{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Euler Bernoulli Beam \"solver\"\n",
    "\n",
    "The Euler-Bernoulli equation describes the relationship between the beam's deflection and the applied load\n",
    "\n",
    "$$\\frac{d^2}{dx^2}\\left(EI\\frac{d^2w}{dx^2}\\right) = q \\enspace .$$\n",
    "\n",
    "The curve $w(x)$ describes the delection of the beam at some point $x$, $q$ is a distributed load.\n",
    "\n",
    "This equation cannot be solve in this form in Sympy. Nevertheless, we can \"trick\" it to do it for us. Let us rewrite the equation as two equations\n",
    "\n",
    "$$\\begin{align}\n",
    "&-\\frac{d^2 M}{dx^2} = q \\enspace ,\\\\\n",
    "&- \\frac{d^2w}{dx^2} = \\frac{M}{EI} \\enspace ,\n",
    "\\end{align}$$\n",
    "\n",
    "where $M$ is the bending moment in the beam. We can, then, solve the two equation as if they have source terms and then couple the two solutions.\n",
    "\n",
    "The code below do this\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib notebook\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols('x')\n",
    "E, I = symbols('E I', positive=True)\n",
    "C1, C2, C3, C4 = symbols('C1 C2 C3 C4')\n",
    "w, M, q, f = symbols('w M q f', cls=Function)\n",
    "EI = symbols('EI', cls=Function, nonnegative=True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - q{\\left(x \\right)} - \\frac{d^{2}}{d x^{2}} M{\\left(x \\right)}$"
      ],
      "text/plain": [
       "          2      \n",
       "         d       \n",
       "-q(x) - ───(M(x))\n",
       "          2      \n",
       "        dx       "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M_eq = -diff(M(x), x, 2) - q(x)\n",
    "\n",
    "M_eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle C_{3} + \\int \\left(C_{4} - \\int q{\\left(x \\right)}\\, dx\\right)\\, dx$"
      ],
      "text/plain": [
       "     ⌠                    \n",
       "     ⎮ ⎛     ⌠        ⎞   \n",
       "C₃ + ⎮ ⎜C₄ - ⎮ q(x) dx⎟ dx\n",
       "     ⎮ ⎝     ⌡        ⎠   \n",
       "     ⌡                    "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M_sol = dsolve(M_eq, M(x)).rhs.subs([(C1, C3), (C2, C4)])\n",
    "\n",
    "M_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} w{\\left(x \\right)}$"
      ],
      "text/plain": [
       "         2      \n",
       "        d       \n",
       "f(x) + ───(w(x))\n",
       "         2      \n",
       "       dx       "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_eq = f(x) + diff(w(x),x,2)\n",
    "w_eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle C_{1} + \\int \\left(C_{2} - \\int \\frac{C_{3} + \\int \\left(C_{4} - \\int q{\\left(x \\right)}\\, dx\\right)\\, dx}{\\operatorname{EI}{\\left(x \\right)}}\\, dx\\right)\\, dx$"
      ],
      "text/plain": [
       "     ⌠                                          \n",
       "     ⎮ ⎛     ⌠                              ⎞   \n",
       "     ⎮ ⎜     ⎮      ⌠                       ⎟   \n",
       "     ⎮ ⎜     ⎮      ⎮ ⎛     ⌠        ⎞      ⎟   \n",
       "     ⎮ ⎜     ⎮ C₃ + ⎮ ⎜C₄ - ⎮ q(x) dx⎟ dx   ⎟   \n",
       "     ⎮ ⎜     ⎮      ⎮ ⎝     ⌡        ⎠      ⎟   \n",
       "     ⎮ ⎜     ⎮      ⌡                       ⎟   \n",
       "C₁ + ⎮ ⎜C₂ - ⎮ ────────────────────────── dx⎟ dx\n",
       "     ⎮ ⎜     ⎮           EI(x)              ⎟   \n",
       "     ⎮ ⎝     ⌡                              ⎠   \n",
       "     ⌡                                          "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_sol = dsolve(w_eq, w(x)).subs(f(x), M_sol/EI(x)).rhs\n",
    "\n",
    "w_sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We want to be sure that this solution is ok. We replaced known values for $E$, $I$ and $q$ to check it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cantilever beam with end load"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "sub_list = [(q(x), 0), (EI(x), E*I)]\n",
    "w_sol1 = w_sol.subs(sub_list).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "L, F = symbols('L F')\n",
    "# Fixed end\n",
    "bc_eq1 = w_sol1.subs(x, 0)\n",
    "bc_eq2 = diff(w_sol1, x).subs(x, 0)\n",
    "# Free end\n",
    "bc_eq3 = diff(w_sol1, x, 2).subs(x, L)\n",
    "bc_eq4 = diff(w_sol1, x, 3).subs(x, L) + F/(E*I)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ C_{1}, \\  C_{2}, \\  - \\frac{C_{3} + C_{4} L}{E I}, \\  - \\frac{C_{4}}{E I} + \\frac{F}{E I}\\right]$"
      ],
      "text/plain": [
       "⎡        -(C₃ + C₄⋅L)      C₄    F ⎤\n",
       "⎢C₁, C₂, ─────────────, - ─── + ───⎥\n",
       "⎣             E⋅I         E⋅I   E⋅I⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[bc_eq1, bc_eq2, bc_eq3, bc_eq4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ C_{1} : 0, \\  C_{2} : 0, \\  C_{3} : - F L, \\  C_{4} : F\\right\\}$"
      ],
      "text/plain": [
       "{C₁: 0, C₂: 0, C₃: -F⋅L, C₄: F}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "constants = solve([bc_eq1, bc_eq2, bc_eq3, bc_eq4], [C1, C2, C3, C4])\n",
    "constants"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{F x^{2} \\left(3 L - x\\right)}{6 E I}$"
      ],
      "text/plain": [
       "   2          \n",
       "F⋅x ⋅(3⋅L - x)\n",
       "──────────────\n",
       "    6⋅E⋅I     "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_sol1.subs(constants).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cantilever beam with uniformly distributed load"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "sub_list = [(q(x), 1), (EI(x), E*I)]\n",
    "w_sol1 = w_sol.subs(sub_list).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "L = symbols('L')\n",
    "# Fixed end\n",
    "bc_eq1 = w_sol1.subs(x, 0)\n",
    "bc_eq2 = diff(w_sol1, x).subs(x, 0)\n",
    "# Free end\n",
    "bc_eq3 = diff(w_sol1, x, 2).subs(x, L)\n",
    "bc_eq4 = diff(w_sol1, x, 3).subs(x, L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "constants = solve([bc_eq1, bc_eq2, bc_eq3, bc_eq4], [C1, C2, C3, C4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALoAAAAxCAYAAABtebPqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAJy0lEQVR4Ae2diZXUOBCGmXkTwNCbwZABRwQLGcBuBEAG8IiAx2bAEgFHBkAEHBlABrCTAft/bpWQ3LIsH93unla9p7aso1Qq/SqV5aNPfv36dW0MnZycvFC9c4ULhe8KT8XrUse9p0OWfe+VuyUBJ48ZQB8a1BdAfmH1FH+r8MXOhxxV76bC3SF1ppRVW7PJPkWOWrccdyVjpjJPcjodDHKYib4phEAHrGT4tFyjlqfy1Htv57s4qr1ZZN+FrLWN9WQoGTOVuavwtktnp8ocS7gsRuayhGmWlzu+UuaDXIEt5YVyjpV9S6JVth0ayI6ZAP5B9b7LxXmUqn+WSuxLE9MbrTJYZujz+tD/K4FeqtRr8TKg9VeaocQcss8gRmUxQAOlY6ZyT4WrbwpvNnClhFHuS1hPMuMOZH2kVnlmJ67OeZjejisff5rwxB1vtstwLmIWc53ACeGLO3+UKh+mqdwg2cO624hLHozGyy7eyhvd1y6eh5aeGzPlgZUN/Z3QySmk2QMQAezjUj59dZR/Ll4fFdjJYUm6pjRWgNs6v8V5ilSGznxXmfaKkyoOz8GyJxnNmCiZmHj04V6O7dC+5ngdUl7fmCkfI/pN+jsJ+3UangyNiynWZRDIXRvUe59pD+v8WcI2IHfl6MDPrjqShYsR6N36kP+dIHue8YRcyYQ1WvWxGNrXPn5j8iXDfSfvmOqj6qi9XrwJM2x1b/jqo4HulH1DjBtLzrmC+eqdHXFlsNghiH15xxfQYm09qZ17BJ+wGbG83ARqark2Bsu+2eR8KZKJicz1CgPVR8V97WN0KPkDxwxsRZsco4CuRgE0QPykODP7vuJPFUoG6bbKXQq0XReh8CG/hJeKemosuuolJ5CVmii7sdnG8bFk/7eQcVFfC3ntfbERY4b7B848nVlMzLCyLA3QHQUAh5UB1H8o/NBA/KMjhP9MeVwMT8rvAq8vowg+dg7EDOJX17m/Ff+hgM/NHmkOxMj5VaGPpsjex3tUvvqK3rkGKaXSvnIdMmRcS9ufXG6gXEPHDByc04bHpCIIjTJeEHfnKJ5Z0eykuPioO5/G047ihWuRvEmkdOSwXZNox0Tp/yncNz7hUelMDur5PoT5+xyXzPTZ71gpzo5RUj/0Q1TcV5Xd2riKN6u4l3uIjrcpl9MRhgBl+V06s+jPlPhcwQgLijW3pRTrXmItrX7uuFJmzqJTF0HfEAmI81eapR/Umcsgnegh+6zP1B/0W0pD+rrLcS2Vn3LblsvwAdYaarYXIxOvZJ1jVX5qAEyp69Iz/Io3KwVgTW5HKp+ZuLFFqHRb3rkojVwYJy+zN9pSCsVVmd/LWJhRGKe+itoSWlILZT+UTJ0GQjyxivTVl3F96dS9yy/qKzKLtw36qHEVD67FWEXahD66jBZ9ii4Gw8pzyBXya8fFHyMNzh5IjmYXrrHooTJcJSzqECvjqhUdULyfaYkaWHs/OIl8OtEm5I3AHxZQx5kk5Of4hlU24k5HnXv4GxV6EhhsFbljA9FTPMwu7usc4yoeSRxIfiYpzzbZdVsoYzY+h1zZBn5n+vE218VnqQMoEuoEzjp7a79Yt5QFsQYjt0fyWtnctuItKdfcMOOz9BG5LyR/dEGvNPR/6dI/hUCa0lfVXXpck/reklwYEchj5ZQzNcYWoWU2wJGCw+X0SZBPlSlE49ZWig/XCs0VcysTa8q2Y3sCmnvVTm+qS25uwuCK7RWpH+8UWFqjICGxQtwsI71tLQf1dcfjWqzfHci1csL8NKFO1SjAxqqYZTRlNmWUj6sAwBiAOeiTmER7nCFTtcMEw6/igqUhycDE+Evh4Tol+mUJZbfIT0zLVT38S0L7wtaK7OMxZwSK+7rAuBbpckdybWCWizcU+0oBAEIs8fi07Lw0llAgmm3Zdx3FzbiemzwOpDbozNDnBmblmcx0yJbk8NZ/lK56nRdGqr8X5PpLX8zg0B/01Ow26Rj1SedGUbr1NdDRVsZV/Ef56NuWC6WojZc6cP3gjfbkh7pM20OOEoQ9cZbmpLsxhFctu4wGxgJ9F9JKNgw0j4B7128poDPjcIeSV/S7UEZtY5oGBCZW1ZXG0F/wTeM4T20nF4Y08hiWAjpL9Ecp6fo83atcqgbWGhDQcbu51xK5q4sAHZEkEBfALC+hb01WpaqB0RoQruxGUbQ5sSTQWfqw6rPdhBmtnVrxSmjAWXMev95wiRcDOpqVYFy5s8wkHwe4EtqvndiJBoQldp94hc7vtIQNLwp0BHFgT90ICuWs8aqBrAaEIzY4Oj+itTjQs9LXzKqBmTRwOhOfyqZqYK81wJ3R5mn+vZayClc1MFED1XWZqMBa/TA0UF2XwxinKuVEDVSgT1RgrX4YGqhAP4xxqlJO1EAF+kQF1uqHoYGzwxBzGSnd3Ta7nXzhpOCmRPQcRVs61eOhNT5IlLzjq3zeeuIOnj17zuPKlwohwYM2v4pP8jEJx4dv8DQvZFBW4bMCn/3Yq6cKJdOyJIVM/pruVeShUeFZnOjD8jrnbSW2Y7P/0KF8Hizq/DaL6cvx6vxejvJ5Em8yH2vvmI/VdRGSOoi3rqJX9wQUrDuWlycvk+Ss7CqZGSSqnFnz10FyO8rbRVnLHPCpL7G0tRecV6AHymhFAWLqpWoAxcvbuBYRKQ1Xg4mQBaerZA8f9QGU1SFHxofX7ip1aKACvUMxSgasfI4CF6aUxnwoNPL31R7uUUNaQZg0fc/rNyuDyvZNGMf1OA9nx9nt/l4LOMkLQNU0Sx5ZbQEUf5on6EoJPhE4xYOL1MiCS46onQRz+ESTJVHm6JMq0AdAQEDEeuKe/OOsbVNb6Vj98wJQWvnGCutkpbrm75MGn6J/64CRk4dozs8n/+ipAn0YBLDYfHjIthyt9tgPhUbfZhRw2YHps+DWJkfzz6OVISxQ42sNVB+9EAkCISDn46jxS7frt6SGWtTGootX2+VgD9yT2jQ3yae1Il18WsXqaQV6AQYEOHxn7jdEN4CUjqtxJwHYPq4b/rmr4FcKx7sBcoZZF59MlePMqq5Lz7gLcNx19P93RHGl4adDAG22D4VqwrDLYvRMEb5DmSTJYJOgbismNRQnVqDH+ojOBCaAnHp5G/Djq7P1t7H9p3p8QKf5UKiOber1q127fFItBP5gPu0Kx3xeX7zoGH2BDauNtQyBjKuyUuBD/J27I6rLYwL48wZqna5JeWwfAuLknxYoH0vNTsyfKtP24ddM9NvHxxeskUYDFegdQBCQuCuKRU/Rxj9yUEh1uNlDHXMrmCRMlr4PhapIs7XI5CIkH+QSf/Jog6PJRhvI4/17nVdqaaACvaWQeno1NfA/+kcmy/Wtbg4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{2} \\left(6 L^{2} - 4 L x + x^{2}\\right)}{24 E I}$"
      ],
      "text/plain": [
       " 2 ⎛   2            2⎞\n",
       "x ⋅⎝6⋅L  - 4⋅L⋅x + x ⎠\n",
       "──────────────────────\n",
       "        24⋅E⋅I        "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_sol1.subs(constants).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cantilever beam with exponential loading"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "sub_list = [(q(x), exp(x)), (EI(x), E*I)]\n",
    "w_sol1 = w_sol.subs(sub_list).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "L = symbols('L')\n",
    "# Fixed end\n",
    "bc_eq1 = w_sol1.subs(x, 0)\n",
    "bc_eq2 = diff(w_sol1, x).subs(x, 0)\n",
    "# Free end\n",
    "bc_eq3 = diff(w_sol1, x, 2).subs(x, L)\n",
    "bc_eq4 = diff(w_sol1, x, 3).subs(x, L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "constants = solve([bc_eq1, bc_eq2, bc_eq3, bc_eq4], [C1, C2, C3, C4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{- \\frac{x^{3} e^{L}}{6} + \\frac{x^{2} \\left(L - 1\\right) e^{L}}{2} - x + e^{x} - 1}{E I}$"
      ],
      "text/plain": [
       "   3  L    2          L             \n",
       "  x ⋅ℯ    x ⋅(L - 1)⋅ℯ         x    \n",
       "- ───── + ───────────── - x + ℯ  - 1\n",
       "    6           2                   \n",
       "────────────────────────────────────\n",
       "                E⋅I                 "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_sol1.subs(constants).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Load written as a Taylor series and constant EI\n",
    "\n",
    "We can prove that the general function is written as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "k = symbols('k', integer=True)\n",
    "C = symbols('C1:4')\n",
    "D = symbols('D', cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle 6 C_{1} + 6 C_{2} x - \\frac{3 C_{3} x^{2} + C_{4} x^{3} - 6 \\sum_{k=0}^{\\infty} \\frac{x^{k + 4} D{\\left(k \\right)}}{\\left(k + 1\\right) \\left(k + 2\\right) \\left(k + 3\\right) \\left(k + 4\\right)}}{E I}$"
      ],
      "text/plain": [
       "                                      ∞                                  \n",
       "                                     ____                                \n",
       "                                     ╲                                   \n",
       "                                      ╲              k + 4               \n",
       "                      2       3        ╲            x     ⋅D(k)          \n",
       "                3⋅C₃⋅x  + C₄⋅x  - 6⋅   ╱  ───────────────────────────────\n",
       "                                      ╱   (k + 1)⋅(k + 2)⋅(k + 3)⋅(k + 4)\n",
       "                                     ╱                                   \n",
       "                                     ‾‾‾‾                                \n",
       "                                    k = 0                                \n",
       "6⋅C₁ + 6⋅C₂⋅x - ─────────────────────────────────────────────────────────\n",
       "                                           E⋅I                           "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_sol1 = 6*(C1 + C2*x) - 1/(E*I)*(3*C3*x**2 + C4*x**3 -\n",
    "                             6*Sum(D(k)*x**(k + 4)/((k + 1)*(k + 2)*(k + 3)*(k + 4)),(k, 0, oo)))\n",
    "\n",
    "w_sol1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Uniform load and varying cross-section"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "Q, alpha = symbols(\"Q alpha\")\n",
    "sub_list = [(q(x), Q), (EI(x), E*x**3/12/tan(alpha))]\n",
    "w_sol1 = w_sol.subs(sub_list).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - Q - \\frac{d^{2}}{d x^{2}} M{\\left(x \\right)}$"
      ],
      "text/plain": [
       "       2      \n",
       "      d       \n",
       "-Q - ───(M(x))\n",
       "       2      \n",
       "     dx       "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M_eq = -diff(M(x), x, 2) - Q\n",
    "\n",
    "M_eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJYAAAAvCAYAAAABxDNfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHWUlEQVR4Ae2ci3HUSBCGvS4CMFwEZ2dgiggwGZgjAiADKEfgMhnARcBBBkAEHGQAjgBDBub/ZqdVI61eM9Kykna6Sp6HZlrTrV/dPS2tV7e3tweZsgZiNbBara4050jHsY7vOl4KS79UOlplYJkqctlXAx5Ur4UdAHWg9jsVx2rfNx6HVsll1kCEBs4rYy/VPhXAsF6OMrBME7mM1UABIk00F1j03YnllsdnDcjlnVS0cOrb/1t/BpZpYmGld0vPJRYBtlkU6l8EjDcji0sgn4P3kZU6OXY+uCYOeiwQfQ0X6APtI/U/CvtT6/5a8APEBeVdYaGK+Vd0k7FIn7wkD3WzzVIVwvkx1+p4o/MvixMJFfF6pmn3q6CCVQZWgkKnOkU3+ovWRgD9dx2obN0aR3rgXGNW1hdbiseZ5jwycPr2jdrOQuZdYaxGJzpeN5Y4hyC6FOs0LNfyT8UurmFcbbeuxXW43mfVzzlUx/o5viqzxUIJcyfdWADyTcd3WYzqjm1DPI1/rU7cGDHYewaoDzdKH/RAB0CBLyD6S8cPjX2lkrE/VTC+RDpfWMC8KyypZrYNC5wdSHpIYZbKLBcguRAwXMwl4ACwDzrIrr9SG9ASrzlgqe+u6q2UgdWqntmctEz4254rJj46EEBsx3ihJtlzI6we4LO0BICzsTamtczBe6t65nFSFsW+JLgrsGBZGkljASHB+3uNfcxA9ZEuKOapzSaAQDw5JXEI40yz1wAu7VcIjhaJCLqhp+vCWa4CVL6PuApXmEwZWMmqm9TEXm5KluiFVo2La9w5agyggj6ui7S/GVhpepvaLHZ5R7g0FkYJiPzhgKI6JdaKxKgLwhkL6RwpAzdXzWr8xXl42Xk3p/OPLkIQl4+Z60A3Gmv0gnspemb3VHXAxC6PE1fWb6X6ABLnSJiqcC5Q1TUm1MbCFfysv6vMwTuqXAjJqmCViLf+9eUPlaQi6HsuMFh64VR15z69JWL8Zx0QO0GAyM6QIB6jY7tDmr2oFlj+YjBnVxAGdh+4iM6DYhbq8h69rrSDQUuRY4jqpAOCcGIqAxL39aPaDmRDeLfN3YixtBBM6rUOdhm8CyI76w713eg8/hwkkzSbLC1FjpEU7BKo0ol98bBVULk1CzQuvlKD4Ax0AxjyGrVxl84BLJwx3zjXjkntF08Ed3HCAB47lyN17duYJ33a/eKeceAGR71vdfxCi/VJFyWQ4zOI0P2pq0Qk17Bm20d96bK9G0uRo7fAHQMBFveKe7rxfVbH3OTTd5jp3RuBHxduAxXDb3T8R2VqtBQ5xtSr7iexVeeL6TGvCS/eRhOI4/76vhnH1dzbhsXyMQAutpRn0fU6SXO3LoeugewEv1DrFwDrIfv7F4tlb8YxmZ3kLVph1aRs8iRsa3kq7ul46seo+kdpkBzhSiUTu9/SezIPqt5fAIT89rFOjGVvxqNT+FI2YPyGhdHBjcVNEoPtgpLlCBcrmeBz5oEUnrpQ4zLoqPsCwL3UDcbsbRWLhZWBegXjKLxikUL/TRrCXIVjWvdHPLBybBSq5Nyszj+pnlAbV91244bK4V6F6Dq4uDq6rMjN+skHOeutsvVbKA9UNhbI2Ifgi/V3+aemCeLLTm+SRHyl9XdvQTWOp7lxu6pzWDDcSNJ21vNPSjdo7mA5xIMcHjeem9WYckE+PyZpran6mdM8XKFzgUI+wW8XkTCtfYL8E/mPGOwqGz9IDq2fXXFfq81YKDp8WE9b/l+AhVuCLPhdtyp/pXjG1YJG53jSMfO4i1rgVdhtozlUjidae6M7k4zjfwGwDS1MhSfmVYSLo7LxFtufw8W1ugbPZ2eucIgcyK3DvUlATh0lV6g28RR9o34BwJqXehQvofVEolCeelwigSPkdnkSvpfJFw/mEuuQaG18+mFcR1gF9SflsYxfrBx+PA+Uy535Nr9CKT7z9X2jfwFga15iya7QkRQLmFrdoR/qCq/sazX4xW3V/dkOLZzyR+qxcmhRWKMTyYO1DelKfe7/HHie1R1pdBI3ZD6VumTEGFiIQx0qvoZYNxP+SmnJ5liXwzoVL6NVJ9ZyvzlL4au5WM2CXwqPoXO4vo6SKxzKc6rzvb7fhetTH14L+c/C/th64QrFKJqEdnZH5JzIvPOjRod+LaLX7krjJ0WSh1gLy4QVw5W/lSzRLl3zZkGSl2T2xpsS9WMcMDidvx9sEnQQsJqY5v55aMADiJ95hUluEsUAjniXL12qYU4v4Q57jcqDlqoBPMuxgEQIMiplizWqOpfBTEBzsbOkKXbGsZJlixWrsYWPF6iIL4mV+bDA0k7RUmeLFa2yZU/w1uqrQFVNr0QJnoEVpa5lDxaoXC5PoOqdz2zSSHaFTZrZs36BihwkKYbBoIJPBhZa2HMSqEgtnISgUh+7RWKtJMrASlLbciYJPCS5+RyqaqkAWzLlGCtZdfOf6C0SvyUN3y6Q0+JdLx90lhKnMRJnYMVoa2FjBSw+Jcdi1VGvX23VTaTvN8stY5p0J8KKAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle C_{3} + C_{4} x - \\frac{Q x^{2}}{2}$"
      ],
      "text/plain": [
       "               2\n",
       "            Q⋅x \n",
       "C₃ + C₄⋅x - ────\n",
       "             2  "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M_sol = dsolve(M_eq, M(x)).rhs.subs([(C1, C3), (C2, C4)])\n",
    "\n",
    "M_sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} w{\\left(x \\right)}$"
      ],
      "text/plain": [
       "         2      \n",
       "        d       \n",
       "f(x) + ───(w(x))\n",
       "         2      \n",
       "       dx       "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_eq = f(x) + diff(w(x),x,2)\n",
    "w_eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "w_sol1 = dsolve(w_eq, w(x)).subs(f(x), M_sol/(E*x**3/tan(alpha)**3)).rhs\n",
    "\n",
    "w_sol1 = w_sol1.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle C_{1} + C_{2} x - \\frac{C_{3} \\tan^{3}{\\left(\\alpha \\right)}}{2 E x} + \\frac{C_{4} \\log{\\left(x \\right)} \\tan^{3}{\\left(\\alpha \\right)}}{E} + \\frac{Q x \\log{\\left(x \\right)} \\tan^{3}{\\left(\\alpha \\right)}}{2 E} - \\frac{Q x \\tan^{3}{\\left(\\alpha \\right)}}{2 E}$"
      ],
      "text/plain": [
       "                  3                   3                    3             3   \n",
       "            C₃⋅tan (α)   C₄⋅log(x)⋅tan (α)   Q⋅x⋅log(x)⋅tan (α)   Q⋅x⋅tan (α)\n",
       "C₁ + C₂⋅x - ────────── + ───────────────── + ────────────────── - ───────────\n",
       "              2⋅E⋅x              E                  2⋅E               2⋅E    "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(w_sol1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\infty \\operatorname{sign}{\\left(C_{3} \\tan^{3}{\\left(\\alpha \\right)} \\right)}$"
      ],
      "text/plain": [
       "       ⎛      3   ⎞\n",
       "-∞⋅sign⎝C₃⋅tan (α)⎠"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(w_sol1, x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "L = symbols('L')\n",
    "# Fixed end\n",
    "bc_eq1 = w_sol1.subs(x, L)\n",
    "bc_eq2 = diff(w_sol1, x).subs(x, L)\n",
    "# Finite solution\n",
    "bc_eq3 = C3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "constants = solve([bc_eq1, bc_eq2, bc_eq3], [C1, C2, C3, C4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{Q \\left(L - x \\left(\\log{\\left(L \\right)} + 1\\right) + x \\log{\\left(x \\right)}\\right) \\tan^{3}{\\left(\\alpha \\right)}}{2 E}$"
      ],
      "text/plain": [
       "                                     3   \n",
       "Q⋅(L - x⋅(log(L) + 1) + x⋅log(x))⋅tan (α)\n",
       "─────────────────────────────────────────\n",
       "                   2⋅E                   "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(w_sol1.subs(constants).subs(C4, 0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The shear stress would be"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{Q x^{2}}{2}$"
      ],
      "text/plain": [
       "    2 \n",
       "-Q⋅x  \n",
       "──────\n",
       "  2   "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = -E*x**3/tan(alpha)**3*diff(w_sol1.subs(constants).subs(C4, 0), x, 2)\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - Q x$"
      ],
      "text/plain": [
       "-Q⋅x"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(M, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "/* Put everything inside the global mpl namespace */\n",
       "window.mpl = {};\n",
       "\n",
       "\n",
       "mpl.get_websocket_type = function() {\n",
       "    if (typeof(WebSocket) !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support. ' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = (this.ws.binaryType != undefined);\n",
       "\n",
       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById(\"mpl-warnings\");\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = $('<div/>');\n",
       "    this._root_extra_style(this.root)\n",
       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen =  function () {\n",
       "            fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n",
       "            fig.send_message(\"send_image_mode\", {});\n",
       "            if (mpl.ratio != 1) {\n",
       "                fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n",
       "            }\n",
       "            fig.send_message(\"refresh\", {});\n",
       "        }\n",
       "\n",
       "    this.imageObj.onload = function() {\n",
       "            if (fig.image_mode == 'full') {\n",
       "                // Full images could contain transparency (where diff images\n",
       "                // almost always do), so we need to clear the canvas so that\n",
       "                // there is no ghosting.\n",
       "                fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n",
       "            }\n",
       "            fig.context.drawImage(fig.imageObj, 0, 0);\n",
       "        };\n",
       "\n",
       "    this.imageObj.onunload = function() {\n",
       "        fig.ws.close();\n",
       "    }\n",
       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
       "    var titletext = $(\n",
       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
       "        'text-align: center; padding: 3px;\"/>');\n",
       "    titlebar.append(titletext)\n",
       "    this.root.append(titlebar);\n",
       "    this.header = titletext[0];\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "    });\n",
       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
       "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
       "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
       "\n",
       "    canvas_div.on(\"wheel\", function (event) {\n",
       "        event = event.originalEvent;\n",
       "        event['data'] = 'scroll'\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        mouse_event_fn(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.append(canvas);\n",
       "    canvas_div.append(rubberband);\n",
       "\n",
       "    this.rubberband = rubberband;\n",
       "    this.rubberband_canvas = rubberband[0];\n",
       "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
       "        // Keep the size of the canvas, canvas container, and rubber band\n",
       "        // canvas in synch.\n",
       "        canvas_div.css('width', width)\n",
       "        canvas_div.css('height', height)\n",
       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width, fig.canvas.height);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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\" width=\"639.8333333333334\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w_plot = w_sol1.subs(constants).subs({C4: 0, L: 1, Q: -1, E: 1, alpha: pi/9})\n",
    "plot(w_plot, (x, 1e-6, 1));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
       "<style>\n",
       "/* Based on Lorena Barba template available at: https://github.com/barbagroup/AeroPython/blob/master/styles/custom.css*/\n",
       "@font-face {\n",
       "font-family: \"Computer Modern\";\n",
       "src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "}\n",
       "div.cell{\n",
       "width:800px;\n",
       "margin-left:16% !important;\n",
       "margin-right:auto;\n",
       "}\n",
       "h1 {\n",
       "font-family: 'Alegreya Sans', sans-serif;\n",
       "}\n",
       "h2 {\n",
       "font-family: 'Fenix', serif;\n",
       "}\n",
       "h3{\n",
       "font-family: 'Fenix', serif;\n",
       "margin-top:12px;\n",
       "margin-bottom: 3px;\n",
       "}\n",
       "h4{\n",
       "font-family: 'Fenix', serif;\n",
       "}\n",
       "h5 {\n",
       "font-family: 'Alegreya Sans', sans-serif;\n",
       "}\t\n",
       "div.text_cell_render{\n",
       "font-family: 'Alegreya Sans',Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
       "line-height: 135%;\n",
       "font-size: 120%;\n",
       "width:600px;\n",
       "margin-left:auto;\n",
       "margin-right:auto;\n",
       "}\n",
       ".CodeMirror{\n",
       "font-family: \"Source Code Pro\";\n",
       "font-size: 90%;\n",
       "}\n",
       "/* .prompt{\n",
       "display: None;\n",
       "}*/\n",
       ".text_cell_render h1 {\n",
       "font-weight: 200;\n",
       "font-size: 50pt;\n",
       "line-height: 100%;\n",
       "color:#CD2305;\n",
       "margin-bottom: 0.5em;\n",
       "margin-top: 0.5em;\n",
       "display: block;\n",
       "}\t\n",
       ".text_cell_render h5 {\n",
       "font-weight: 300;\n",
       "font-size: 16pt;\n",
       "color: #CD2305;\n",
       "font-style: italic;\n",
       "margin-bottom: .5em;\n",
       "margin-top: 0.5em;\n",
       "display: block;\n",
       "}\n",
       ".warning{\n",
       "color: rgb( 240, 20, 20 )\n",
       "}\n",
       "</style>\n",
       "<script>\n",
       "MathJax.Hub.Config({\n",
       "TeX: {\n",
       "extensions: [\"AMSmath.js\"]\n",
       "},\n",
       "tex2jax: {\n",
       "inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "},\n",
       "displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "\"HTML-CSS\": {\n",
       "styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "}\n",
       "});\n",
       "</script>\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = open('./styles/custom_barba.css', 'r').read()\n",
    "    return HTML(styles)\n",
    "css_styling()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.9"
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}