1.1 Solving lucrative problems with math and software
Predicting financial market movements
Building 3D graphics and animations
Slogging through math textbooks
1.3 Using your well-trained left brain
Building abstractions with functions
Part 1. Vectors and graphics
Multiplying vectors by numbers
Subtraction, displacement, and distance
2.3 Angles and trigonometry in the plane
Radians and trigonometry in Python
From components back to angles
2.4 Transforming collections of vectors
Combining vector transformations
3.1 Picturing vectors in 3D space
Representing 3D vectors with coordinates
Computing lengths and distances
Computing angles and directions
3.3 The dot product: Measuring vector alignment
Measuring angles with the dot product
3.4 The cross product: Measuring oriented area
Finding the direction of the cross product
Finding the length of the cross product
Computing the cross product of 3D vectors
3.5 Rendering a 3D object in 2D
Defining a 3D object with vectors
4 Transforming vectors and graphics
Composing vector transformations
Rotating an object about an axis
Inventing your own geometric transformations
Picturing linear transformations
Computing linear transformations
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
5.2 Interpreting matrices of different shapes
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
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
6 Generalizing to higher dimensions
6.1 Generalizing our definition of vectors
Creating a class for 2D coordinate vectors
Repeating the process with 3D vectors
Unit testing vector space classes
6.2 Exploring different vector spaces
Enumerating all coordinate vector spaces
Identifying vector spaces in the wild
Manipulating images with vector operations
6.3 Looking for smaller vector spaces
Finding subspaces of the vector space of functions
7 Solving systems of linear equations
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
7.3 Generalizing linear equations to higher dimensions
Solving linear equations in 3D
Studying hyperplanes algebraically
Counting dimensions, equations, and solutions
7.4 Changing basis by solving linear equations
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
8.2 Plotting the average flow rate over time
Finding the average flow rate in different time intervals
Plotting the interval flow rates
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
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
8.5 Plotting the volume over time
Picturing Riemann sums for the volume function
Definite and indefinite integrals
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
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
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
Translating the expression tree to Python
10.3 Putting a symbolic expression to work
Finding all the variables in an expression
10.4 Finding the derivative of a function
Derivatives of transformed functions
Derivatives of some special functions
Derivatives of products and compositions
10.5 Taking derivatives automatically
Implementing a derivative method for expressions
Implementing the product rule and chain rule
10.6 Integrating functions symbolically
11.1 Modeling gravity with a vector field
Modeling gravity with a potential energy function
11.2 Modeling gravitational fields
11.3 Adding gravity to the asteroid game
Making game objects feel gravity
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
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
12.2 Calculating the optimal range
Finding the projectile range as a function of the launch angle
Modeling terrain around the cannon
Solving for the range of the projectile in 3D
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
13 Analyzing sound waves with a Fourier series
13.1 Combining sound waves and decomposing them
13.2 Playing sound waves in Python
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
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
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
Part 3. Machine learning applications
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
14.2 Exploring spaces of functions
Picturing cost for lines through the origin
The space of all linear functions
14.3 Finding the line of best fit using gradient descent
Finding and plotting the line of best fit
14.4 Fitting a nonlinear function
Understanding the behavior of exponential functions
Finding the exponential function of best fit
15 Classifying data with logistic regression
15.1 Testing a classification function on real data
Testing the classification function
15.2 Picturing a decision boundary
Drawing a better decision boundary
Implementing the classification function
15.3 Framing classification as a regression problem
Measuring the “BMWness” of a car
Introducing the sigmoid function
Composing the sigmoid function with other functions
15.4 Exploring possible logistic functions
Parameterizing logistic functions
Measuring the quality of fit for a logistic function
Testing different logistic functions
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
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
16.3 Designing a neural network
Organizing neurons and connections
Data flow through a neural network
Calculating activations in matrix notation
16.4 Building a neural network in Python
Implementing an MLP class in Python
Testing the classification performance of an MLP
16.5 Training a neural network using gradient descent
Framing training as a minimization problem
Calculating gradients with backpropagation
Automatic training with scikit-learn
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
appendix A. Getting set up with Python
appendix B. Python tips and tricks
appendix C. Loading and rendering 3D Models with OpenGL and PyGame