{ "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 :)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.1" } }, "nbformat": 4, "nbformat_minor": 1 }