The writing of textbooks and making them freely available on the web is an idea whose time has arrived. Most college mathematics textbooks attempt to be all things to all people and, as a result, are much too big and expensive. This perhaps made some sense when these books were rather expensive to produce and distribute--but this time has passed.
Professor Jim Herod and I have written Multivariable Calculus ,a book which we and a few others have used here at Georgia Tech for two years. We have also proposed that this be the first calculus course in the curriculum here, but that is another story....
Although it is still in print, Calculus,by Gilbert Strang is made available through MIT's OpenCourseWare electronic publishing initiative.
Here is one that has also been used here at Georgia Tech. Linear Methods of Applied Mathematics, by Evans Harrell and James Herod.
Yet another one produced at Georgia Tech is Linear Algebra, Infinite Dimensions, and Maple, by James Herod.
I have also written a modest book, Complex Analysis, which I have used in our introductory undergraduate complex analysis course here.
Professor E.H. Connell of the University of Miami has made available on the web his book Elements of Abstract and Linear Algebra. You should read his insightful comments about textbooks.
An introductory algebraic topology book, Algebraic Topology I, by Professor Allen Hatcher, of Cornell University, is available, and Professor Hatcher promises the second volume, Algebraic Topology II, will be ready soon.
The Geometry and Topology of Three-Manifolds, by William Thurston. This is an electronic edition of the 1980 lecture notes distributed by Princeton University.
Professor Jim Hefferon of Saint Michaels's College has made available his undergraduate textbook Linear Algebra.
Another elementary linear algebra textbook is Elementary Linear Algebra, by Keith Matthews.
Introduction to Probability, by Charles Grinstead & J. Laurie Snell.
An Introduction to Probability and Random Processes, by Gian-Carlo Rota and Kenneth Baclawski. This is the 1979 manuscript of the work Professor Rota had been working on for some time. It is made available through the efforts of David Ellerman.
Professor Herbert Wilf has made available his book generatingfunctionology.
Another one by Professor Wilf is Algorithms and Complexity.
A = B, by Marko Petkovsek, Herbert Wilf, and Doron Zeilberger.
Furnished by David Joyce is a version of perhaps the greatest textbook of them all, Euclid's Elements.
Daniel Callahan has also provided what he calls a "an open textbook based on Euclid's Elements", Euclid's 'Elements' Redux.
Originally published by Springer-Verlag, the book A Course in Universal Algebra, by Stanley Burris, and H. P. Sankappanavar, is available online.
Professor Robert Ash has written and made available Abstract Algebra:The Basic Graduate Year.
Another one by Professor Ash is A Course In Algebraic Number Theory.
Professor Ash has also completed and made available A Course in Commutative Algebra.
Originally published by Prindle, Weber & Schmidt but currently out of print, Elementary Calculus: An Approach Using Infinitesimals, by Professor H. Jerome Keisler, is now freely available online.
Graph Theory, by Reinhard Diestel.
Available for self-study from The Trillia Group is Basic Concepts of Mathematics, by Elias Zakon.
Another one from The Trillia Group is An Introduction to the Theory of Numbers by Leo Moser.
Also from The Trillia Group are Mathematical Analysis I, and Mathematical Analysis II, by Elias Zakon.
Thanks to Malaspina Great Books, Mechanism of the Heavens (1831), by Mary Somerville, is available online. This second edition was prepared by Russell McNeil.
Lecture Notes on Optimization, by Pravin Varaiya. This is a re-issue of a book out of print since 1975. It is an introduction to mathematical programming, optimal control, and dynamic programming.
Published by Cambridge Press, but still freely available is Mathematical Ilustrations, by Bill Casselman.
A Problem Course in Mathematical Logic, by Stefan Bilaniuk .
Concepts and Applications of Inferential Statistics, by Richard Lowry.
Published by Cambridge University Press, A Computational Introduction to Number Theory and Algebra, by Victor Shoup is nevertheless freely available on-line.
Yet another one out of print, but now freely available is Convergence of Stochastic Processes, by David Pollard.
Designed for undergraduate physics students is
Mathematical Tools for Physics, by James Nearing.
A First Course in Linear Algebra, by Rob Beezer.
Group Theory, by Pedrag Civitanovic.
Shlomo Sternberg has written Theory of Functions of a Real Variable.
Lie Algebras is another one by Professor Sternberg
Yet another one by Professor Sternberg is Semi-Riemann Geometry and General Relativity
Advanced Calculus, by Lynn Loomis and Schlomo Sternberg
Difference Equations to Differential Equations, by Dan Sloughter.
The Calculus of Functions of Several Variables is another one by Professor Sloughter.
Notes on Differential Equations, by Bob Terrell.
Sets, Relations, Functions, by Ivo Düntsch and Günther Gediga.
Another one by Düntsch and Gediga is Rough Set Data Analysis.
Predicative Arithmetic, by Edward Nelson.
Toposes, Triples and Theories, by Michael Barr and Charles Wells. This was orinally published by Springer-Verlag, 1985
Also by Barr and Wells is Category Theory for Computing Science.
Information Theory, Inference, and Learning Algorithms, by David J. C. MacKay is published by Cambridge University Press, but is, nevertheless, freely available online.
Numerical Methods and Analysis for Engineers, by Douglas Wilhelm Harder.
Analysis of Functions of a Single Variable, by Lawerence Baggett, was originally written to be used for a one semester senior course, but the author suggests that it is more appropriate for first year graduate students.
Another one by Baggett ia Functional Analysis.
Convex Optimization, by Stephen Boyd, and Lieven Vandenberghe is freely available thanks to Cambridge University Press.
Introduction to Statistical Signal Processing, by R. M. Gray and L. D. Davisson is, according to Professor Gray, a "...much revised version of the earlier text Random Processes: An Introduction for Engineers, Prentice-Hall, 1986, which is long out of print." The current book is published by Cambridge University Press.
Abelian Categories, by Peter Freyd. This was published originally by Harper & Row.
Published by Van Nostrand in 1971 and now available is Categories and Groupoids, by P. J. Higgins.
Basic Concepts of Enriched Category Theory, by G. M. Kelley, was published by Cambridge University Press.
Abstract and Concrete Categories:The Joy of Cats, by Jiri Adamek, Horst Herrlich, and George Strecker.
A Problem Text in Advanced Calculus, by John Erdman.
Abstract Algebra:Theory and Applications, by Tom Judson.
Originally published by Pearson in 2003, Introduction to Real Analysis, by William F. Trench is now freely available.
Another by Professor Trench is Elementary Differential Equations , originally published by Brooks/Cole Thompson Learning in 2000.
Elementary Differential Equations With BoundaryValue Problems, also by Professor Trench, and also originally published by Brooks/Cole Thomson Learning in 2000.
Notes on Diffy Qs: Differential Equations for Engineers, by Jiří Lebl .
According to the authors, Proofs and Concepts: the fundamentals of abstract mathematics, by Dave Morris, and Joy Morris, provides an introduction to proofs, logic, sets, functions, and other fundamental topics of abstract mathematics.
Mathematical Biology, by Jeffrey Chasnov.
Originally published in 1970, Calculus and Linear Algebra. Vol. 1 and also Calculus and Linear Algebra. Vol. 2, by Wilfred Kaplan and Donald J. Lewis, are available thanks to the Scholarly Publishing Office of the University Library at University of Michigan.
Introduction to Calculus I and II, by J.H. Heinbockel.
Active Calculus, by Matt Boelkins.
Analytic Combinatorics, by Philippe Flajolet and Robert Sedgewick.
Mathematical Reasoning: Writing and Proof, by Ted Sundstrom. Earlier editions of this one were published by Pearson Education, Inc., but this most recent edition is now freely available.
Linear Algebra, by David Cherney, Tom Denton, and Andrew Waldron
Yet Another Introductory Number Theory Textbook, by Jonathan Poritz.
Introduction to Vectors and Tensors, Vol. 1, Linear and Multilinear Algebra and Vol. 2, Vector and Tensor Analysis, by Ray M. Bowen and C.-C. Wang are revisons of works originally published by Plenum Press. They are available from Dover Press, but made freely available here.
Introduction to Continuum Mechanics, by Ray. M. Bowen, also originally published by Plenum Press and now available from Dover and made freely available here.
Not simply an online textbook, but certainly in the same spirit is the Topology Webcourse project undertaken by Topology Atlas.