Taming math and physics using SymPy

Tutorial based on the No bullshit guide series of textbooks by Ivan Savov

Abstract

Most people consider math and physics to be scary beasts from which it is best to keep one's distance. Computers, however, can help us tame the complexity and tedious arithmetic manipulations associated with these subjects. Indeed, math and physics are much more approachable once you have the power of computers on your side.

This tutorial serves a dual purpose. On one hand, it serves as a review of the fundamental concepts of mathematics for computer-literate people. On the other hand, this tutorial serves to demonstrate to students how a computer algebra system can help them with their classwork. A word of warning is in order. Please don't use SymPy to avoid the suffering associated with your homework! Teachers assign homework problems to you because they want you to learn. Do your homework by hand, but if you want, you can check your answers using SymPy. Better yet, use SymPy to invent extra practice problems for yourself.

Contents

Introduction

You can use a computer algebra system (CAS) to compute complicated math expressions, solve equations, perform calculus procedures, and simulate physics systems.

All computer algebra systems offer essentially the same functionality, so it doesn't matter which system you use: there are free systems like SymPy, Magma, or Octave, and commercial systems like Maple, MATLAB, and Mathematica. This tutorial is an introduction to SymPy, which is a symbolic computer algebra system written in the programming language Python. In a symbolic CAS, numbers and operations are represented symbolically, so the answers obtained are exact. For example, the number √2 is represented in SymPy as the object Pow(2,1/2), whereas in numerical computer algebra systems like Octave, the number √2 is represented as the approximation 1.41421356237310 (a float). For most purposes the approximation is okay, but sometimes approximations can lead to problems: float(sqrt(2))*float(sqrt(2)) = 2.00000000000000044 ≠ 2. Because SymPy uses exact representations, you'll never run into such problems: Pow(2,1/2)*Pow(2,1/2) = 2.

This tutorial is organized as follows. We'll begin by introducing the SymPy basics and the bread-and-butter functions used for manipulating expressions and solving equations. Afterward, we'll discuss the SymPy functions that implement calculus operations like differentiation and integration. We'll also introduce the functions used to deal with vectors and complex numbers. Later we'll see how to use vectors and integrals to understand Newtonian mechanics. In the last section, we'll introduce the linear algebra functions available in SymPy.

This tutorial presents many explanations as blocks of code. Be sure to try the code examples on your own by typing the commands into SymPy. It's always important to verify for yourself!

Using SymPy

The easiest way to use SymPy, provided you're connected to the Internet, is to visit http://live.sympy.org. You'll be presented with an interactive prompt into which you can enter your commands—right in your browser.

If you want to use SymPy on your own computer, you must install Python and the python package sympy. You can then open a command prompt and start a SymPy session using:

you@host$ python
Python X.Y.Z
[GCC a.b.c (Build Info)] on platform
Type "help", "copyright", or "license" for more information.
>>> from sympy import *
>>>

The >>> prompt indicates you're in the Python shell which accepts Python commands. The command from sympy import * imports all the SymPy functions into the current namespace. All SymPy functions are now available to you. To exit the python shell press CTRL+D.

I highly recommend you also install ipython, which is an improved interactive python shell. If you have ipython and SymPy installed, you can start an ipython shell with SymPy pre-imported using the command isympy. For an even better experience, you can try ipython notebook, which is a web frontend for the ipython shell.

You can start your session the same way as isympy do, by running following commands, which will be detaily described latter.

In [1]:
from sympy import init_session
init_session()

IPython console for SymPy 0.7.6 (Python 3.4.2-64-bit) (ground types: gmpy)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)
>>> init_printing()

Documentation can be found at http://www.sympy.org

Conclusion

I would like to conclude with some words of caution about the overuse of computers. Computer technology is very powerful and is everywhere around us, but let's not forget that computers are actually very dumb: computers are mere calculators and they depend on your knowledge to direct them. It's important that you learn how to do complicated math by hand in order to be able to instruct computers to do math for you and to check the results of your computer calculations. I don't want you to use the tricks you learned in this tutorial to avoid math problems from now on and simply rely blindly on SymPy for all your math needs. I want both you and the computer to become math powerhouses! The computer will help you with tedious calculations (they're good at that) and you'll help the computer by guiding it when it gets stuck (humans are good at that).

Book plug

Cover

The examples and math explanations in this tutorial are sourced from the No bullshit guide series of books published by Minireference Co. We publish textbooks that make math and physics accessible and affordable for everyone. If you're interested in learning more about the math, physics, and calculus topics discussed in this tutorial, check out the No bullshit guide to math and physics. The book contains the distilled information that normally comes in two first-year university books: the introductory physics book (1000+ pages) and the first-year calculus book (1000+ pages). Would you believe me if I told you that you can learn the same material from a single book that is 1/7th the size and 1/10th of the price of mainstream textbooks?

This book contains short lessons on math and physics, calculus. Often calculus and mechanics are taught as separate subjects. It shouldn't be like that. If you learn calculus without mechanics, it will be boring. If you learn mechanics without calculus, you won't truly understand what is going on. This textbook covers both subjects in an integrated manner.

Contents:

  • High school math
  • Vectors
  • Mechanics
  • Differential calculus
  • Integral calculus
  • 250+ practice problems

For more information, see the book's website at minireference.com

The presented linear algebra examples are sourced from the No bullshit guide to linear algebra. Check out the book if you're taking a linear algebra course of if you're missing the prerequisites for learning machine learning, computer graphics, or quantum mechanics.

I'll close on a note for potential readers who suffer from math-phobia. Both books start with an introductory chapter that reviews all high school math concepts needed to make math and physics accessible to everyone. Don't worry, we'll fix this math-phobia thing right up for you; when you've got SymPy skills, math fears you!

To stay informed about upcoming titles, follow @minireference on twitter and check out the facebook page at fb.me/noBSguide.