284406*^9, 3.7311576403*^9}, 3.7311577130726*^9, {3.7311635888262*^9, 3.731163609237*^9}, {3.7311636676014*^9, 3.731163695252*^9}, { 3.7311638759049997`*^9, 3.7311638797748003`*^9}, {3.7311694181802*^9, 3.731169422517*^9}, {3.7311695093982*^9, 3.731169518572*^9}, { 3.7312385533458*^9, 3.7312385541902*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Ahora puede hacerse la animaci\[OAcute]n con todo ya calculado y guardado \ previamente:\ \>", "Subchapter", CellChangeTimes->{{3.73115765366*^9, 3.7311576771438*^9}, { 3.7311695284206*^9, 3.731169529733*^9}}, FontSize->18], Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"tgraf", "[", RowBox[{"[", "i", "]"}], "]"}], ",", " ", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"i", ",", " ", "1", ",", " ", RowBox[{"Length", "[", "tgraf", "]"}], ",", " ", "1", ",", " ", RowBox[{"Appearance", "\[Rule]", " ", "\"\\""}]}], "}"}]}], "\[IndentingNewLine]", "]"}]], "Input", CellChangeTimes->{{3.731157679676*^9, 3.7311576986756*^9}}], Cell["\<\ El movimiento es m\[AAcute]s fluido pero los gr\[AAcute]ficos siguen siendo \ espantosos.\ \>", "Text", CellChangeTimes->{{3.731157751565*^9, 3.7311577792218*^9}, { 3.7311607378266*^9, 3.731160739293*^9}, {3.7312386459021997`*^9, 3.7312386488294*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Para obtener buenos gr\[AAcute]ficos 3D hay que hacer primero los gr\ \[AAcute]ficos en un tama\[NTilde]o exagerado y luego reducirlos. As\[IAcute] \ obtienen una resoluci\[OAcute]n aceptable.\n\nEso es lo que hace la funci\ \[OAcute]n ", StyleBox["antialias", FontWeight->"Bold"], " definida aqu\[IAcute] abajo. Hace un imagen 2D ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " veces m\[AAcute]s grande de lo que queremos, v\[IAcute]a el comando ", StyleBox["Rasterize", FontWeight->"Bold"], ", y luego la reduce. " }], "Subchapter", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.7310982793352003`*^9, 3.7310984112855997`*^9}, { 3.7311019906099997`*^9, 3.7311020169184*^9}, {3.7311020687696*^9, 3.7311020977396*^9}, {3.731156888818*^9, 3.7311568910986*^9}, { 3.731156923365*^9, 3.731157003677*^9}, {3.7311570511935997`*^9, 3.7311570521140003`*^9}, {3.7311577638008003`*^9, 3.731157885175*^9}, { 3.7311579268004*^9, 3.7311579494086*^9}, 3.7311579798297997`*^9}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"antialias", "[", RowBox[{"g_", ",", RowBox[{"n_:", " ", "1"}]}], "]"}], ":=", RowBox[{"ImageResize", "[", RowBox[{ RowBox[{"Rasterize", "[", RowBox[{"g", ",", "\"\\"", ",", RowBox[{"ImageResolution", "\[Rule]", RowBox[{"n", " ", "72"}]}], ",", " ", RowBox[{"Background", "\[Rule]", "None"}]}], "]"}], ",", RowBox[{"Scaled", "[", RowBox[{"1", "/", "n"}], "]"}]}], "]"}]}], ";"}], " ", RowBox[{"(*", RowBox[{"ejecutar", " ", "esta", " ", "celda"}], "*)"}]}]], "Input", CellChangeTimes->{{3.7311032746092*^9, 3.7311032799132*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Ejemplo: primero va el gr\[AAcute]fico en bruto y luego el antialiasiado. \ Comparen.\ \>", "Subchapter", CellChangeTimes->{{3.7310984416949997`*^9, 3.7310984455794*^9}, { 3.7310986974198*^9, 3.7310987098016*^9}, {3.7310987502146*^9, 3.7310987521022*^9}, {3.731102121543*^9, 3.7311021232352*^9}, { 3.7311579941288*^9, 3.7311580044398003`*^9}, {3.7312387764049997`*^9, 3.7312387797398*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"antialias", "[", RowBox[{"g_", ",", RowBox[{"n_:", " ", "1"}]}], "]"}], ":=", RowBox[{"ImageResize", "[", RowBox[{ RowBox[{"Rasterize", "[", RowBox[{"g", ",", "\"\\"", ",", RowBox[{"ImageResolution", "\[Rule]", RowBox[{"n", " ", "72"}]}], ",", " ", RowBox[{"Background", "\[Rule]", "None"}]}], "]"}], ",", RowBox[{"Scaled", "[", RowBox[{"1", "/", "n"}], "]"}]}], "]"}]}], ";"}], " ", RowBox[{"(*", RowBox[{ "repito", " ", "la", " ", "definici\[OAcute]n", " ", "por", " ", "las", " ", "dudas"}], "*)"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Module", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"y", ",", " ", "\[Phi]", ",", " ", RowBox[{"t", " ", "=", "2"}], ",", " ", RowBox[{"ran", " ", "=", " ", "2"}], ",", " ", "orb", ",", " ", "ejes"}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"y", " ", "=", " ", RowBox[{"Y", "[", "t", "]"}]}], ",", " ", RowBox[{"\[Phi]", " ", "=", " ", RowBox[{"Phi", "[", "t", "]"}]}]}], "}"}], ";", "\[IndentingNewLine]", RowBox[{"orb", " ", "=", " ", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{ RowBox[{"Y", "[", "\[Tau]", "]"}], "+", RowBox[{"Sin", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Tau]", ",", " ", "0", ",", " ", "T"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", " ", RowBox[{ RowBox[{"AbsoluteDashing", "[", RowBox[{"{", RowBox[{"5", ",", "8"}], "}"}], "]"}], ",", "Orange"}], "}"}]}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"ejes", " ", "=", " ", RowBox[{"Graphics3D", "[", RowBox[{"{", " ", RowBox[{"White", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "ran"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "ran", ",", " ", "0"}], "}"}]}], "}"}], ",", " ", "0.012"}], "]"}], ",", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", " ", RowBox[{"-", "ran"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "ran"}], "}"}]}], "}"}], ",", " ", "0.012"}], "]"}]}], "}"}], "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"bruto", " ", "=", " ", RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{ "orb", ",", "\[IndentingNewLine]", "ejes", ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "1", "]"}], ",", " ", "Blue", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "20"}], "]"}], ",", " ", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "y", ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}]}], "}"}], ",", " ", "0.03"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Green", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Sphere", "[", " ", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}], ",", " ", "0.08"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Sphere", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "y", ",", " ", "0"}], "}"}], ",", " ", "0.08"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"AspectRatio", "\[Rule]", " ", "Automatic"}], ",", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"ran", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "0.5"}], ",", " ", "0.5"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"BoxStyle", "\[Rule]", " ", RowBox[{"{", RowBox[{"White", ",", " ", RowBox[{"AbsoluteThickness", "[", "3", "]"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "600"}], ",", "\[IndentingNewLine]", RowBox[{"ViewVector", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"8", RowBox[{"{", RowBox[{"2", ",", " ", RowBox[{"-", "0.9"}], ",", " ", "0.5"}], "}"}]}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"ViewAngle", "\[Rule]", " ", RowBox[{"17", "\[Degree]"}]}], ",", " ", "\[IndentingNewLine]", RowBox[{"PlotRangePadding", "\[Rule]", " ", "None"}]}], "\[IndentingNewLine]", "]"}]}]}]}], "\[IndentingNewLine]", "]"}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"antibruto", " ", "=", " ", RowBox[{"antialias", "[", RowBox[{"bruto", ",", " ", "3"}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"ImageAssemble", "[", RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"Image", "[", "bruto", "]"}], ",", " ", "antibruto"}], "}"}], "}"}], "]"}]}], "Input", CellChangeTimes->{{3.7310984527251997`*^9, 3.7310985845042*^9}, { 3.731098615995*^9, 3.731098740682*^9}, {3.7311021323929996`*^9, 3.731102156118*^9}, {3.7311032445011997`*^9, 3.7311032723627996`*^9}, { 3.7311582778*^9, 3.7311582882368*^9}, {3.7311595401774*^9, 3.731159558703*^9}, {3.7311607458606*^9, 3.7311607611566*^9}, { 3.7311635430802*^9, 3.7311635707178*^9}, {3.731163625175*^9, 3.7311636504393997`*^9}, {3.7311637062464*^9, 3.7311637078552*^9}, { 3.7311638132662*^9, 3.7311638656382*^9}, {3.731169580472*^9, 3.7311696187637997`*^9}, {3.7311696857167997`*^9, 3.7311697133554*^9}, { 3.7311697871034*^9, 3.7311697908473997`*^9}, {3.7311698501826*^9, 3.7311698785146*^9}}], Cell[TextData[{ "Noten sobre todo c\[OAcute]mo cambian las l\[IAcute]neas rectas entre una \ figura y la otra. Noten tambi\[EAcute]n que rescalear y antialiazar el gr\ \[AAcute]fico no es un proceso r\[AAcute]pido.\n\nNo exageren con la ampliaci\ \[OAcute]n auxiliar. Aqu\[IAcute] un valor de n = 3 significa que en los \ pasos intermedios el ", StyleBox["Mathematica", FontSlant->"Italic"], " maneja im\[AAcute]genes de 1800 px, ya que el original tiene 600 px" }], "Text", CellChangeTimes->{{3.7311021734102*^9, 3.731102250598*^9}, { 3.7311581805966*^9, 3.731158183079*^9}, {3.7311607795666*^9, 3.7311608053624*^9}, {3.7311699095404*^9, 3.7311699170188*^9}, { 3.7311917340002003`*^9, 3.7311917548752003`*^9}, {3.7314179150998*^9, 3.731417917824*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "L\[IAcute]nea de producci\[OAcute]n para hacer un gif como la gente.\n\n\ Cada gr\[AAcute]fico es sometido al proceso de antialias.\n\nPresten atenci\ \[OAcute]n al uso del comando ", StyleBox["Monitor", FontWeight->"Bold"], ", que les permite ir viendo los resultados, y, en caso de que est\[EAcute] \ dando cualquier cosa, pueden interrumpir la evaluaci\[OAcute]n (\"Alt + .\") \ sin necesidad de esperar a que llegue a su t\[EAcute]rmino natural. \n\nPuede \ tardar varios minutos." }], "Subchapter", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.7310987987164*^9, 3.7310988097953997`*^9}, { 3.7310988573386*^9, 3.7310989645994*^9}, {3.7310994739962*^9, 3.7310994981470003`*^9}, {3.7311023940467997`*^9, 3.7311024126085997`*^9}, { 3.7311570665982*^9, 3.7311570678638*^9}, {3.7311576225982*^9, 3.7311576235341997`*^9}, {3.7311582020488*^9, 3.7311582025656*^9}, { 3.7311608231982*^9, 3.7311608348096*^9}, {3.7311917877132*^9, 3.7311918053012*^9}}, FontSize->24], Cell[TextData[StyleBox["ATENCI\[CapitalOAcute]N: A esta altura ya sabr\ \[AAcute]n si el movimiento que van a animar es de rotaci\[OAcute]n o de \ oscilaci\[OAcute]n. Para no calcular cosas de m\[AAcute]s, en el primer caso \ habr\[AAcute] que redefinir Tverdadero = T/2.", FontSize->18]], "Text", CellDingbat->"\[WatchIcon]", CellChangeTimes->{{3.7311639382852*^9, 3.7311640106806*^9}, { 3.731164075072*^9, 3.7311640765861998`*^9}}, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"antialias", "[", RowBox[{"g_", ",", RowBox[{"n_:", " ", "1"}]}], "]"}], ":=", RowBox[{"ImageResize", "[", RowBox[{ RowBox[{"Rasterize", "[", RowBox[{"g", ",", "\"\\"", ",", RowBox[{"ImageResolution", "\[Rule]", RowBox[{"n", " ", "72"}]}], ",", " ", RowBox[{"Background", "\[Rule]", "None"}]}], "]"}], ",", RowBox[{"Scaled", "[", RowBox[{"1", "/", "n"}], "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"$HistoryLength", " ", "=", " ", "0"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "Prevenci\[OAcute]n", " ", "v\[AAcute]lida", " ", "para", " ", "que", " ", "no", " ", "queden", " ", "resultados", " ", "innecesarios", " ", "almacenados", " ", "en", " ", "el", " ", "Historial"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Clear", "[", "AnimaTabla", "]"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "En", " ", "prevenci\[OAcute]n", " ", "de", " ", "que", " ", "se", " ", "haya", " ", "calculado", " ", "antes", " ", "y", " ", "ocupe", " ", "mucha", " ", RowBox[{"memoria", "."}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Module", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Tverdadero", " ", "=", " ", RowBox[{"T", "/", "2"}]}], ",", " ", RowBox[{"(*", RowBox[{ "dejar", " ", "\"\\"", " ", "si", " ", "el", " ", "movimiento", " ", "es", " ", "de", " ", "oscilaci\[OAcute]n"}], "*)"}], "\[IndentingNewLine]", " ", RowBox[{"Ncuadros", " ", "=", " ", "100"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "Disminuyan", " ", "el", " ", "n\[UAcute]mero", " ", "de", " ", "cuadros", " ", "si", " ", "tienen", " ", "poca", " ", RowBox[{"paciencia", "."}]}], " ", "*)"}], "\[IndentingNewLine]", " ", "y", ",", " ", "\[Phi]", ",", " ", RowBox[{"ran", " ", "=", " ", "2"}], ",", " ", "\[CapitalDelta]", ",", " ", RowBox[{"pasos", " ", "=", " ", "0"}], ",", " ", "orb", ",", "ejes", ",", " ", RowBox[{"bruto", "=", RowBox[{"Graphics", "[", "]"}]}], ",", " ", RowBox[{"rescaleado", "=", RowBox[{"Graphics", "[", "]"}]}]}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"\[CapitalDelta]", " ", "=", " ", RowBox[{"Tverdadero", "/", "Ncuadros"}]}], ";", " ", RowBox[{"(*", RowBox[{ "el", " ", "paso", " ", "temporal", " ", "entre", " ", "cuadros", " ", "sucesivos"}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"orb", " ", "=", " ", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{ RowBox[{"Y", "[", "\[Tau]", "]"}], "+", RowBox[{"Sin", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", RowBox[{"Phi", "[", "\[Tau]", "]"}], "]"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Tau]", ",", " ", "0", ",", " ", "Tverdadero"}], "}"}], ",", " ", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", " ", RowBox[{ RowBox[{"AbsoluteDashing", "[", RowBox[{"{", RowBox[{"5", ",", " ", "8"}], "}"}], "]"}], ",", "Orange"}], "}"}]}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"ejes", " ", "=", " ", RowBox[{"Graphics3D", "[", RowBox[{"{", " ", RowBox[{"White", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "ran"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "ran", ",", " ", "0"}], "}"}]}], "}"}], ",", " ", "0.012"}], "]"}], ",", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0", ",", " ", RowBox[{"-", "ran"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "ran"}], "}"}]}], "}"}], ",", " ", "0.012"}], "]"}]}], "}"}], "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Monitor", " ", "[", " ", RowBox[{"(*", RowBox[{ "este", " ", "comando", " ", "permite", " ", "ir", " ", "viendo", " ", "las", " ", "variables", " ", "que", " ", "uno", " ", RowBox[{"elija", " ", "[", RowBox[{ "que", " ", "se", " ", "indican", " ", "al", " ", "final", " ", "de", " ", "todo"}], "]"}], " ", "a", " ", "medida", " ", "que", " ", "se", " ", "ejecuta", " ", "la", " ", "celda"}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"AnimaTabla", " ", "=", " ", RowBox[{"(*", RowBox[{ "en", " ", "esta", " ", "tabla", " ", "voy", " ", "almacenando", " ", "los", " ", "cuadros", " ", "para", " ", "hacer", " ", "luego", " ", "la", " ", "animaci\[OAcute]n"}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", " ", RowBox[{"Table", "[", " ", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"y", " ", "=", " ", RowBox[{"Y", "[", "t", "]"}]}], ",", " ", RowBox[{"\[Phi]", " ", "=", " ", RowBox[{"Phi", "[", "t", "]"}]}]}], "}"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"bruto", " ", "=", " ", RowBox[{"Show", "[", "\[IndentingNewLine]", RowBox[{ "orb", ",", "\[IndentingNewLine]", "ejes", ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{ RowBox[{"AbsoluteThickness", "[", "1", "]"}], ",", " ", "Blue", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "20"}], "]"}], ",", " ", RowBox[{"Tube", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "y", ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}]}], "}"}], ",", " ", "0.02"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Green", ",", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Sphere", "[", " ", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"y", " ", "+", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}], ",", " ", "0.08"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"Graphics3D", "[", RowBox[{"{", RowBox[{"Red", ",", " ", RowBox[{"Specularity", "[", RowBox[{"White", ",", "100"}], "]"}], ",", RowBox[{"Sphere", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "y", ",", " ", "0"}], "}"}], ",", " ", "0.08"}], "]"}]}], "}"}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"AspectRatio", "\[Rule]", " ", "Automatic"}], ",", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"ran", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "0.5"}], ",", " ", "0.5"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}]}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", " ", "False"}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", " ", "Black"}], ",", "\[IndentingNewLine]", RowBox[{"BoxStyle", "\[Rule]", " ", RowBox[{"{", RowBox[{"White", ",", " ", RowBox[{"AbsoluteThickness", "[", "3", "]"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", " ", "600"}], ",", "\[IndentingNewLine]", RowBox[{"ViewVector", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"8", RowBox[{"{", RowBox[{"2", ",", " ", RowBox[{"-", "0.9"}], ",", " ", "0.5"}], "}"}]}], ",", " ", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"ViewAngle", "\[Rule]", " ", RowBox[{"17", "\[Degree]"}]}], ",", " ", "\[IndentingNewLine]", RowBox[{"PlotRangePadding", "\[Rule]", " ", "None"}]}], "\[IndentingNewLine]", "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"++", "pasos"}], ";", " ", RowBox[{"(*", RowBox[{ "voy", " ", "llevando", " ", "la", " ", "cuenta", " ", "de", " ", "los", " ", "cuadros", " ", "para", " ", "que", " ", "los", " ", "muestre", " ", "Monitor"}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"rescaleado", " ", "=", " ", RowBox[{"antialias", "[", RowBox[{"bruto", ",", " ", "3"}], "]"}]}]}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", RowBox[{"Tverdadero", "-", "\[CapitalDelta]"}], ",", "\[CapitalDelta]"}], "}"}]}], "\[IndentingNewLine]", "]"}]}], ";"}], " ", ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"Abajo", ":", " ", RowBox[{ "las", " ", "variables", " ", "que", " ", "muestra", " ", "el", " ", "comando", " ", RowBox[{"Monitor", ".", " ", "Incluyo"}], " ", "la", " ", "Memoria", " ", "en", " ", "Uso", " ", "para", " ", "tener", " ", "una", " ", "idea", " ", "de", " ", "si", " ", "va", " ", "a", " ", "alcanzar", " ", "la", " ", "memoria", " ", RowBox[{"RAM", ".", " ", "Caso"}], " ", "contrario"}]}], ",", " ", RowBox[{ "habr\[AAcute]", " ", "que", " ", "ir", " ", "guardando", " ", "las", " ", "figuras", " ", "en", " ", RowBox[{"archivos", "."}]}]}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"pasos", ",", " ", RowBox[{"Show", "[", RowBox[{"bruto", ",", " ", RowBox[{"ImageSize", "\[Rule]", " ", "400"}]}], "]"}], ",", " ", RowBox[{"Show", "[", RowBox[{"rescaleado", ",", " ", RowBox[{"ImageSize", "\[Rule]", " ", "400"}]}], "]"}], ",", " ", RowBox[{"MemoryInUse", "[", "]"}]}], "}"}]}], " ", "\[IndentingNewLine]", "]"}]}]}], "\[IndentingNewLine]", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"$HistoryLength", " ", "=", " ", "\[Infinity]"}], ";"}]}], "Input", CellChangeTimes->{{3.731098827521*^9, 3.7310988553417997`*^9}, { 3.7310989786902*^9, 3.7310992668464003`*^9}, {3.7310993137958*^9, 3.731099317572*^9}, {3.73109935475*^9, 3.7310994540866003`*^9}, { 3.7311022933532*^9, 3.731102345531*^9}, {3.7311033047952003`*^9, 3.7311033049824*^9}, {3.73110334626*^9, 3.73110337044*^9}, 3.7311034553196*^9, {3.7311044481036*^9, 3.7311044614728003`*^9}, { 3.7311582251738*^9, 3.7311582635518*^9}, {3.7311583023318*^9, 3.7311583919166*^9}, {3.7311584930629997`*^9, 3.7311584934091997`*^9}, { 3.7311585308602*^9, 3.731158613002*^9}, {3.7311586923774*^9, 3.7311587092530003`*^9}, {3.7311595785868*^9, 3.7311596076196003`*^9}, 3.7311596467223997`*^9, {3.731160865377*^9, 3.7311608756468*^9}, { 3.7311613852539997`*^9, 3.7311613881108*^9}, {3.731161612039*^9, 3.731161670799*^9}, {3.7311617059236*^9, 3.7311617360358*^9}, { 3.7311638468066*^9, 3.7311638703202*^9}, {3.7311640290193996`*^9, 3.7311642360722*^9}, {3.7311918597911997`*^9, 3.7311918779512*^9}, { 3.7312388489868*^9, 3.7312388491428003`*^9}, {3.7312388792886*^9, 3.731238921374*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Abajo: la animaci\[OAcute]n con los gr\[AAcute]ficos ", StyleBox["remasterizados.", FontSlant->"Italic"] }], "Subchapter", CellChangeTimes->{{3.731158465172*^9, 3.7311584869231997`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"AnimaTabla", "[", RowBox[{"[", "i", "]"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"i", ",", " ", "1", ",", " ", RowBox[{"Length", "[", "AnimaTabla", "]"}], ",", " ", "1", ",", " ", RowBox[{"Appearance", "\[Rule]", " ", "\"\\""}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.7311033781776*^9, 3.7311034032*^9}}], Cell["\[DownExclamation]Qu\[EAcute] diferencia, se\[NTilde]ores!", "Text", CellDingbat->"\[HappySmiley]", CellChangeTimes->{{3.7311920400662003`*^9, 3.7311920492412*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Exporto un gif con los cuadros de la animaci\[OAcute]n guardados en \ AnimaTabla. Puede tomar de 1 a 3 minutos.\ \>", "Subchapter", CellDingbat->"\[WatchIcon]", CellChangeTimes->{{3.7311024321198*^9, 3.7311024595692*^9}, { 3.7311619874676*^9, 3.731161991587*^9}, {3.7311919264312*^9, 3.7311919501672*^9}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{"SetDirectory", "[", RowBox[{"NotebookDirectory", "[", "]"}], "]"}], ";"}], "\n", RowBox[{"AbsoluteTiming", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Export", "[", RowBox[{ "\"\\"", ",", " ", "AnimaTabla", ",", " ", "\"\\"", ",", " ", RowBox[{"\"\\"", "\[Rule]", " ", RowBox[{"1", "/", "25"}]}]}], "]"}], ";"}], "\[IndentingNewLine]", "]"}]}], "Input", CellChangeTimes->{{3.7311024687022*^9, 3.7311025099578*^9}, { 3.7311038846784*^9, 3.73110389967*^9}, {3.7311618458964*^9, 3.7311618654611998`*^9}, {3.7311619989*^9, 3.7311620021926003`*^9}, { 3.73116284906*^9, 3.7311628523282003`*^9}, {3.7311628911736*^9, 3.7311629239546003`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Alternativamente, para poder editar los cuadros por fuera del ", StyleBox["Mathematica", FontSlant->"Italic"], ", los guardo en un directorio aparte. Luego puedo manipularlos en otro \ programa." }], "Subchapter", CellChangeTimes->{{3.7311047137332*^9, 3.7311047416008*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Presten", " ", "atenci\[OAcute]n", " ", "al", " ", "uso", " ", "de", " ", "las", " ", "dobles", " ", "barras", " ", "para", " ", "indicar", " ", "la", " ", RowBox[{"ruta", ".", " ", "Si"}], " ", "est\[AAcute]n", " ", "usando", " ", "Linux", " ", "deber\[AAcute]n", " ", "cambiarlas", " ", RowBox[{"por", " ", "/", " ", "o"}], " ", "quiz\[AAcute]"}], " ", "//"}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ "SetDirectory", "[", "\"\\"", "]"}], ";"}], " ", "\[IndentingNewLine]", "\n", RowBox[{"Monitor", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"For", "[", RowBox[{ RowBox[{"i", " ", "=", " ", "1"}], ",", " ", RowBox[{"i", " ", "\[LessEqual]", " ", RowBox[{"Length", "[", "AnimaTabla", "]"}]}], ",", " ", RowBox[{"++", "i"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "El", " ", "comando", " ", "For", " ", "tiene", " ", "una", " ", "sint\[AAcute]xis", " ", "transparente"}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"Export", "[", RowBox[{ RowBox[{"StringJoin", "[", RowBox[{"\"\\"", ",", " ", RowBox[{"If", "[", RowBox[{ RowBox[{"i", " ", "<", " ", "10"}], ",", " ", "\"\<000\>\"", ",", " ", RowBox[{"If", "[", RowBox[{ RowBox[{"i", " ", "<", " ", "100"}], ",", " ", "\"\<00\>\"", ",", " ", "\"\<0\>\""}], "]"}]}], "]"}], ",", " ", RowBox[{"ToString", "[", "i", "]"}], ",", " ", "\"\<.png\>\""}], "]"}], ",", " ", RowBox[{"AnimaTabla", "[", RowBox[{"[", "i", "]"}], "]"}], ",", " ", "\"\\""}], "]"}], ";"}]}], "\[IndentingNewLine]", "]"}], ";"}], ",", " ", "\[IndentingNewLine]", "i"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "\"\\"", " ", "es", " ", "la", " ", "variable", " ", "que", " ", "muestra", " ", "el", " ", "comando", " ", "Monitor"}], ";", " ", RowBox[{ "la", " ", "cosa", " ", "no", " ", "se", " ", "acaba", " ", "hasta", " ", "que", " ", "\"\\"", " ", "no", " ", "llega", " ", "al", " ", "n\[UAcute]mero", " ", "de", " ", "cuadros", " ", "guardados", " ", "en", " ", "AnimaTabla"}]}], "*)"}], "\[IndentingNewLine]", "]"}]}]}]], "Input", CellChangeTimes->{{3.7311039599639997`*^9, 3.7311040046268*^9}, 3.7311047128908*^9, {3.7311047498375998`*^9, 3.7311047555492*^9}, { 3.7311618673176003`*^9, 3.7311618840332003`*^9}, {3.7311620362106*^9, 3.7311620414230003`*^9}, {3.7311621160366*^9, 3.7311621205304003`*^9}}], Cell[TextData[{ "Dentro del ", StyleBox["Export", FontWeight->"Bold"], ", la funci\[OAcute]n del StringJoin es construir el nombre de cada archivo. \ El par de If agrega ceros antes del n\[UAcute]mero de cuadro para que luego \ se ordenen correctamente. El comando Monitor nos permite planificar en qu\ \[EAcute] usaremos los pr\[OAcute]ximos segundos, minutos u horas, en el peor \ de los casos. " }], "Text", CellChangeTimes->{{3.7311041039208*^9, 3.7311042686256*^9}, 3.7311619295620003`*^9}, FontSize->18] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Detr\[AAcute]s de la escena: el Lagrangiano y las ecuaciones de \ movimiento a trav\[EAcute]s del ", FontSize->24], StyleBox["Mathematica.", FontSize->24, FontSlant->"Italic"], " ", StyleBox["(expandir la celda)", FontSize->18] }], "Chapter", CellFrame->{{0, 0}, {0, 2}}, CellDingbat->"\[GrayCircle]", CellChangeTimes->{{3.7312389855204*^9, 3.731239015547*^9}, { 3.7312422120722*^9, 3.7312422167858*^9}}], Cell[CellGroupData[{ Cell["Problema 5:", "Subchapter", CellChangeTimes->{{3.7312390232774*^9, 3.7312390264114*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Quit", "[", "]"}], ";"}]], "Input", CellChangeTimes->{{3.7312391720797997`*^9, 3.731239187289*^9}, { 3.73123924446*^9, 3.7312392447408*^9}, {3.7312399033532*^9, 3.7312399086284*^9}, {3.7312409373746*^9, 3.7312409384695997`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "Posiciones", " ", "en", " ", "t\[EAcute]rminos", " ", "de", " ", "la", " ", "coordenada", " ", "generalizada", " ", "\[Phi]"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"r1", "[", "\[Phi]_", "]"}], ":=", RowBox[{"l", " ", RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], ",", " ", "0"}], "}"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"r2", "[", "\[Phi]_", "]"}], ":=", " ", RowBox[{"l", RowBox[{"{", RowBox[{"0", ",", " ", RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], "}"}]}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Energ\[IAcute]a", " ", "cin\[EAcute]tica", " ", "y", " ", "energ\[IAcute]a", " ", "potencial"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"T", "[", RowBox[{"\[Phi]_", ",", " ", "\[Phi]p_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "Phi", "}"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "variable", " ", "auxiliar", " ", "para", " ", "hacer", " ", "reemplazos"}], " ", "*)"}], "\[IndentingNewLine]", " ", RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "2"], "m1", " ", RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"r1", "[", RowBox[{"Phi", "[", "t", "]"}], "]"}], ",", " ", "t"}], "]"}], " ", ".", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"r1", "[", RowBox[{"Phi", "[", "t", "]"}], "]"}], ",", " ", "t"}], "]"}]}], ")"}]}], " ", "+", " ", RowBox[{ FractionBox["1", "2"], "m2", " ", RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"r2", "[", RowBox[{"Phi", "[", "t", "]"}], "]"}], ",", " ", "t"}], "]"}], " ", ".", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"r2", "[", RowBox[{"Phi", "[", "t", "]"}], "]"}], ",", " ", "t"}], "]"}]}], ")"}]}]}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Phi", "[", "t", "]"}], "\[Rule]", " ", "\[Phi]"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"Phi", "'"}], "[", "t", "]"}], "\[Rule]", " ", "\[Phi]p"}]}], "}"}]}]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"V", "[", "\[Phi]_", "]"}], ":=", " ", RowBox[{"m2", " ", "g", " ", "l", " ", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", "Lagrangiano", " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"L", "[", RowBox[{"\[Phi]_", ",", " ", "\[Phi]p_"}], "]"}], ":=", " ", RowBox[{ RowBox[{"T", "[", RowBox[{"\[Phi]", ",", " ", "\[Phi]p"}], "]"}], " ", "-", " ", RowBox[{"V", "[", "\[Phi]", "]"}]}]}], ";"}]}]}]], "Input", CellChangeTimes->{{3.731239029712*^9, 3.731239134333*^9}, { 3.7312392064318*^9, 3.7312392505192003`*^9}, {3.7312392863157997`*^9, 3.731239354568*^9}, {3.7312397538742*^9, 3.7312397946893997`*^9}, { 3.7312420611486*^9, 3.7312420674086*^9}}], Cell[BoxData[ RowBox[{"Collect", "[", RowBox[{ RowBox[{"L", "[", RowBox[{"\[Phi]", ",", " ", "\[Phi]p"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"\[Phi]p", ",", " ", "m1", ",", " ", "m2"}], "}"}], ",", " ", "Simplify"}], "]"}]], "Input", CellChangeTimes->{{3.7312420941142*^9, 3.7312421307827997`*^9}}], Cell[CellGroupData[{ Cell["\<\ Ecuaci\[OAcute]n de movimiento en forma directa. Simplificada paso a paso \ como hicimos en clase.\ \>", "Subsection", CellChangeTimes->{{3.7312405028696003`*^9, 3.731240534801*^9}}], Cell[BoxData[ RowBox[{"ec", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"L", "[", RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], ",", " ", RowBox[{"\[Phi]p", "[", "t", "]"}]}], "]"}], ",", " ", RowBox[{"\[Phi]p", "[", "t", "]"}]}], "]"}], ",", " ", "t"}], "]"}], " ", "-", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"L", "[", RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], ",", " ", RowBox[{"\[Phi]p", "[", "t", "]"}]}], "]"}], ",", " ", RowBox[{"\[Phi]", "[", "t", "]"}]}], "]"}]}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"\[Phi]p", "[", "t", "]"}], "\[Rule]", " ", RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]p", "'"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{ RowBox[{"\[Phi]", "''"}], "[", "t", "]"}]}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.731239299729*^9, 3.7312393065858*^9}, { 3.7312393604556*^9, 3.731239393809*^9}, {3.7312394518938*^9, 3.7312395375708*^9}, {3.7312398405634003`*^9, 3.7312398976870003`*^9}, 3.7312400498526*^9, {3.7312404996918*^9, 3.731240511617*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Para", " ", "hacerla", " ", "m\[AAcute]s", " ", "legible", " ", "abrevio", " ", RowBox[{"\[Phi]", "[", "t", "]"}], " ", "como", " ", "\[Phi]"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}], " ", "como", " ", RowBox[{"\[Phi]", "'"}], " ", "y", " ", RowBox[{ RowBox[{"\[Phi]", "''"}], "[", "t", "]"}], " ", "como", " ", RowBox[{"\[Phi]", "''"}]}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"ecsimple", " ", "=", " ", RowBox[{"ec", " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], "\[Rule]", " ", "\[Phi]"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{"\[Phi]", "'"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "''"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{"\[Phi]", "''"}]}]}], "}"}]}]}]}]], "Input", CellChangeTimes->{{3.7312400592436*^9, 3.7312401311400003`*^9}, { 3.7312402160274*^9, 3.7312402621434*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"Defino", " ", "la", " ", "frecuencia", " ", "g"}], " ", "=", " ", RowBox[{ SuperscriptBox["\[Omega]", RowBox[{"2", " "}]], "l"}]}], ",", " ", RowBox[{ "y", " ", "divido", " ", "la", " ", "ecuaci\[OAcute]n", " ", "por", " ", RowBox[{ SuperscriptBox["l", "2"], ".", " ", "Se"}], " ", "va", " ", RowBox[{"l", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"ecsimple2", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ FractionBox["1", SuperscriptBox["l", "2"]], RowBox[{"(", RowBox[{"ecsimple", "/.", " ", RowBox[{"g", "\[Rule]", " ", RowBox[{"l", " ", SuperscriptBox["\[Omega]", "2"]}]}]}], ")"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7312399234546003`*^9, 3.7312400458938*^9}, { 3.7312401400484*^9, 3.7312402092194*^9}, {3.7312402806858*^9, 3.7312402898096*^9}, {3.731240325351*^9, 3.731240332007*^9}, { 3.73124045345*^9, 3.7312404568598003`*^9}, {3.7312405580629997`*^9, 3.7312406036152*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{"Luego", " ", "defino", " ", "m1"}], " ", "=", " ", RowBox[{ "\[Mu]", " ", "m2", " ", "y", " ", "divido", " ", "la", " ", "ecuaci\[OAcute]n", " ", "por", " ", RowBox[{"m2", ".", " ", "Se"}], " ", "va", " ", "m2"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"ecsimple3", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ FractionBox["1", "m2"], RowBox[{"(", RowBox[{"ecsimple2", "/.", " ", RowBox[{"{", RowBox[{"m1", "\[Rule]", " ", RowBox[{"\[Mu]", " ", "m2"}]}], "}"}]}], ")"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7312399234546003`*^9, 3.7312400458938*^9}, { 3.7312401400484*^9, 3.7312402092194*^9}, {3.7312402806858*^9, 3.7312403393304*^9}, {3.7312404586688004`*^9, 3.7312404598262*^9}, { 3.7312405824558*^9, 3.7312406021896*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{"Por", " ", "\[UAcute]ltimo"}], ",", " ", RowBox[{ RowBox[{"usar", " ", "\[Tau]"}], " ", "=", " ", RowBox[{ RowBox[{ "\[Omega]", " ", "t", " ", "en", " ", "lugar", " ", "de", " ", "t", " ", "equivale", " ", "a", " ", "definir", " ", "\[Omega]"}], " ", "=", " ", RowBox[{"1.", " ", "Se", " ", "va", " ", RowBox[{"\[Omega]", "."}]}]}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"ecfinal", "=", RowBox[{"ecsimple3", " ", "/.", " ", RowBox[{"\[Omega]", "\[Rule]", " ", "1"}]}]}]}]], "Input", CellChangeTimes->{{3.7312403619175997`*^9, 3.7312404682812*^9}, { 3.7312405872142*^9, 3.7312406049598*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Ecuaci\[OAcute]n de movimiento obtenida usando el comando EulerEquations:\ \>", "Subsection", CellChangeTimes->{{3.7312406915875998`*^9, 3.7312407197966003`*^9}}], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.731240619472*^9, 3.7312406498062*^9}}], Cell[BoxData[ RowBox[{"ecEE", " ", "=", " ", RowBox[{ RowBox[{"EulerEquations", "[", RowBox[{ RowBox[{"L", "[", RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], ",", " ", RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}]}], "]"}], ",", " ", RowBox[{"\[Phi]", "[", "t", "]"}], ",", " ", "t"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.7312406527314*^9, 3.7312406847162*^9}, { 3.7312407357278*^9, 3.731240745227*^9}, {3.7312408298198*^9, 3.7312408305861998`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"La", " ", "simplificamos", " ", "como", " ", "antes"}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"ecEEfinal", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ FractionBox["1", RowBox[{ SuperscriptBox["l", "2"], "m2"}]], RowBox[{"(", RowBox[{"ecEE", " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], "\[Rule]", " ", "\[Phi]"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{"\[Phi]", "'"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "''"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{"\[Phi]", "''"}]}], ",", " ", RowBox[{"g", "\[Rule]", " ", RowBox[{"l", " ", SuperscriptBox["\[Omega]", "2"]}]}], ",", " ", RowBox[{"m1", "\[Rule]", " ", RowBox[{"\[Mu]", " ", "m2"}]}]}], "}"}]}], ")"}]}], "/.", " ", RowBox[{"\[Omega]", "\[Rule]", " ", "1"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7312407499084*^9, 3.7312408160584*^9}, { 3.7312408554736*^9, 3.731240880358*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ RowBox[{"Salvo", " ", "el", " ", "signo"}], ",", " ", RowBox[{ "es", " ", "el", " ", "mismo", " ", "resultado", " ", "de", " ", "antes"}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{"ecEEfinal", " ", "+", " ", "ecfinal"}], "]"}]}]], "Input", CellChangeTimes->{{3.731240881531*^9, 3.731240912611*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Problema 17:", "Subchapter", CellChangeTimes->{{3.7312390232774*^9, 3.7312390264114*^9}, { 3.7312409307528*^9, 3.7312409309712*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Quit", "[", "]"}], ";"}]], "Input", CellChangeTimes->{{3.7312391720797997`*^9, 3.731239187289*^9}, { 3.73123924446*^9, 3.7312392447408*^9}, {3.7312399033532*^9, 3.7312399086284*^9}, {3.7312409435794*^9, 3.7312409462872*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "Posiciones", " ", "en", " ", "t\[EAcute]rminos", " ", "de", " ", "las", " ", "coordenadas", " ", "generalizadas", " ", RowBox[{"{", RowBox[{"y", ",", " ", "\[Phi]"}], "}"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"r1", "[", RowBox[{"y_", ",", " ", "\[Phi]_"}], "]"}], ":=", " ", RowBox[{"{", RowBox[{"y", ",", " ", "0"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"r2", "[", RowBox[{"y_", ",", " ", "\[Phi]_"}], "]"}], ":=", " ", RowBox[{ RowBox[{"r1", "[", RowBox[{"y", ",", " ", "\[Phi]"}], "]"}], " ", "+", " ", RowBox[{"l", RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Cos", "[", "\[Phi]", "]"}]}], ",", " ", RowBox[{"Sin", "[", "\[Phi]", "]"}]}], "}"}]}]}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Energ\[IAcute]a", " ", "cin\[EAcute]tica", " ", "y", " ", "energ\[IAcute]a", " ", "potencial"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"T", "[", RowBox[{ "y_", ",", " ", "\[Phi]_", ",", " ", "yp_", ",", " ", "\[Phi]p_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"Phi", ",", " ", "Y"}], "}"}], ",", " ", RowBox[{"(*", " ", RowBox[{ "variables", " ", "auxiliares", " ", "para", " ", "hacer", " ", "reemplazos"}], " ", "*)"}], "\[IndentingNewLine]", " ", RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "2"], "m1", " ", RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"r1", "[", RowBox[{ RowBox[{"Y", "[", "t", "]"}], ",", " ", RowBox[{"Phi", "[", "t", "]"}]}], "]"}], ",", " ", "t"}], "]"}], " ", ".", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"r1", "[", RowBox[{ RowBox[{"Y", "[", "t", "]"}], ",", " ", RowBox[{"Phi", "[", "t", "]"}]}], "]"}], ",", " ", "t"}], "]"}]}], ")"}]}], " ", "+", " ", RowBox[{ FractionBox["1", "2"], "m2", " ", RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"r2", "[", RowBox[{ RowBox[{"Y", "[", "t", "]"}], ",", " ", RowBox[{"Phi", "[", "t", "]"}]}], "]"}], ",", " ", "t"}], "]"}], " ", ".", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"r2", "[", RowBox[{ RowBox[{"Y", "[", "t", "]"}], ",", " ", RowBox[{"Phi", "[", "t", "]"}]}], "]"}], ",", " ", "t"}], "]"}]}], ")"}]}]}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Y", "[", "t", "]"}], "\[Rule]", " ", "y"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"Y", "'"}], "[", "t", "]"}], "\[Rule]", " ", "yp"}], ",", " ", RowBox[{ RowBox[{"Phi", "[", "t", "]"}], "\[Rule]", " ", "\[Phi]"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"Phi", "'"}], "[", "t", "]"}], "\[Rule]", " ", "\[Phi]p"}]}], "}"}]}]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"V", "[", RowBox[{"y_", ",", " ", "\[Phi]_"}], "]"}], ":=", " ", RowBox[{"m2", " ", "g", " ", "l", " ", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", "Lagrangiano", " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"L", "[", RowBox[{ "y_", ",", " ", "\[Phi]_", ",", " ", "yp_", ",", " ", "\[Phi]p_"}], "]"}], ":=", " ", RowBox[{ RowBox[{"T", "[", RowBox[{"y", ",", " ", "\[Phi]", ",", " ", "yp", ",", " ", "\[Phi]p"}], "]"}], " ", "-", " ", RowBox[{"V", "[", RowBox[{"y", ",", " ", "\[Phi]"}], "]"}]}]}], ";"}]}]}]], "Input", CellChangeTimes->{{3.731239029712*^9, 3.731239134333*^9}, { 3.7312392064318*^9, 3.7312392505192003`*^9}, {3.7312392863157997`*^9, 3.731239354568*^9}, {3.7312397538742*^9, 3.7312397946893997`*^9}, { 3.7312409582602*^9, 3.731241166757*^9}, 3.7312412150698*^9, { 3.731242018222*^9, 3.7312420519955997`*^9}}], Cell[BoxData[ RowBox[{"Collect", "[", RowBox[{ RowBox[{"L", "[", RowBox[{"y", ",", " ", "\[Phi]", ",", " ", "yp", ",", " ", "\[Phi]p"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"m1", ",", " ", "m2"}], "}"}], ",", " ", "Simplify"}], "]"}]], "Input", CellChangeTimes->{{3.7312411495797997`*^9, 3.7312412036865997`*^9}, { 3.7312412452792*^9, 3.731241247126*^9}}], Cell[CellGroupData[{ Cell["\<\ Ecuaci\[OAcute]n de movimiento obtenida usando directamente EulerEquations:\ \>", "Subsection", CellChangeTimes->{{3.7312406915875998`*^9, 3.7312407197966003`*^9}, { 3.7312419257332*^9, 3.7312419307265997`*^9}}], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.731240619472*^9, 3.7312406498062*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"Ahora", " ", "son", " ", "dos", " ", "ecuaciones"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"ecEE", " ", "=", " ", RowBox[{ RowBox[{"EulerEquations", "[", RowBox[{ RowBox[{"L", "[", RowBox[{ RowBox[{"y", "[", "t", "]"}], ",", " ", RowBox[{"\[Phi]", "[", "t", "]"}], ",", RowBox[{ RowBox[{"y", "'"}], "[", "t", "]"}], ",", " ", RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"y", "[", "t", "]"}], ",", " ", RowBox[{"\[Phi]", "[", "t", "]"}]}], "}"}], ",", " ", "t"}], "]"}], "[", RowBox[{"[", RowBox[{"All", ",", " ", "1"}], "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7312406527314*^9, 3.7312406847162*^9}, { 3.7312407357278*^9, 3.731240745227*^9}, {3.7312408298198*^9, 3.7312408305861998`*^9}, {3.7312412992644*^9, 3.7312413373602*^9}, { 3.7312414983227997`*^9, 3.7312415115186*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{"Las", " ", "simplificamos", " ", "como", " ", "antes"}], ",", " ", RowBox[{ RowBox[{"definiendo", " ", "adicionalmente", " ", "l"}], " ", "=", " ", "1."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"ecEEfinal", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{ FractionBox["1", "m2"], RowBox[{"(", RowBox[{"ecEE", " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"y", "[", "t", "]"}], "\[Rule]", " ", "y"}], ",", " ", RowBox[{ RowBox[{"\[Phi]", "[", "t", "]"}], "\[Rule]", " ", "\[Phi]"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{"y", "'"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "'"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{"\[Phi]", "'"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"y", "''"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{"y", "''"}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"\[Phi]", "''"}], "[", "t", "]"}], "\[Rule]", " ", RowBox[{"\[Phi]", "''"}]}], ",", " ", RowBox[{"g", "\[Rule]", " ", RowBox[{"l", " ", SuperscriptBox["\[Omega]", "2"]}]}], ",", " ", RowBox[{"m1", "\[Rule]", " ", RowBox[{"\[Mu]", " ", "m2"}]}]}], "}"}]}], ")"}]}], "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"\[Omega]", "\[Rule]", " ", "1"}], ",", " ", RowBox[{"l", "\[Rule]", " ", "1"}]}], "}"}]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7312407499084*^9, 3.7312408160584*^9}, { 3.7312408554736*^9, 3.731240880358*^9}, {3.7312413590648003`*^9, 3.7312413949886*^9}, {3.7312414374849997`*^9, 3.7312415454038*^9}, { 3.7312421836029997`*^9, 3.7312421846376*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{"Despejando", " ", RowBox[{"y", "''"}], " ", "de", " ", "la", " ", "primera", " ", "ecuaci\[OAcute]n", " ", "y", " ", "reemplazando", " ", "en", " ", "la", " ", "segunda"}], ",", " ", RowBox[{ "se", " ", "obtiene", " ", "una", " ", "ecuaci\[OAcute]n", " ", "s\[OAcute]lo", " ", "para", " ", RowBox[{"\[Phi]", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"ypp", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"ecEEfinal", "[", RowBox[{"[", "1", "]"}], "]"}], " ", "\[Equal]", " ", "0"}], ",", " ", RowBox[{"y", "''"}]}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ec\[Phi]", " ", "=", RowBox[{"Collect", "[", RowBox[{ RowBox[{ RowBox[{"ecEEfinal", "[", RowBox[{"[", "2", "]"}], "]"}], " ", "/.", " ", RowBox[{ RowBox[{"y", "''"}], " ", "\[Rule]", " ", "ypp"}]}], ",", " ", RowBox[{"\[Phi]", "''"}], ",", " ", "FullSimplify"}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.7312415540576*^9, 3.731241671528*^9}, { 3.7312417969382*^9, 3.7312418077372*^9}, {3.7312419506144*^9, 3.7312419515222*^9}}], Cell["\<\ Identidades trigonom\[EAcute]tricas mediante, con \[Mu] = m1, es la misma \ ecuaci\[OAcute]n que usamos al resolver num\[EAcute]ricamente el problema.\ \>", "Text", CellChangeTimes->{{3.7312418524358*^9, 3.7312419091944*^9}}] }, Open ]] }, Open ]] }, Closed]] }, Open ]] }, WindowSize->{1440, 830}, WindowMargins->{{-8, Automatic}, {Automatic, -8}}, PrintingCopies->1, PrintingPageRange->{32000, 32000}, PrintingOptions->{"Magnification"->1., "PaperOrientation"->"Portrait", "PaperSize"->{595.2, 841.8000000000001}}, FrontEndVersion->"10.0 for Microsoft Windows (32-bit) (July 1, 2014)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[558, 20, 136, 3, 31, "Input"], Cell[697, 25, 180, 2, 64, "Chapter"], Cell[CellGroupData[{ Cell[902, 31, 1096, 32, 171, "Title"], Cell[2001, 65, 26263, 444, 800, "Text"], Cell[CellGroupData[{ Cell[28289, 513, 292, 6, 43, "Subsection"], Cell[CellGroupData[{ Cell[28606, 523, 342, 6, 57, "Subsubsection"], Cell[28951, 531, 594, 11, 52, "Input"], Cell[29548, 544, 931, 19, 52, "Input"], Cell[30482, 565, 632, 12, 54, "Input"], Cell[31117, 579, 1162, 25, 52, "Input"], Cell[32282, 606, 817, 14, 52, "Input"], Cell[33102, 622, 907, 17, 72, "Input"], Cell[34012, 641, 1232, 25, 72, "Input"], Cell[35247, 668, 423, 10, 123, "Input"], Cell[35673, 680, 1958, 41, 250, "Input"], Cell[37634, 723, 571, 11, 67, "Input"], Cell[38208, 736, 850, 20, 67, "Input"], Cell[39061, 758, 1407, 34, 90, "Input"], Cell[40471, 794, 1812, 44, 148, "Input"], Cell[42286, 840, 381, 8, 35, "Text"], Cell[42670, 850, 561, 13, 52, "Input"], Cell[43234, 865, 743, 17, 54, "Input"], Cell[43980, 884, 609, 16, 72, "Input"], Cell[44592, 902, 1670, 39, 155, "Input"], Cell[46265, 943, 766, 17, 54, "Input"], Cell[47034, 962, 685, 17, 31, "Input"], Cell[47722, 981, 1100, 30, 92, "Input"], Cell[48825, 1013, 1063, 24, 112, "Input"], Cell[49891, 1039, 864, 22, 115, "Input"], Cell[50758, 1063, 872, 29, 241, "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[51679, 1098, 32953, 563, 878, "Chapter"], Cell[CellGroupData[{ Cell[84657, 1665, 953, 28, 84, "Subchapter"], Cell[85613, 1695, 2592, 58, 415, "Output"] }, Open ]], Cell[88220, 1756, 5459, 163, 614, "Subchapter"], Cell[CellGroupData[{ Cell[93704, 1923, 811, 16, 189, "Subchapter"], Cell[94518, 1941, 505, 11, 83, "Text"], Cell[95026, 1954, 6419, 147, 560, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[101482, 2106, 12465, 242, 395, "Subchapter"], Cell[113950, 2350, 645, 18, 72, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[114632, 2373, 421, 8, 83, "Subchapter"], Cell[115056, 2383, 289, 9, 31, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[115382, 2397, 248, 4, 56, "Subchapter"], Cell[115633, 2403, 1884, 41, 292, "Input"], Cell[117520, 2446, 383, 7, 30, "Text"], Cell[117906, 2455, 252, 6, 31, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[118195, 2466, 624, 15, 140, "Subchapter"], Cell[118822, 2483, 7168, 172, 672, "Input"], Cell[125993, 2657, 509, 12, 30, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[126539, 2674, 555, 14, 85, "Subchapter"], Cell[127097, 2690, 4581, 99, 503, "Input"], Cell[131681, 2791, 500, 13, 30, "Text"], Cell[132184, 2806, 1072, 19, 131, "Text"] }, Open ]], Cell[133271, 2828, 2383, 46, 389, "Subchapter"], Cell[CellGroupData[{ Cell[135679, 2878, 822, 17, 149, "Subchapter"], Cell[136504, 2897, 3682, 88, 364, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[140223, 2990, 12517, 247, 255, "Subchapter"], Cell[152743, 3239, 589, 16, 52, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[153369, 3260, 180, 4, 56, "Subchapter"], Cell[153552, 3266, 167, 3, 31, "Input"], Cell[153722, 3271, 924, 23, 75, "Input"], Cell[154649, 3296, 237, 4, 30, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[154923, 3305, 571, 10, 83, "Subchapter"], Cell[155497, 3317, 6561, 162, 728, "Input"], Cell[162061, 3481, 288, 6, 30, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[162386, 3492, 496, 17, 115, "Subchapter"], Cell[162885, 3511, 4424, 103, 468, "Input"], Cell[CellGroupData[{ Cell[167334, 3618, 333, 8, 44, "Subsection"], Cell[167670, 3628, 660, 14, 31, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[168379, 3648, 355, 8, 55, "Subchapter"], Cell[168737, 3658, 511, 12, 52, "Input"], Cell[169251, 3672, 280, 5, 30, "Text"], Cell[169534, 3679, 139, 2, 35, "Text"], Cell[169676, 3683, 217, 5, 31, "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[169942, 3694, 3642, 80, 662, "Chapter"], Cell[CellGroupData[{ Cell[173609, 3778, 562, 13, 144, "Section"], Cell[174174, 3793, 515, 17, 36, "Text"], Cell[174692, 3812, 218, 4, 31, "Input"], Cell[174913, 3818, 946, 26, 107, "Text"], Cell[175862, 3846, 11098, 250, 689, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[186997, 4101, 131, 2, 56, "Subchapter"], Cell[187131, 4105, 8581, 191, 752, "Input"], Cell[195715, 4298, 239, 4, 30, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[195991, 4307, 471, 12, 56, "Subchapter"], Cell[196465, 4321, 7607, 174, 632, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[204109, 4500, 1509, 34, 118, "Subchapter"], Cell[205621, 4536, 7744, 173, 692, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[213402, 4714, 237, 6, 50, "Subchapter"], Cell[213642, 4722, 478, 10, 92, "Input"], Cell[214123, 4734, 267, 6, 30, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[214427, 4745, 1035, 22, 179, "Subchapter"], Cell[215465, 4769, 684, 18, 31, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[216186, 4792, 425, 9, 56, "Subchapter"], Cell[216614, 4803, 8370, 200, 632, "Input"], Cell[224987, 5005, 768, 14, 68, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[225792, 5024, 1008, 19, 261, "Subchapter"], Cell[226803, 5045, 470, 8, 59, "Text"], Cell[227276, 5055, 13803, 305, 1092, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[241116, 5365, 223, 6, 57, "Subchapter"], Cell[241342, 5373, 440, 10, 52, "Input"], Cell[241785, 5385, 172, 2, 30, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[241994, 5392, 337, 8, 55, "Subchapter"], Cell[242334, 5402, 772, 17, 92, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[243143, 5424, 312, 8, 84, "Subchapter"], Cell[243458, 5434, 2973, 65, 232, "Input"], Cell[246434, 5501, 520, 12, 59, "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[247003, 5519, 447, 14, 63, "Chapter"], Cell[CellGroupData[{ Cell[247475, 5537, 95, 1, 62, "Subchapter"], Cell[247573, 5540, 269, 5, 31, "Input"], Cell[247845, 5547, 3585, 99, 256, "Input"], Cell[251433, 5648, 327, 8, 31, "Input"], Cell[CellGroupData[{ Cell[251785, 5660, 193, 4, 43, "Subsection"], Cell[251981, 5666, 1377, 37, 31, "Input"], Cell[253361, 5705, 1188, 33, 72, "Input"], Cell[254552, 5740, 1121, 29, 67, "Input"], Cell[255676, 5771, 903, 22, 67, "Input"], Cell[256582, 5795, 744, 18, 52, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[257363, 5818, 173, 3, 43, "Subsection"], Cell[257539, 5823, 145, 2, 31, "Input"], Cell[257687, 5827, 555, 14, 31, "Input"], Cell[258245, 5843, 1315, 34, 88, "Input"], Cell[259563, 5879, 431, 10, 72, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[260043, 5895, 141, 2, 62, "Subchapter"], Cell[260187, 5899, 265, 5, 31, "Input"], Cell[260455, 5906, 4644, 126, 284, "Input"], Cell[265102, 6034, 389, 10, 31, "Input"], Cell[CellGroupData[{ Cell[265516, 6048, 224, 4, 43, "Subsection"], Cell[265743, 6054, 145, 2, 31, "Input"], Cell[265891, 6058, 1049, 27, 52, "Input"], Cell[266943, 6087, 2009, 51, 67, "Input"], Cell[268955, 6140, 1514, 41, 92, "Input"], Cell[270472, 6183, 236, 4, 30, "Text"] }, Open ]] }, Open ]] }, Closed]] }, Open ]] } ] *) (* End of internal cache information *)