Guide to Free Math References

You can study lots of math without buying materials, but it's not always easy to know which references are good in different areas, so I maintain a list of books, notes, and videos I recommend for different topics, which, I hope, will be of some use to students or anyone interested in mathematics!

The list is short by design because I only want to recommend books I know well and have used in teaching or in my studies for quality control. If there are multiple free sources for the same materials, I chose my favorite.

Typical U.S. Undergraduate Topics

Calculus Sequence -- not rigorous :(, but useful

Calculus, Volume 1 by Strang and Herman. 

Differential calculus (similar to popular books from Stewart or Thomas)

Calculus, Volume 2 by Strang and Herman.

Integral calculus (similar to popular books from Stewart or Thomas)

Calculus, Volume 3 by Strang and Herman.

Multivariate calculus (similar to popular books from Stewart or Thomas)

The Essence of Calculus video playlist by Grant Sanderson

Tremendous animated series on single-variable calculus. Great supplements to books.

Differential Equations

Elementary Differential Equations with Boundary Value Problems by Trench

Mostly consists of the basics of ordinary differential equations for a first differential equations course.

Linear Algebra

Linear Algebra by Hefferon

Linear systems of equations, vector spaces, bases, projection, determinants, similarity.

Linear Algebra lectures by Gilbert Strang

Strang's lectures on intro linear algebra at MIT (multivariate calculus prerequisite, but it's not entirely necessary).

The Essence of Linear Algebra video playlist by Grant Sanderson

Beautiful animated videos, the perfect supplement for lectures and/or a book!

Probability

Probability and Statistics: The Science of Uncertainty by Evans and Rosenthal.

Slightly more advanced than the typical calculus-based intro probability book, but it is suitable for a first probability course.

Probability and Statistics playlist by me

Lectures from my intro course (single-variable calculus prerequisite)

Machine Learning Resources

Knowledge equivalent to introductory courses on multivariable calculus, linear algebra, and probability (e.g. at the level of the books above) are recommended to understand these topics well, but all is not necessarily required for all topics.

Neural Networks and Deep Learning

Neural Networks video playlist by Grant Sanderson

Beautiful animated videos on feedforward neural nets, stochastic gradient descent, backpropagation corresponding to Nielsen's book. The best place to start for neural nets in my view.

Neural Networks and Deep Learning by Michael Nielsen (2015)

Introduction to feedforward neural networks, stochastic gradient descent, and backpropagation.

Deep Learning. Ian Goodfellow, Yoshua Bengio, and Aaron Courville (2016)

Relatively mathematical treatment of deep learning. Includes a useful first 5 chapters of prerequisites for deep learning (minimal coverage of probability, linear algebra, numerical analysis, machine learning basics). It then covers MLPs, CNNs, RNNs, before moving on to some more modern techniques.

CS231n: Convolutional Neural Networks for Visual Recognition course by Fei-Fei Li, Ranjay Krishna, Danfei Xu and their army of TAs.

Notes, lectures, slides, assignments for a Stanford course, mostly on image recognition using CNNs and other models, image preprocessing, and computer vision. 


More Topics

The Elements of Statistical Learning by Trevor Hastie, Robert Tibshirani, and Jerome Friedman (2009)

Linear models for regression and classification, tree methods, support vector machines, unsupervised learning.

Machine Learning lectures by Andrew Ng

Ng's lectures from Stanford. It covers a wide array of basics: linear models, gradient descent, intro neural nets, regularization, support vector machines, principal component analysis, anomaly detection.

Other Topics

generatingfunctionology by Herbert Wilf

My favorite book on generating functions.

Graph Theory by Reinhard Diestel

Standard proof-based graph theory book.

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