{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "x, y, z = symbols('x,y,z')"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "solve(x**2 - 4, x)"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "solve(x**2 - 9 == 0, x)"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "height, width, area = symbols('height,width,area')\n",
    "solve(area - height*width, height)"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solve for the radius of a sphere, given the volume\n",
    "\n"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Replace x with y\n",
    "(x**2).subs({x: y})"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Replace x with sin(x)\n",
    "\n"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solve for the height of a rectangle given area and width\n",
    "\n",
    "soln = solve(area - height*width, height)[0]\n",
    "soln"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define perimeter of rectangle in terms of height and width\n",
    "\n",
    "perimeter = 2*(height + width)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Substitute the solution for height into the expression for perimeter\n",
    "\n",
    "perimeter.subs({height: soln})"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "V, r = symbols('V,r', real=True)\n",
    "4*pi/3 * r**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "solve(V - 4*pi/3 * r**3, r)[0]"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(?).subs(?)"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot(x**2)"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot(?)"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "textplot(x**2, -3, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "Play with `textplot` and enjoy :)"
   ]
  }
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