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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Tutorial Brief"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "SymPy is symbolic mathematics library written completely in Python and doesn't require any dependencies.\n",
      "\n",
      "Finding Help:\n",
      "\n",
      "- http://docs.sympy.org/latest/index.html\n",
      "- http://sympy.org/en/features.html\n",
      "- http://nbviewer.ipython.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-5-Sympy.ipynb"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<table>\n",
      "<tr>\n",
      "    <td><img src=\"http://www.scipy.org/_static/images/numpylogo_med.png\"  style=\"width:50px;height:50px;\" /></td>\n",
      "    <td><h4>NumPy</h4> Base N-dimensional array package </td>\n",
      "    <td><img src=\"http://www.scipy.org/_static/images/scipy_med.png\" style=\"width:50px;height:50px;\" /></td>\n",
      "    <td><h4>SciPy</h4> Fundamental library for scientific computing </td>\n",
      "    <td><img src=\"http://www.scipy.org/_static/images/matplotlib_med.png\" style=\"width:50px;height:50px;\" /></td>\n",
      "    <td><h4>Matplotlib</h4> Comprehensive 2D Plotting </td>\n",
      "</tr>\n",
      "<tr>\n",
      "    <td><img src=\"http://www.scipy.org/_static/images/ipython.png\" style=\"width:50px;height:50px;\" /></td>\n",
      "    <td><h4>IPython</h4> Enhanced Interactive Console </td>\n",
      "    <td style=\"background:Lavender;\"><img src=\"http://www.scipy.org/_static/images/sympy_logo.png\" style=\"width:50px;height:50px;\" /></td>\n",
      "    <td style=\"background:Lavender;\"><h4>SymPy</h4> Symbolic mathematics </td>\n",
      "    <td><img src=\"http://www.scipy.org/_static/images/pandas_badge2.jpg\" style=\"width:50px;height:50px;\" /></td>\n",
      "    <td><h4>Pandas</h4> Data structures & analysis </td>\n",
      "</tr>\n",
      "</table>\n",
      "\n",
      "> ###SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python and does not require any external libraries.\n",
      "> **http://sympy.org/**"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Import SymPy"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "3 + math.sqrt(3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 2,
       "text": [
        "4.732050807568877"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = 3 * sqrt(3)\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "3*sqrt(3)"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "init_printing(use_latex='mathjax')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3 \\sqrt{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "    ___\n",
        "3\u22c5\u2572\u2571 3 "
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = sqrt(8)\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 \\sqrt{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "    ___\n",
        "2\u22c5\u2572\u2571 2 "
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`symbols()` & `Symbol()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x, y = symbols(\"x y\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = x**2 + y**2\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{2} + y^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        " 2    2\n",
        "x  + y "
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = (x+y)**3\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + y\\right)^{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "       3\n",
        "(x + y) "
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Assumptions for symbols"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = Symbol(\"a\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a.is_imaginary"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b = Symbol(\"b\", integer=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b.is_imaginary"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c = Symbol(\"c\", positive=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c.is_positive"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c.is_imaginary"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Imaginary Numbers"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "I"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$i$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "\u2148"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "I ** 2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "-1"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`Rational()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Rational(1,3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "1/3"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Rational(1,3) + Rational(1,2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{5}{6}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "5/6"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Numerical evaluation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = Rational(1,3) + Rational(1,2)\n",
      "N(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0.833333333333333$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": [
        "0.833333333333333"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N(pi, 100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": [
        "3.1415926535897932384626433832795028841971693993751058209749445923078164062862\n",
        "08998628034825342117068"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pi.evalf(100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 23,
       "text": [
        "3.1415926535897932384626433832795028841971693993751058209749445923078164062862\n",
        "08998628034825342117068"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`subs()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = x**2 + 2*x + 1\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{2} + 2 x + 1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 24,
       "text": [
        " 2          \n",
        "x  + 2\u22c5x + 1"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr.subs(x, 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$4$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": [
        "4"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = pi * x**2\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\pi x^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 26,
       "text": [
        "   2\n",
        "\u03c0\u22c5x "
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr.subs(x, 3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$9 \\pi$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": [
        "9\u22c5\u03c0"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N(_)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$28.2743338823081$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 28,
       "text": [
        "28.2743338823081"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`factor()` and `expand()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = (x + y) ** 2\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + y\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 29,
       "text": [
        "       2\n",
        "(x + y) "
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{2} + 2 x y + y^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 30,
       "text": [
        " 2            2\n",
        "x  + 2\u22c5x\u22c5y + y "
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "factor(_)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + y\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "       2\n",
        "(x + y) "
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`simplify()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = (2*x + Rational(1,3)*x + 4) / x\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{x} \\left(\\frac{7 x}{3} + 4\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "7\u22c5x    \n",
        "\u2500\u2500\u2500 + 4\n",
        " 3     \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "   x   "
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "simplify(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{7}{3} + \\frac{4}{x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 33,
       "text": [
        "7   4\n",
        "\u2500 + \u2500\n",
        "3   x"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = \"(2*x + 1/3*x + 4)/x\"\n",
      "simplify(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{7}{3} + \\frac{4}{x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": [
        "7   4\n",
        "\u2500 + \u2500\n",
        "3   x"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = sin(x)/cos(x)\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\sin{\\left (x \\right )}}{\\cos{\\left (x \\right )}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 35,
       "text": [
        "sin(x)\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "cos(x)"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "simplify(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\tan{\\left (x \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 36,
       "text": [
        "tan(x)"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`apart()` and `together()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = 1/(x**2 + 2*x)\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{x^{2} + 2 x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 37,
       "text": [
        "   1    \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " 2      \n",
        "x  + 2\u22c5x"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "apart(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{2 x + 4} + \\frac{1}{2 x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": [
        "      1        1 \n",
        "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\n",
        "  2\u22c5(x + 2)   2\u22c5x"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "together(_)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{x \\left(x + 2\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": [
        "    1    \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "x\u22c5(x + 2)"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Calculus"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(sin(x), x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\cos{\\left (x \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 40,
       "text": [
        "cos(x)"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(log(x**2 + 1) + 2*x, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 x}{x^{2} + 1} + 2$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 41,
       "text": [
        " 2\u22c5x      \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500 + 2\n",
        " 2        \n",
        "x  + 1    "
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(cos(x), x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (x \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "sin(x)"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Integral(sin(x), (x,0,pi))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\int_{0}^{\\pi} \\sin{\\left (x \\right )}\\, dx$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 43,
       "text": [
        "\u03c0          \n",
        "\u2320          \n",
        "\u23ae sin(x) dx\n",
        "\u2321          \n",
        "0          "
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N(_)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2.0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 44,
       "text": [
        "2.00000000000000"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`Sum()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = Sum(1/(x**2 + 2*x), (x, 1, 10))\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sum_{x=1}^{10} \\frac{1}{x^{2} + 2 x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": [
        "  10          \n",
        " ____         \n",
        " \u2572            \n",
        "  \u2572      1    \n",
        "   \u2572  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "   \u2571   2      \n",
        "  \u2571   x  + 2\u22c5x\n",
        " \u2571            \n",
        " \u203e\u203e\u203e\u203e         \n",
        "x = 1         "
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr.doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{175}{264}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 46,
       "text": [
        "175\n",
        "\u2500\u2500\u2500\n",
        "264"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`Product()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = Product(1/(x**2 + 2*x), (x, 1, 10))\n",
      "expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\prod_{x=1}^{10} \\frac{1}{x^{2} + 2 x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 47,
       "text": [
        "    10             \n",
        "\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c         \n",
        "\u2502        \u2502    1    \n",
        "\u2502        \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u2502        \u2502  2      \n",
        "\u2502        \u2502 x  + 2\u22c5x\n",
        "\u2502        \u2502         \n",
        "  x = 1            "
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr.doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{869100503040000}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 48,
       "text": [
        "1/869100503040000"
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "`Solve()`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = 2*x + 1\n",
      "solve(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}- \\frac{1}{2}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 49,
       "text": [
        "[-1/2]"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = x**2 - 1\n",
      "solve(expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}-1, & 1\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 50,
       "text": [
        "[-1, 1]"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr_1 = 2*x + y + 3\n",
      "expr_2 = 2*y - x\n",
      "solve([expr_1, expr_2],(x,y))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}x : - \\frac{6}{5}, & y : - \\frac{3}{5}\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 51,
       "text": [
        "{x: -6/5, y: -3/5}"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Units"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics import units as u"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "5. * u.milligram"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$5.0 \\cdot 10^{-6} kg$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 53,
       "text": [
        "5.0e-6\u22c5kg"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "1./2 * u.inch"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0.0127 m$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 54,
       "text": [
        "0.0127\u22c5m"
       ]
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "1. * u.nano"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1.0 \\cdot 10^{-9}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 55,
       "text": [
        "1.00000000000000e-9"
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u.watt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{kg m^{2}}{s^{3}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 56,
       "text": [
        "    2\n",
        "kg\u22c5m \n",
        "\u2500\u2500\u2500\u2500\u2500\n",
        "   3 \n",
        "  s  "
       ]
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u.ohm"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{kg m^{2}}{A^{2} s^{3}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 57,
       "text": [
        "    2\n",
        "kg\u22c5m \n",
        "\u2500\u2500\u2500\u2500\u2500\n",
        " 2  3\n",
        "A \u22c5s "
       ]
      }
     ],
     "prompt_number": 57
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Converting from Kilometers/hours to Miles/hours"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "kmph = u.km / u.hour\n",
      "mph = u.mile / u.hour\n",
      "\n",
      "N(mph / kmph)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1.609344$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 58,
       "text": [
        "1.60934400000000"
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "80 * N(mph / kmph)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$128.74752$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 69,
       "text": [
        "128.747520000000"
       ]
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Working with NumPy / Pandas and Matplotlib"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def sympy_expr(x_val):\n",
      "    expr = x**2 + sqrt(3)*x - Rational(1,3)\n",
      "    return expr.subs(x, x_val)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 59
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy_expr(3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3 \\sqrt{3} + \\frac{26}{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 60,
       "text": [
        "    ___   26\n",
        "3\u22c5\u2572\u2571 3  + \u2500\u2500\n",
        "          3 "
       ]
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import pandas as pd\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 61
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "list1 = np.arange(1,1000)\n",
      "list2 = pd.Series(list1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 62
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit [sympy_expr(item) for item in list1]\n",
      "%timeit [sympy_expr(item) for item in list2]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1 loops, best of 3: 103 ms per loop\n",
        "10 loops, best of 3: 105 ms per loop"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit np.vectorize(sympy_expr)(list1)\n",
      "%timeit list2.apply(sympy_expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "10 loops, best of 3: 776 ms per loop\n",
        "1 loops, best of 3: 1.09 s per loop"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expr = x**2 + sqrt(3)*x - Rational(1,3)\n",
      "\n",
      "lf = lambdify(x, expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 65
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit lf(list1)\n",
      "%timeit lf(list2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1000 loops, best of 3: 242 \u00b5s per loop\n",
        "100 loops, best of 3: 4.8 ms per loop"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n"
       ]
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig = plt.figure()\n",
      "axes = fig.add_subplot(111)\n",
      "\n",
      "x_vals = np.linspace(-5.,5.)\n",
      "y_vals = lf(x_vals)\n",
      "\n",
      "axes.grid()\n",
      "axes.plot(x_vals, y_vals)\n",
      "\n",
      "plt.show();"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7fc397382a90>"
       ]
      }
     ],
     "prompt_number": 67
    }
   ],
   "metadata": {}
  }
 ]
}