Return to Math for Programmers by Paul Orland

Fair Use Sources:

• B09781CX29 (MthPrg 2021)

• contents

• preface

• acknowledgments

about this book

about the author

about the cover illustration

• 1 Learning math with code
• 1.1 Solving lucrative problems with math and software
• Predicting financial market movements
• Finding a good deal
• Building 3D graphics and animations
• Modeling the physical world

1.2 How not to learn math

• Jane wants to learn some math
• Slogging through math textbooks

1.3 Using your well-trained left brain

• Using a formal language
• Build your own calculator
• Building abstractions with functions

Part 1. Vectors and graphics

2 Drawing with 2D vectors

2.1 Picturing 2D vectors

• Representing 2D vectors
• 2D drawing in Python
• Exercises

2.2 Plane vector arithmetic

• Vector components and lengths
• Multiplying vectors by numbers
• Subtraction, displacement, and distance
• Exercises

2.3 Angles and trigonometry in the plane

• From angles to components
• Radians and trigonometry in Python
• From components back to angles
• Exercises

2.4 Transforming collections of vectors

• Combining vector transformations
• Exercises

2.5 Drawing with Matplotlib

3 Ascending to the 3D world

3.1 Picturing vectors in 3D space

• Representing 3D vectors with coordinates
• 3D drawing with Python
• Exercises

3.2 Vector arithmetic in 3D

• Adding 3D vectors
• Scalar multiplication in 3D
• Subtracting 3D vectors
• Computing lengths and distances
• Computing angles and directions
• Exercises

3.3 The dot product: Measuring vector alignment

• Picturing the dot product
• Computing the dot product
• Dot products by example
• Measuring angles with the dot product
• Exercises

3.4 The cross product: Measuring oriented area

• Orienting ourselves in 3D
• Finding the direction of the cross product
• Finding the length of the cross product
• Computing the cross product of 3D vectors
• Exercises

3.5 Rendering a 3D object in 2D

• Defining a 3D object with vectors
• Projecting to 2D
• Orienting faces and shading
• Exercises

4 Transforming vectors and graphics

4.1 Transforming 3D objects

• Drawing a transformed object
• Composing vector transformations
• Rotating an object about an axis
• Inventing your own geometric transformations
• Exercises

4.2 Linear transformations

• Preserving vector arithmetic
• Picturing linear transformations
• Why linear transformations?
• Computing linear transformations
• Exercises

5 Computing transformations with matrices

5.1 Representing linear transformations with matrices

• Writing vectors and linear transformations as matrices
• Multiplying a matrix with a vector
• Composing linear transformations by matrix multiplication
• Implementing matrix multiplication
• 3D animation with matrix transformations
• Exercises

5.2 Interpreting matrices of different shapes

• Column vectors as matrices
• What pairs of matrices can be multiplied?
• Viewing square and non-square matrices as vector functions
• Projection as a linear map from 3D to 2D
• Composing linear maps
• Exercises

5.3 Translating vectors with matrices

• Making plane translations linear
• Finding a 3D matrix for a 2D translation
• Combining translation with other linear transformations
• Translating 3D objects in a 4D world
• Exercises

6 Generalizing to higher dimensions

6.1 Generalizing our definition of vectors

• Creating a class for 2D coordinate vectors
• Improving the Vec2 class
• Repeating the process with 3D vectors
• Building a vector base class
• Defining vector spaces
• Unit testing vector space classes
• Exercises

6.2 Exploring different vector spaces

• Enumerating all coordinate vector spaces
• Identifying vector spaces in the wild
• Treating functions as vectors
• Treating matrices as vectors
• Manipulating images with vector operations
• Exercises

6.3 Looking for smaller vector spaces

• Identifying subspaces
• Starting with a single vector
• Spanning a bigger space
• Defining the word dimension
• Finding subspaces of the vector space of functions
• Subspaces of images
• Exercises

7 Solving systems of linear equations

7.1 Designing an arcade game

• Modeling the game
• Rendering the game
• Shooting the laser
• Exercises

7.2 Finding intersection points of lines

• Choosing the right formula for a line
• Finding the standard form equation for a line
• Linear equations in matrix notation
• Solving linear equations with NumPy
• Deciding whether the laser hits an asteroid
• Identifying unsolvable systems
• Exercises

7.3 Generalizing linear equations to higher dimensions

• Representing planes in 3D
• Solving linear equations in 3D
• Studying hyperplanes algebraically
• Counting dimensions, equations, and solutions
• Exercises

7.4 Changing basis by solving linear equations

• Solving a 3D example
• Exercises

Part 2. Calculus and physical simulation

8 Understanding rates of change

8.1 Calculating average flow rate from volume

• Implementing an average_flow_rate function
• Picturing the average flow rate with a secant line
• Negative rates of change
• Exercises

8.2 Plotting the average flow rate over time

• Finding the average flow rate in different time intervals
• Plotting the interval flow rates
• Exercises

8.3 Approximating instantaneous flow rates

• Finding the slope of small secant lines
• Building the instantaneous flow rate function
• Currying and plotting the instantaneous flow rate function
• Exercises

8.4 Approximating the change in volume

• Finding the change in volume for a short time interval
• Breaking up time into smaller intervals
• Picturing the volume change on the flow rate graph
• Exercises

8.5 Plotting the volume over time

• Finding the volume over time
• Picturing Riemann sums for the volume function
• Improving the approximation
• Definite and indefinite integrals

9 Simulating moving objects

9.1 Simulating a constant velocity motion

• Adding velocities to the asteroids
• Updating the game engine to move the asteroids
• Keeping the asteroids on the screen
• Exercises

9.2 Simulating acceleration

• Accelerating the spaceship

9.3 Digging deeper into Euler’s method

• Carrying out Euler’s method by hand
• Implementing the algorithm in Python

9.4 Running Euler’s method with smaller time steps

• Exercises

10 Working with symbolic expressions

10.1 Finding an exact derivative with a computer algebra system

• Doing symbolic algebra in Python

10.2 Modeling algebraic expressions

• Breaking an expression into pieces
• Building an expression tree
• Translating the expression tree to Python
• Exercises

10.3 Putting a symbolic expression to work

• Finding all the variables in an expression
• Evaluating an expression
• Expanding an expression
• Exercises

10.4 Finding the derivative of a function

• Derivatives of powers
• Derivatives of transformed functions
• Derivatives of some special functions
• Derivatives of products and compositions
• Exercises

10.5 Taking derivatives automatically

• Implementing a derivative method for expressions
• Implementing the product rule and chain rule
• Implementing the power rule
• Exercises

10.6 Integrating functions symbolically

• Integrals as antiderivatives
• Introducing the SymPy library
• Exercises

11 Simulating force fields

• 11.1 Modeling gravity with a vector field
• Modeling gravity with a potential energy function

11.2 Modeling gravitational fields

• Defining a vector field
• Defining a simple force field

11.3 Adding gravity to the asteroid game

• Making game objects feel gravity
• Exercises

11.4 Introducing potential energy

• Defining a potential energy scalar field
• Plotting a scalar field as a heatmap
• Plotting a scalar field as a contour map

11.5 Connecting energy and forces with the gradient

• Measuring steepness with cross sections
• Calculating partial derivatives
• Finding the steepness of a graph with the gradient
• Calculating force fields from potential energy with the gradient
• Exercises

12 Optimizing a physical system

12.1 Testing a projectile simulation

• Building a simulation with Euler’s method
• Measuring properties of the trajectory
• Exploring different launch angles
• Exercises

12.2 Calculating the optimal range

• Finding the projectile range as a function of the launch angle
• Solving for the maximum range
• Identifying maxima and minima
• Exercises

12.3 Enhancing our simulation

• Adding another dimension
• Modeling terrain around the cannon
• Solving for the range of the projectile in 3D
• Exercises

12.4 Optimizing range using gradient ascent

• Plotting range versus launch parameters
• The gradient of the range function
• Finding the uphill direction with the gradient
• Implementing gradient ascent
• Exercises

13 Analyzing sound waves with a Fourier series

13.1 Combining sound waves and decomposing them

13.2 Playing sound waves in Python

• Producing our first sound
• Playing a musical note
• Exercises

13.3 Turning a sinusoidal wave into a sound

• Making audio from sinusoidal functions
• Changing the frequency of a sinusoid
• Sampling and playing the sound wave
• Exercises

13.4 Combining sound waves to make new ones

• Adding sampled sound waves to build a chord
• Picturing the sum of two sound waves
• Building a linear combination of sinusoids
• Building a familiar function with sinusoids
• Exercises

13.5 Decomposing a sound wave into its Fourier series

• Finding vector components with an inner product
• Defining an inner product for periodic functions
• Writing a function to find Fourier coefficients
• Finding the Fourier coefficients for the square wave
• Fourier coefficients for other waveforms
• Exercises

Part 3. Machine learning applications

14 Fitting functions to data

14.1 Measuring the quality of fit for a function

• Measuring distance from a function
• Summing the squares of the errors
• Calculating cost for car price functions
• Exercises

14.2 Exploring spaces of functions

• Picturing cost for lines through the origin
• The space of all linear functions
• Exercises
• 14.3 Finding the line of best fit using gradient descent
• Rescaling the data
• Finding and plotting the line of best fit
• Exercises

14.4 Fitting a nonlinear function

• Understanding the behavior of exponential functions
• Finding the exponential function of best fit
• Exercises

15 Classifying data with logistic regression

15.1 Testing a classification function on real data

• Loading the car data
• Testing the classification function
• Exercises

15.2 Picturing a decision boundary

• Picturing the space of cars
• Drawing a better decision boundary
• Implementing the classification function
• Exercises

15.3 Framing classification as a regression problem

• Scaling the raw car data
• Measuring the “BMWness” of a car
• Introducing the sigmoid function
• Composing the sigmoid function with other functions
• Exercises

15.4 Exploring possible logistic functions

• Parameterizing logistic functions
• Measuring the quality of fit for a logistic function
• Testing different logistic functions
• Exercises

15.5 Finding the best logistic function

• Gradient descent in three dimensions
• Using gradient descent to find the best fit
• Testing and understanding the best logistic classifier
• Exercises

16 Training neural networks

16.1 Classifying data with neural networks

16.2 Classifying images of handwritten digits

• Building the 64-dimensional image vectors
• Building a random digit classifier
• Measuring performance of the digit classifier
• Exercises

16.3 Designing a neural network

• Organizing neurons and connections
• Data flow through a neural network
• Calculating activations
• Calculating activations in matrix notation
• Exercises

16.4 Building a neural network in Python

• Implementing an MLP class in Python
• Evaluating the MLP
• Testing the classification performance of an MLP
• Exercises

16.5 Training a neural network using gradient descent

• Framing training as a minimization problem
• Calculating gradients with backpropagation
• Automatic training with scikit-learn
• Exercises

16.6 Calculating gradients with backpropagation

• Finding the cost in terms of the last layer weights
• Calculating the partial derivatives for the last layer weights using the chain rule
• Exercises

• appendix A. Getting set up with Python
• appendix B. Python tips and tricks
• appendix C. Loading and rendering 3D Models with OpenGL and PyGame

• Index

Fair Use Sources:

• B09781CX29 (MthPrg 2021)