{
 "metadata": {
  "name": "",
  "signature": "sha256:daa6eac8936225c392c04e7bc0306c213e669afac73b2ee5b23df1a02748c721"
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "init_printing()\n",
      "x, y, z = symbols('x,y,z')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Solve\n",
      "\n",
      "Equation solving is both a common need also a common building block for more complicated symbolic algorithms.  \n",
      "\n",
      "Here we introduce the `solve` function"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(x**2 - 4, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}-2, & 2\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "[-2, 2]"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solve takes two arguments, an equation like $x^2 - 4$ and a variable on which we want to solve, like $x$.  \n",
      "\n",
      "Solve returns the values of the variable, $x$, for which the equation, $x^2 - 4$ equals 0."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise\n",
      "\n",
      "What would the following code produce?  Are you sure?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(x**2 - 9 == 0, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Symbolic use of `solve`\n",
      "\n",
      "Results of `solve` don't need to be numeric, like `[-2, 2]`.  We can use solve to perform algebraic manipulations.  For example if we know a simple equation for the area of a square\n",
      "\n",
      "    area = height * width\n",
      "    \n",
      "we can solve this equation for any of the variables.  For example how would we solve this system for the `height`, given the `area` and `width`?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "height, width, area = symbols('height,width,area')\n",
      "solve(area - height*width, height)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\frac{area}{width}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "\u23a1 area\u23a4\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a3width\u23a6"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that we would have liked to have written\n",
      "\n",
      "    solve(area == height * width, height)\n",
      "    \n",
      "But the `==` gotcha bites us.  Instead we remember that `solve` expects an expression that is equal to zero, so we rewrite the equation\n",
      "\n",
      "    area = height * width\n",
      "    \n",
      "into the equation\n",
      "\n",
      "    0 = height * width - area\n",
      "    \n",
      "and that is what we give to solve."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise\n",
      "\n",
      "Compute the radius of a sphere, given the volume.  Reminder, the volume of a sphere of radius `r` is given by\n",
      "\n",
      "$$ V = \\frac{4}{3}\\pi r^3 $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Solve for the radius of a sphere, given the volume\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "You will probably get several solutions, this is fine.  The first one is probably the one that you want."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Substitution\n",
      "\n",
      "We often want to substitute in one expression for another.  For this we use the subs method"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAXCAYAAAALHW+jAAAABHNCSVQICAgIfAhkiAAAASRJREFU\nOI3t0z8oRWEcxvHPvRbEpZRFhDtQom4GcQcGJWVTsrBQMtsZpCRGMiqjshkNZJHRpmx3Mypl8Gd4\nz63jb87bHT116jznPed7fr/397zUWLnI70YwhhaUsYnL2CKasJ3yc3hCRyxwCK8oJr6AtwQcpZzQ\ncnW7BhJgKRb4WcfYqxVsCTvih/tBMwkQ6tENdZGwcXThTJj6hFBlJQbWi0dhEOmrwNfeh7GIF6GF\nZaygVcjZBu6z/Hkf+cQf4U44EWUhe2tZWjlAc8qf4Dq578Qu2rIAez75CrayAH5Tn7DRk7UCruIZ\njalnxR/e/VYNQuIHE3+Km9R6HodZgLNCi/Poxy2uUuvrGP0LqHpSHtAu5K2EBSEu05jCOS6yVPiv\neL0DViMs6ugPO8EAAAAASUVORK5CYII=\n",
       "prompt_number": 14,
       "text": [
        " 2\n",
        "x "
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Replace x with y\n",
      "(x**2).subs({x: y})"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$y^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAbCAYAAACeA7ShAAAABHNCSVQICAgIfAhkiAAAAVtJREFU\nSInt1LFLHFEQx/GPZ/ACygmpJEI8tFFEIYJIVIxNiIXNIVhoaZki5F/Q5gikOkhlpYX+A3aCaNLY\nSbqkEIQUgatEEBTUFO8drKshu8ulyw+WZXbe+87svJlHG9WRc/0UptGLGWzgqEjgHtQT9jIu0V8E\nNo5bDEW7grsIza0O4TdbpRmNsJdFYGlt41M7QGv4KP8BPtBihMFTVFuOzpyg13iBPeF052N2P/Nm\nNIgLoejJp/LY4ioaMepKyvcO+3kif0YX3uNbyneM3b8BSvE9i6+4xgJ+JNZ0YwKHWbPqQ1kYjRvU\nEr43Qm1Gs8Ja+oDzCG5pHU0ZeqqUst/iAFeJb3P4ErPLBRvA94RdFuYx0zWThp3hWcKuC12eqfjp\nOgxjEyfCXfUKYzHAbRbgn1TCL2wV2bzjfrPWhL4bKQJrCuMEz3GK1TyAZM2WMIknQhM3hDH6r3+o\n38LwOEuimErTAAAAAElFTkSuQmCC\n",
       "prompt_number": 13,
       "text": [
        " 2\n",
        "y "
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise\n",
      "\n",
      "Subsitute $x$ for $sin(x)$ in the equation $x^2 + 2\\cdot x + 1$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Replace x with sin(x)\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Subs + Solve\n",
      "\n",
      "We can use subs and solve together to plug the solution of one equation into another"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Solve for the height of a rectangle given area and width\n",
      "\n",
      "soln = solve(area - height*width, height)[0]\n",
      "soln"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{area}{width}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGYAAAAZCAYAAADDq1t2AAAABHNCSVQICAgIfAhkiAAABKtJREFU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\nAAEAAAAASUVORK5CYII=\n",
       "prompt_number": 25,
       "text": [
        " area\n",
        "\u2500\u2500\u2500\u2500\u2500\n",
        "width"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Define perimeter of rectangle in terms of height and width\n",
      "\n",
      "perimeter = 2*(height + width)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Substitute the solution for height into the expression for perimeter\n",
      "\n",
      "perimeter.subs({height: soln})"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 area}{width} + 2 width$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMAAAAAZCAYAAABn7SHgAAAABHNCSVQICAgIfAhkiAAABhBJREFU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\nHR0dHR0dU/kfLywOMJQIn3wAAAAASUVORK5CYII=\n",
       "prompt_number": 27,
       "text": [
        "2\u22c5area          \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500 + 2\u22c5width\n",
        "width           "
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise\n",
      "\n",
      "In the last section you solved for the radius of a sphere given its volume"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "V, r = symbols('V,r', real=True)\n",
      "4*pi/3 * r**3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{4 \\pi}{3} r^{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAcCAYAAAA5pQx5AAAABHNCSVQICAgIfAhkiAAAAytJREFU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\noBhjjDFGMX8BujKdmTfmfkUAAAAASUVORK5CYII=\n",
       "prompt_number": 34,
       "text": [
        "     3\n",
        "4\u22c5\u03c0\u22c5r \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "  3   "
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(V - 4*pi/3 * r**3, r)[0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\sqrt[3]{6} \\sqrt[3]{V}}{2 \\sqrt[3]{\\pi}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAgCAYAAADzCU3nAAAABHNCSVQICAgIfAhkiAAABONJREFU\naIHt2XmMnVMYx/HPdFrdGM1Eqb0YYi9FI5Zah1qTWmNJk1ZFbSFSO6FaSxsECUktIZYSe1GiJCjV\nRoJGhVgSuwiiEqUI5Y/nbea979zlvXfunda43+Qm55z3Oe/vfd9zzvM851yaNGmyelgHp+FUzEZr\nzn7/1OGXV3P3Oumt+vUJjsKNSflt7JKz3/MY1EuaPdHqs/QTq24glmBojj7jcHEvafZUq0+zCW5A\nZ077Z9DWS5r10OrTtOBlDK9gtzdm9JJmPbUaQa35QV1oSZXnYHwF+8dUHtx6adZDqxqOqdK+1vyg\nZvqlyjNxZlLeDF+W6TcK3+KHHurn0ayXVl62xYQq+8zDNBGrW/FJCbtpGpAR74iTcQYuqPCg94sP\nXYptxAq6B3fgJuFOsuTRLKc1G9+IF/kLi3FcEbv5WJnYLcGeZZ79GpyYaRuDJ/EClor32jhjUylW\n74SJZXQbTgfuLnN9e7Fy9krqI/AZrm6AFuwgBmROBbt3cUgOzXcwOFUfLQZ+WFJfGwvwPUZm+paL\n1bMwIId+w5gt3Ekx+uMjnJ9q21S4ufPqrLWKwWLgXi1jMx7n5tAbgwcybfPEBEqza6L5SFKvFKu3\nE16loRRzaavYWPmZPQl/6pqdjdRK8x2+KHGtDU8ojOeluBWHZdqWCw+yfqZ9GX5MyrN0xeo3sFvG\ndqaIfw1hCB7C7WVsbiryUGlewge9pJVmkYhxxVzRLSLBqUQr3hNeI81S/IEtMu3f4dekXC5Wd+Cc\nHPo10R9XYDp+UXyj2y4CdClasAKvYazYd92Cp4VrqadWljnCdW2Zad8T1+e8R6fiE2moiNNpNkr0\nXslx3+sVxsw0bcLdrlA6e1yJfSuJtOM3xePRNBxQpu96idCHmJJq318M0A511MpyXaJ9UKptAOaK\n1Z2He8VGPw834G9dCVg7ti5iN1JhvE/TIib1VThUxPPJOBjP4fCkvI98bt5dIsFIB9y1RZAuxwbi\n4/2u+wz7OnmYemllOT3Rnpxqu1R8kDwMEtuElkqGwvUt13WSs7MY9IeL2F4r3qcYExTG08d0nbg8\nm+M5ujFKfIRxqbapOLpCvwFJv/eLXFss4kQ2QNeqlaUzuc+1Sb1D7LXycqx8LnUg3sLNqbZWkYwt\nU7i32wQX5tQfhqeS8hDxvWpiga5ZPxAvyjcbv8frRdpfEx92wzpqpelI7v9QUn9UeIC8PC4SjHK0\niFh6TYnrt+uaOEQMz3sofjYuSsqjsTBnv24cL4Jih8iUTsnZ7xkR47IsFi40m7H1RCvNWiLmvClc\n0OTy5gWsK1ZRJWbgykxb+mhsGzFxB4lk5pIqnmEJ9kjKR4jwURP9RVy6TaT4eU+7TxIJRzrGteBn\nsQrqqZXlK/wk9mzVrNhJKn/kiYqvtDsz9Xli0lwtEpY8jBVudtV7nySyzJr/ML5crIQplQxT9BN7\nqqmpthPEycnIOmtlWSDiaKWTlizzsXmZ6weKjfaDmd8juicknfhYbHfyMlfhpN5PuP0jq7hHAcPx\nqepHvh33iSxpjni5rRqkleZOXFZlnxEi/pZjmdJ7rOlF7Bep7i+ozxWeobaJU6BqE7T/FefjrNX9\nEE2qZ6E4OPjPkWtH3kfpEEnTj5UM10T+zwN3suKnHU3WcJ5V+jiqSZMmTZo0abKa+BdhmUdtJ3/Y\n5QAAAABJRU5ErkJggg==\n",
       "prompt_number": 36,
       "text": [
        "3 ___ 3 ___\n",
        "\u2572\u2571 6 \u22c5\u2572\u2571 V \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "    3 ___  \n",
        "  2\u22c5\u2572\u2571 \u03c0   "
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now lets compute the surface area of a sphere in terms of the volume.  Recall that the surface area of a sphere is given by\n",
      "\n",
      "$$ 4 \\pi r^2 $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(?).subs(?)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "SyntaxError",
       "evalue": "invalid syntax (<ipython-input-49-df38c236f23a>, line 1)",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[1;36m  File \u001b[1;32m\"<ipython-input-49-df38c236f23a>\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m    (?).subs(?)\u001b[0m\n\u001b[1;37m     ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Does the expression look right?  How would you expect the surface area to scale with respect to the volume?  What is the exponent on $V$?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Plotting\n",
      "\n",
      "SymPy can plot expressions easily using the `plot` function.  By default this links against matplotlib."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(x**2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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uKmwXWrcO7rsPxo835R0qDh06xA033MATTzxB586dufjii88o6LCwMHJycs54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\nQoUtIuISKmwREZf4f3hrutPlcLgYAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f58e010c190>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "<sympy.plotting.plot.Plot at 0x7f58e1de6710>"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise\n",
      "\n",
      "In the last exercise you derived a relationship between the volume of a sphere and the surface area.  Plot this relationship using `plot`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(?)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Low dependencies\n",
      "\n",
      "You may know that SymPy tries to be a very low-dependency project.  Our user base is very broad.  Some entertaining aspects result.  For example, `textplot`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "textplot(x**2, -3, 3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "      9 |                                                        \n",
        "        |  .                                                    /\n",
        "        |   \\                                                  / \n",
        "        |    \\                                                /  \n",
        "        |     \\                                              /   \n",
        "        |      \\                                            .    \n",
        "        |       \\                                                \n",
        "        |        \\                                        ..     \n",
        "4.50149 | --------\\--------------------------------------/-------\n",
        "        |          \\                                    /        \n",
        "        |           \\                                  /         \n",
        "        |            \\                                /          \n",
        "        |             ..                            ..           \n",
        "        |               \\                          /             \n",
        "        |                ..                      ..              \n",
        "        |                  ..                  ..                \n",
        "        |                    ..              ..                  \n",
        "0.00297 |                      ..............                    \n",
        "          -3                     0                          3\n"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise\n",
      "\n",
      "Play with `textplot` and enjoy :)"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}