2. Resources
3. Textbooks

Here is a partial list of free and/or open-source textbooks. If you find one of these more helpful than another, please let me know. Also, take a peek at Rob Beezer’s selection on this page. Moreover, the American Institute of Mathematics maintains a list of approved open-source textbooks. Also, check out the free computer science books by Allen B. Downey (Olin College) at Green Tea Press. Downey’s Textbook manifesto is definitely worth reading. There are also lots of free textbooks available at LibreTexts.

Calculus

• Calculus for Team-Based Inquiry Learning by TBIL Institute Fellows. Activities and exercises for easily implementing Team-Based Inquiry Learning in a single-variable calculus classroom. PDF and HTML versions.
• CLP Calculus Textbooks by Joel Feldman, Andrew Rechnitzer, and Elyse Yeager were written for standard university calculus courses at the University of British Colombia. Free PDF and web-based textbooks.
• APEX Calculus is an open-source textbook by a group of faculty from Virginia Military Institute. It has the look and feel of a traditional calculus (e.g., Stewart’s Calculus). Includes interactive 3D graphics.
• Open Source Calculus and Analysis by a group of faculty at Ghent University. This book substantially builds on APEX Calculus. As compared to the APEX Calculus, it comes with precalculus, more formalism, proofs of most theorems (and more theorems), differential equations, computer laboratories, review exercises, built-in code for symbolic computation, and many other things. It’s tailored to students in engineering programs. Multiple free PDFs available for download.
• MOOCulus Textbook is a free textbook from the Ximera folks at Ohio State University. This text is mostly an adaptation of the Community Calculus text listed below.
• Community Calculus by David Guichard (Whitman College). This one is also available as a free PDF. You can also purchase a very inexpensive hardcopy from Lulu.
• Active Calculus by Matt Boelkins (Grand Valley State College). This is one of the newer additions to the list of free calculus texts.
• Coordinated Calculus by Nathan Wakefield (University of Nebraska) et al. This is the calculus textbook used at the Univerisity of Nebraska. Based upon Active Calculus by Matthew Boelkins (see above).
• Calculus by Gilbert Strang (MIT). This book is available as a free PDF from the MIT Open Courseware Project. There are also some corresponding videos, which can be found here.
• Calculus Refresher by Paul Garrett (University of Minnesota). Another free PDF. This short book seems designed for students who have had some experience with calculus and need some review.
• Funny Little Calculus Text by Robert Ghrist (University of Pennsylvania). Currently only first semester calculus, very short, and no exercises, but a free PDF.
• Paul’s Online Math Notes by Paul Dawkins (Lamar University). Paul’s notes cover Calculus 1-3, Linear Algebra, and more. The notes are available as free HTML-based material. These notes seem to be popular with students.
• Differential Calculus with Sage by David Joyner (United States Naval Academy, retired) and William Granville. This free book is based on Granville’s classic text book Elements of the Differential and Integral Calculus, which fell into the public domain. The book covers first semester calculus and incorporates Sage, which is an open-source mathematics software package.
• Integral Calculus with Sage by Dale Hoffman (Bellevue Community College), William Stein (SageMath), and David Joyner (United States Naval Academy, retired). The book picks up where the previous book left off.
• Contemporary Calculus by Dale Hoffman (Bellevue Community College). A free online calculus text that is being developed for the Open Course Library Project of the Washington State Colleges. This book is part of the foundation for the previous book. It covers three semesters of calculus and seems to do a thorough job of it, but the formatting is awful.
• Differential Calculus: From Practice to Theory by Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia). This book covers all of the topics in a typical first course in differential calculus. The authors claim that their approach is more historically accurate than the usual development of calculus and, more importantly, pedagogically sound.

Differential Equations

• Elementary Differential Equations and Elementary Differential Equations with Boundary Value Problems by William Trench (Trinity College). Both books were previously published by Brooks/Cole Thomson Learning. PDF and LaTeX source files are available for free.
• Introductory Differential Equations using Sage (PDF) by David Joyner (United States Naval Academy, retired).

Discrete Mathematics

• Discrete Mathematics: An Open Introduction by Oscar Levin (University of Northern Colorado) is a free, open-source textbook appropriate for a first or second year undergraduate course for math majors. There is a really nice interactive ebook version.
• Applied Discrete Structures by Al Doerr (UMass Lowell) and Kenneth M. Levasseur (UMass Lowell).

Logic

• Open Logic Project. An open-source, collaborative textbook of logic and formal methods, starting at an intermediate level, aimed at a non-mathematical audience.

Introduction to Proof

• An Introduction to Proof via Inquiry-Based Learning by Dana C. Ernst (Northern Arizona University). Textbook for the transition to proof course for mathematics majors. Designed to promote active learning through inquiry. Open-source project, free PDF download with low-cost paperback available via AMS/MAA Press.
• Introduction to Proof by Ron Taylor (Berry College). These IBL notes are available from the Journal of Inquiry-Based Learning in Mathematics.
• Notes for a Course on Proofs by Jacqueline A. Jensen-Vallin (Slippery Rock University). Another set of IBL notes available from the Journal of Inquiry-Based Learning in Mathematics.
• Mathematical Reasoning: Writing and Proof by Ted Sundstrom (Grand Valley State University). This textbook is designed for the first course in the college mathematics curriculum that introduces students to the process of constructing and writing proofs. This book is included in the approved list of textbooks for the Open Textbook Initiative by the American Institute of Mathematics. There are also over 100 screencasts that accompany the book.

Linear Algebra

• Linear Algebra for Team-Based Inquiry Learning by Steven Clontz (University of South Alabama) and Drew Lewis (University of South Alabama) with contributions from others. Activities and exercises for easily implementing Team-Based Inquiry Learning in a sophomore linear algebra classroom. PDF and HTML versions.
• Understanding Linear Algebra by David Austin (Grand Valley State University). An open-access linear algebra textbook with both PDF and HTML formats.
• Interactive Linear Algebra by Dan Margalit (Georgia Institute of Technology) and Joseph Rabinoff (Duke University). An HTML based book with beautiful interactive figures.
• A First Course in Linear Algebra by Rob Beezer (Puget Sound University). FCLA is a free and open-source introductory textbook designed for university sophomores and juniors. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The book is available for free in PDF and HTML form.
• Linear Algebra by Jim Hefferon (St. Michael’s College). This is another free and open-source textbook.
• Linear Algebra by David Clark (SUNY New Paltz). These notes are available for free from the Journal of Inquiry-Based Learning in Mathematics. The notes are designed for an inquiry-based learning (IBL) approach to the subject.
• Linear Algebra by Gilbert Strang (MIT). This isn’t really a textbook, but rather free course materials, including videos, available from the MIT Open Courseware Project.
• Paul’s Online Math Notes by Paul Dawkins (Lamar University).
• Computational Linear Algebra by Chad Topaz (Williams College). This is more of a course than a book, but an excellent resource nonetheless.

Combinatorics

• Applied Combinatorics by Mitch Keller (Morningside College) and William T. Trotter (Georgia Institute of Technology). An open-source textbook for a course covering the fundamental enumeration techniques (permutations, combinations, subsets, pigeon hole principle), recursion and mathematical induction, more advanced enumeration techniques (inclusion-exclusion, generating functions, recurrence relations, Polyá theory), discrete structures (graphs, digraphs, posets, interval orders), and discrete optimization (minimum weight spanning trees, shortest paths, network flows). Available in PDF, HTML, and ePub formats.
• Combinatorics Through Guided Discovery by Kenneth Bogart. An introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially, but not exclusively, on the part of combinatorics that mathematicians refer to as “counting.”

Abstract Algebra

• An Inquiry-Based Approach to Abstract Algebra by Dana C. Ernst (Northern Arizona University). Free and open-source course materials for an abstract algebra course that utilizes an inquiry-based learning approach.
• Abstract Algebra: Theory and Applications by Tom Judson (Stephen F. Austin University).
• Elementary Abstract Algebra: Examples and Applications by Justin Hill (Temple College) and Chris Thron (Texas A\&M University-Central Texas) with contributions from others. The text is designed for students who are planning to become secondary-school teachers. The authors particularly emphasize material that has relevance to high school math, as well as practical applications. Much of the content is derived from the Judson book mentioned above.
• Essential Group Theory by Michael Batty (University of Durham).
• Rings with Inquiry by Michael Janssen (Dordt University) and Melissa Lindsey (University of Wisconsin-Madison). The goal of this book is to explore the idea of factorization from an abstract perspective.
• Group Theory: Birdtracks, Lie’s, and Exceptional Groups by Predrag Cvitanović (Georgia Tech).

Real Analysis

• Introduction to Real Analysis by Dana C. Ernst (Northern Arizona University). Free course materials for an undergraduate real analysis course that utilizes an inquiry-based learning approach. These notes are a modification of materials written by Karl-Dieter Crisman (Gordon College), which are an adaptation of the materials by W. Ted Mahavier (Lamar University) listed below.
• Analysis by W. Ted Mahavier (Lamar University). These notes are for an IBL real analysis course and are available from the Journal of Inquiry-Based Learning in Mathematics.
• How We Got from There to Here: A Story of Real Analysis by Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia). This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.
• Real Analysis by Gary Towsley (SUNY Geneseo). This text is a conventional coverage of Real Analysis for undergraduate students. The real numbers are developed via the Completeness Axiom. The topology of the real numbers is also explored. The coverage culminates in proving the two parts of the Fundamental Theorem of Calculus.
• Analysis WebNotes by John Lindsay Orr (University of Nebraska at Lincoln).
• Classical Real Analysis contains links to a few real analysis texts. The word on the street is that Elementary Real Analysis by Bruckner and Bruckner is excellent.
• Measure, Integration & Real Analysis by Sheldon Axler (San Francisco State University). This book is in Springer’s Open Access program. Thus the electronic version of the book is available without cost. The content and level of this book fit well with the first-year graduate course on these topics at most American universities. This textbook features a reader-friendly style and format that will appeal to today’s students.

Complex Analysis

• A First Course in Complex Analysis by Matthias Beck (San Francisco State University), Gerald Marchesi (Binghamton University), Dennis Pixton (Binghamton University), and Lucas Sabalka (Ocuvera). Written for a one-semester undergraduate complex analysis course. Free PDF and a low-cost paperback is available.

Topology

• Topology without Tears by Sidney A. Morris. An introduction to point set topology.

Homotopy Type Theory

• Homotopy Type Theory: Univalent Foundations of Mathematics by The Univalent Foundations Program (Institute for Advanced Study). Intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning. Open-source project hosted on GitHub, low-cost paperback available.

Differential Geometry

• Differential Geometry and Its Applications by John Oprea (Cleveland State University). This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. Book is published by MAA Press, but PDF is available for free.

Statistics

• OpenIntro Stats. Offers a traditional introduction to statistics at the college level. There is a suggested minimum price, but you choose to bypass this and obtain book for free.
• An Introduction to Statistical Learning by Gareth James (University of Southern California), Daniela Witten (University of Washington), Trevor Hastie (Stanford University), and Rob Tibshirani (Stanford University). Provides a broad and less technical treatment of key topics in statistical learning. Each chapter includes an R lab. This book is appropriate for anyone who wishes to use contemporary tools for data analysis. Book is published by Springer, but PDF is available for free.
• Introduction to Modern Statistics by Mine Çetinkaya-Rundel and Johanna Hardin. Book puts a heavy emphasis on exploratory data analysis (specifically exploring multivariate relationships using visualization, summarization, and descriptive models) and provides a thorough discussion of simulation-based inference using randomization and bootstrapping, followed by a presentation of the related Central Limit Theorem based approaches. There is a suggested minimum price, but you choose to bypass this and obtain book for free.
• Team Based Statistics by Tien Chih (Oxford College of Emory University). Activities and exercises for easily implementing Team-Based Learning in an introductory statistics classroom.
• Bayes Rules! An Introduction to Applied Bayesian Modeling by Alicia A. Johnson, Miles Q. Ott, Mine Dogucu. The primary goal of the book is to make modern Bayesian thinking, modeling, and computing accessible to a broader audience.

Computing & Programming

• An Introduction to Functional Programming Through Lambda Calculus by Greg Michaelson (Heriot-Watt University). PDF available for free.
• Sage for Undergraduates by Gregory V. Bard (University of Wisconsin Stout). PDF available for free.
• Computational Mathematics with SageMath by Paul Zimmermann et al. PDF available for free.
• Peering into Mathematics through Sage-colored Glasses by John Harris (University of Southern Mississippi), Karen Kohl (University of Southern Mississippi), and John Perry (University of Southern Mississippi). This free book serves as an introduction to mathematical computation and incorporates Sage.