{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction\n",
    "\n",
    "In this section we learn to do the following:\n",
    "\n",
    "* Import SymPy and set up pretty printing\n",
    "* Use mathematical operations like `sqrt` and `sin`\n",
    "* Make SymPy Symbols\n",
    "* Take derivatives of expressions\n",
    "* Simplify expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preamble"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just like NumPy and Pandas replace functions like `sin`, `cos`, `exp`, and `log` to powerful numeric implementations, SymPy replaces `sin`, `cos`, `exp` and `log` with powerful mathematical implementations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()  # Set up fancy printing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "math.sqrt(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sqrt(2)  # This `sqrt` comes from SymPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cos(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "Use the function `acos` on `-1` to find when cosine equals `-1`.  Try this same function with the math library.  Do you get the same result?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Call acos on -1 to find where on the circle the x coordinate equals -1\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Call `math.acos` on -1 to find the same result using the builtin math module.  \n",
    "# Is the result the same?  \n",
    "# What do you think `numpy.acos` give you?\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbols\n",
    "\n",
    "Just like the NumPy `ndarray` or the Pandas `DataFrame`, SymPy has the `Symbol`, which represents a mathematical variable.\n",
    "\n",
    "We create symbols using the function `symbols`.  Operations on these symbols don't do numeric work like with NumPy or Pandas, instead they build up mathematical expressions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, z = symbols('x,y,z')\n",
    "alpha, beta, gamma = symbols('alpha,beta,gamma')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "log(alpha ** beta) + gamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sin(x)**2 + cos(x)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "Use `symbols` to create two variables, `mu` and `sigma`.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "?, ? = symbols('?')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "Use `exp`, `sqrt`, and Python's arithmetic operators like `+, -, *, **` to create the standard bell curve with SymPy objects\n",
    "\n",
    "$$ e^{\\frac{(x - \\mu)^2}{ \\sigma^2}} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "exp(?)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derivatives\n",
    "\n",
    "One of the most commonly requested operations in SymPy is the derivative.  To take the derivative of an expression use the `.diff` method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(x**2).diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sin(x).diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(x**2 + x*y + y**2).diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(x**2 + x*y + y**2).diff(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "In the last section you made a normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu, sigma = symbols('mu,sigma')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "bell = exp((x - mu)**2 / sigma**2)\n",
    "bell"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Take the derivative of this expression with respect to $x$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "?.diff(?)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "There are three symbols in that expression.  We normally are interested in the derivative with repspect to `x`, but we could just as easily ask for the derivative with respect to `sigma`.  Try this now"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Derivative of bell curve with respect to sigma\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "The second derivative of an expression is just the derivative of the derivative.  Chain `.diff( )` calls to find the second and third derivatives of your expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#  Find the second and third derivative of `bell`\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functions\n",
    "\n",
    "SymPy has a number of useful routines to manipulate expressions.  The most commonly used function is `simplify`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sin(x)**2 + cos(x)**2\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "simplify(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "In the last section you found the third derivative of the bell curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "bell.diff(x).diff(x).diff(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You might notice that this expression has lots of shared structure.  We can factor out some terms to simplify this expression.  \n",
    "\n",
    "Call `simplify` on this expression and observe the result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Call simplify on the third derivative of the bell expression\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sympify\n",
    "\n",
    "The `sympify` function transforms Python objects (ints, floats, strings) into SymPy objects (Integers, Reals, Symbols). \n",
    "\n",
    "*note the difference between `sympify` and `simplify`.  These are not the same function.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sympify('r * cos(theta)^2')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's useful whenever you interact with the real world, or for quickly copy-pasting an expression from an external source."
   ]
  }
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