{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "\n",
    "# See: http://docs.sympy.org/latest/tutorial/printing.html\n",
    "sp.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Live shell"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "http://live.sympy.org/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Make symbols"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make one symbol:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sp.symbols(\"x\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make several symbols at once:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x, y, z = sp.symbols(\"x y z\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derivatives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "See: http://docs.sympy.org/latest/tutorial/calculus.html#derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x, y, z = sp.symbols(\"x y z\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create an unevaluated derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sp.Derivative(sp.cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Evaluate an unevaluated derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(sp.cos(x), x)\n",
    "diff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Directly compute an integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sp.diff(sp.cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Print the equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sp.exp(x*y*z)\n",
    "diff = sp.Derivative(expr, x, y, y, z, z, z, z)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### First derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(sp.cos(x), x)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(3*sp.cos(x)**2, x)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(sp.exp(x**2), x)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Second derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(x**4, x, 2)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(x**4, x, x)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Third derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(x**4, x, 3)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(x**4, x, x, x)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Derivatives with respect to several variables at once"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(sp.exp(x*y), x, y)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiple derivatives with respect to several variables at once"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff = sp.Derivative(sp.exp(x*y*z), x, y, y, z, z, z, z)\n",
    "sp.Eq(diff, diff.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integrals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "See: http://docs.sympy.org/latest/tutorial/calculus.html#integrals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x, y, z = sp.symbols(\"x y z\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create an unevaluated derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "sp.Integral(sp.cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Evaluate an unevaluated derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "integ = sp.Integral(sp.cos(x), x)\n",
    "integ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "integ.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Directly compute an integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sp.integrate(sp.cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Print the equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "integ = sp.Integral(sp.cos(x), x)\n",
    "sp.Eq(integ, integ.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create an indefinite integral (i.e. an antiderivative or primitive)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "integ = sp.Integral(sp.cos(x), x)\n",
    "sp.Eq(integ, integ.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create a definite integral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`sp.oo` means infinity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "integ = sp.Integral(sp.cos(x), (x, -sp.oo, sp.oo))\n",
    "sp.Eq(integ, integ.doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "integ = sp.Integral(sp.cos(x), (x, -sp.pi, sp.pi))\n",
    "sp.Eq(integ, integ.doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "integ = sp.Integral(sp.exp(-x), (x, 0, sp.oo))\n",
    "sp.Eq(integ, integ.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiple integrals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "integ = sp.Integral(sp.cos(x), (x, -sp.oo, sp.oo), (x, -sp.oo, sp.oo))\n",
    "sp.Eq(integ, integ.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiple variables integrals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "integ = sp.Integral(sp.cos(x**2 + y**2), (x, -sp.oo, sp.oo), (y, -sp.oo, sp.oo))\n",
    "sp.Eq(integ, integ.doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "See: http://docs.sympy.org/latest/tutorial/calculus.html#limits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Series expansion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "See: http://docs.sympy.org/latest/tutorial/calculus.html#series-expansion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finite differences"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "See: http://docs.sympy.org/latest/tutorial/calculus.html#finite-differences"
   ]
  }
 ],
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