Math for Programmers
3D graphics, machine learning, and simulations with Python
Paul Orland
  • ISBN 9781617295355
  • 550 pages (estimated)
  • printed in black & white

It feels like you're learning exactly what you need to know to get stuff done.

Jens Christian Bredahl Madsen
To score a job in data science, machine learning, computer graphics, and cryptography, you need to bring strong math skills to the party. Math for Programmers teaches the math you need for these hot careers, concentrating on what you need to know as a developer. Filled with lots of helpful graphics and more than 200 exercises and mini-projects, this book unlocks the door to interesting–and lucrative!–careers in some of today’s hottest programming fields.
Table of Contents detailed table of contents

1 Learning Math in Code

1.1 Solving lucrative problems with math and software

1.1.1 Predicting financial market movements

1.1.2 Finding a good deal

1.1.3 Building 3D graphics and animations

1.1.4 Modeling the physical world

1.2 How not to learn math

1.2.1 Jane wants to learn some math

1.2.2 Slogging through math textbooks

1.3 Using your well-trained left brain

1.3.1 Using a formal language

1.3.2 Build your own calculator

1.3.3 Building abstractions with functions

1.4 Summary

Part I: Vectors and Graphics

2 Drawing with 2D Vectors

2.1 Drawing with 2D Vectors

2.1.1 Representing 2D vectors

2.1.2 2D Drawing in Python

2.1.3 Exercises

2.2 Plane vector arithmetic

2.2.1 Vector components and lengths

2.2.2 Multiplying Vectors by Numbers

2.2.3 Subtraction, displacement, and distance

2.2.4 Exercises

2.3 Angles and Trigonometry in the Plane

2.3.1 From angles to components

2.3.2 Radians and trigonometry in Python

2.3.3 From components back to angles

2.3.4 Exercises

2.4 Transforming collections of vectors

2.4.1 Combining vector transformations

2.4.2 Exercises

2.5 Drawing with Matplotlib

2.6 Summary

3 Ascending to the 3D World

3.1 Picturing vectors in three-dimensional space

3.1.1 Representing 3D vectors with coordinates

3.1.2 3D Drawing in Python

3.1.3 Exercises

3.2 Vector arithmetic in 3D

3.2.1 Adding 3D vectors

3.2.2 Scalar Multiplication in 3D

3.2.3 Subtracting 3D vectors

3.2.4 Computing lengths and distances

3.2.5 Computing angles and directions

3.2.6 Exercises

3.3 The dot product: measuring alignment of vectors

3.3.1 Picturing the dot product

3.3.2 Computing the dot product

3.3.3 Dot products by example

3.3.4 Measuring angles with the dot product

3.3.5 Exercises

3.4 The cross product: measuring oriented area

3.4.1 Orienting ourselves in 3D

3.4.2 Finding the direction of the cross product

3.4.3 Finding the length of the cross product

3.4.4 Computing the cross product of 3D vectors

3.4.5 Exercises

3.5 Rendering a 3D object in 2D

3.5.1 Defining a 3D object with vectors

3.5.2 Projecting to 2D

3.5.3 Orienting faces and shading

3.5.4 Exercises

3.6 Summary

4 Transforming Vectors and Graphics

4.1 Transforming 3D objects

4.1.1 Drawing a transformed object

4.1.2 Composing vector transformations

4.1.3 Rotating an object about an axis

4.1.4 Inventing your own geometric transformations

4.1.5 Exercises

4.2 Linear transformations

4.2.1 Preserving vector arithmetic

4.2.2 Picturing linear transformations

4.2.3 Why linear transformations?

4.2.4 Computing linear transformations

4.2.5 Exercises

4.3 Summary

5 Computing Transformations with Matrices

5.1 Representing linear transformations with matrices

5.1.1 Writing vectors and linear transformations as matrices

5.1.2 Multiplying a matrix with a vector

5.1.3 Composing linear transformations by matrix multiplication

5.1.4 Implementing matrix multiplication

5.1.5 3D Animation with matrix transformations

5.1.6 Exercises

5.2 Interpreting matrices of different shapes

5.2.1 Column vectors as matrices

5.2.2 What pairs of matrices can be multiplied?

5.2.3 Viewing square and non-square matrices as vector functions

5.2.4 Projection as a linear map from 3D to 2D

5.2.5 Composing linear maps

5.2.6 Exercises

5.3 Translating vectors with matrices

5.3.1 Making plane translations linear

5.3.2 Finding a 3D matrix for a 2D translation

5.3.3 Combining translation with other linear transformations

5.3.4 Translating 3D objects in a 4D world

5.3.5 Exercises

5.4 Summary

6 Generalizing to Higher Dimensions

6.1 Generalizing our definition of vectors

6.1.1 Creating a class for 2D coordinate vectors

6.1.2 Improving the Vec2 class

6.1.3 Repeating the process with 3D vectors

6.1.4 Building a Vector base class

6.1.5 Defining vector spaces

6.1.6 Unit testing vector space classes

6.1.7 Exercises

6.2 Exploring different vector spaces

6.2.1 Enumerating all coordinate vector spaces

6.2.2 Identifying vector spaces in the wild

6.2.3 Treating functions as vectors

6.2.4 Treating matrices as vectors

6.2.5 Manipulating images with vector operations

6.2.6 Exercises

6.3 Looking for smaller vector spaces

6.3.1 Identifying subspaces

6.3.2 Starting with a single vector

6.3.3 Spanning a bigger space

6.3.4 Defining the word “dimension”

6.3.5 Finding subspaces of the vector space of functions

6.3.6 Subspaces of images

6.3.7 Exercises

6.4 Summary

7 Solving Systems of Linear Equations

7.1 Designing an arcade game

7.1.1 Modeling the game

7.1.2 Rendering the game

7.1.3 Shooting the laser

7.1.4 Exercises

7.2 Finding intersection points of lines

7.2.1 Choosing the right formula for a line

7.2.2 Finding the standard form equation for a line

7.2.3 Linear equations in matrix notation

7.2.4 Solving linear equations with numpy

7.2.5 Deciding whether the laser hits an asteroid

7.2.6 Identifying unsolvable systems

7.2.7 Exercises

7.3 Generalizing linear equations to higher dimensions

7.3.1 Representing planes in 3D

7.3.2 Solving linear equations in 3D

7.3.3 Studying hyperplanes algebraically

7.3.4 Counting dimensions, equations, and solutions

7.3.5 Exercises

7.4 Changing basis by solving linear equations

7.4.1 Solving a 3D example

7.4.2 Exercises

7.5 Summary

Part 2: Calculus and Physical Simulation

8 Simulating Moving Objects

8.1 Simulating constant-speed motion

8.1.1 Intuiting speed

8.1.2 Thinking of velocity as a vector

8.1.3 Animating the asteroids

8.1.4 Exercises

8.2 Simulating acceleration

8.2.1 Picturing acceleration in various directions

8.2.2 Quantifying Acceleration

8.2.3 Accelerating the spaceship

8.2.4 Exercises

8.3 Digging deeper into Euler’s method

8.3.1 Stepping through Euler’s method

8.3.2 Implementing the algorithm in Python

8.3.3 Picturing the approximation

8.3.4 Applying Euler’s method to other problems

8.3.5 Exercises

8.4 Calculating exact trajectories

8.4.1 Writing position, velocity, and acceleration as functions of time

8.4.2 Using the terminology of integration

8.4.3 Calculating integrals

8.4.4 Exercises

8.5 Summary

9 Reverse-engineering the laws of physics

9.1 Measuring motion from a video

9.1.1 Calculating position from pixels

9.1.2 Finding pixel position using computer vision

9.1.3 Plotting position over time

9.1.4 Exercises

9.2 Approximating velocity

9.2.1 Picturing velocity as the slope of the position graph

9.2.2 Calculating and plotting velocity over time

9.2.3 Exercises

9.3 Working with a position function

9.3.1 Writing position as a function of time

9.3.2 Rethinking our velocity calculation

9.3.3 Finding the slope of secant lines

9.3.4 Approximating instantaneous velocity

9.3.5 Exercises

9.4 Finding exact rates of change with derivatives

9.4.1 Finding the derivative at a point

9.4.2 Currying the derivative

9.4.3 Calculating instantaneous acceleration

9.4.4 Finding a formula for the derivative

9.4.5 Exercises

9.5 Inferring force laws

9.5.1 Testing a solution to a differential equation

9.5.2 The differential equation for the ball on a spring

9.5.3 Finding a force law for the spring

9.5.4 Exercises

9.6 Summary

10 Working with symbolic expressions

11 Approximating functions

[[chapter_id_12] == 12 Simulating Force Fields

13 Optimizing a physical system

Part 3: Machine Learning Applications

14 Quantifying uncertainty

15 Fitting functions to data

16 Classifying data

17 Training neural networks


Appendix A: A Loading and Rendering 3D Models with OpenGL and PyGame

A.1 Recreating the octahedron from Chapter 3

A.2 Changing our perspective

A.3 Loading and rendering the Utah teapot

A.4 Exercises

About the Technology

Most businesses realize they need to apply data science and effective machine learning to gain and maintain a competitive edge. To build these applications, they need developers comfortable writing code and using tools steeped in statistics, linear algebra, and calculus. Math also plays an integral role in other modern applications like game development, computer graphics and animation, image and signal processing, pricing engines, and stock market analysis. Whether you’re a self-taught programmer without a core university math foundation or you just need to rekindle the glowing math embers, this book is a great way to fire up your skills.

About the book

Math for Programmers teaches you to solve mathematical problems in code. Thanks to the author’s fun and engaging style, you’ll enjoy thinking about math like a programmer. With accessible examples, scenarios, and exercises perfect for the working developer, you’ll start by exploring functions and geometry in 2D and 3D. With those basic building blocks behind you, you’ll move into the bread and butter math for machine learning and game programming, including matrices and linear transformations, derivatives and integrals, differential equations, probability, classification algorithms, and more. Don’t worry if it sounds intimidating or, worse yet, boring! Coder and mathematician Paul Orland makes learning these vital concepts painless, relevant, and fun!

Real-world examples in this practical tutorial include building and rendering 3D models, animations with matrix transformations, manipulating images and sound waves, and building a physics engine for a video game. Along the way, you’ll test yourself with lots of exercises to ensure you’ve got a firm grasp of the concepts. Hands-on mini-projects throughout lock in all you’ve learned. When you’re done, you’ll have a solid foundation of the math skills essential in today’s most popular tech trends.

What's inside

  • 2D and 3D vector math
  • Matrices and linear transformations
  • Core concepts from linear algebra
  • Calculus with one or more variables
  • Algorithms for regression, classification, and clustering
  • Interesting real-world examples
  • More than 200 exercises and mini-projects

About the reader

Written for programmers with solid algebra skills (even if they need some dusting off). No formal coursework in linear algebra or calculus is required.

About the author

Paul Orland is CEO of Tachyus, a Silicon Valley startup building predictive analytics software to optimize energy production in the oil and gas industry. As founding CTO, he led the engineering team to productize hybrid machine learning and physics models, distributed optimization algorithms, and custom web-based data visualizations. He has a B.S. in mathematics from Yale University and a M.S. in physics from the University of Washington.

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