Useful Online Advanced Math Textbooks?

On LinkedIn just recently a mathematics student, Benjamin Bowden,  posted this question.

Does anybody know of any useful, accessible, and manageable (preferably free) online textbooks for undergraduate math courses? I found this one (http://www.math-cs.gordon.edu/~kcrisman/mat338/index.html) for Number Theory very good when I used it and I'm wondering if anybody knows of any others.

I think a bit of time googling this question might have led somewhere, but perhaps not.  One professional mathematician made this suggestion:

There is Diestel's book on graph theory. Starts from the elementary but soon goes to the advanced.
http://diestel-graph-theory.com/

11 hours ago

Readers of this blog (if any) will know about the following.  But I repeat it here in case there is anyone out there asking the same question and stumbling, for the first time on this blog.

I believe that this is the future of textbooks, especially in mathematics. Having published with a major commercial publisher of textbooks who shall remain nameless (Pearson!) who priced our textbooks in the stratosphere, I find now it much more sensible to offer free PDFs and inexpensive paperbacks directly.

All of our textbooks are online at 
http://classicalrealanalysis.info/com/Home.php 
and PDF copies are free. Mostly these are real analysis textbooks from an elementary level up to graduate.

Also there is a more general source of free mathematical textbooks available 
at 
http://aimath.org/textbooks/approved-textbooks/ 
collected and authenticated by the American Institute of Mathematics.

 Students need all the financial assistance they can get nowadays.  Free textbooks are an excellent way to help in a small way.  We academic mathematicians can make better use of our expository talents than enriching the commercial publishing industry.

Sample pages from Chinese Edition

To Download sample pages!

Link to Amazon for the Chinese translation of TBB

Here is a link to the Amazon listing for this Chinese language textbook, published today.

Chinese Language Edition of "Elementary Real Analysis"

Just published today!

This is a Chinese language translation of the first eight chapters of our elementary real analysis textbook.

It can be purchased directly from CreateSpace or from Amazon.com by following the links below.  I will shortly put up some sample pages on our web site ClassicalRealAnalysis.com.

This excellent and accurate translation was done by two accomplished mathematicians: Hongjian Shi and Lei Zhang.

 

• Publication Date:  Nov 13 2013
• ISBN/EAN13:  1492864188 / 9781492864188
• Page Count:  332
• Binding Type:  US Trade Paper
• Trim Size:  8" x 10"
• Language:  Chinese
• Color:  Black and White
• Related Categories:Mathematics / Mathematical Analysis

 

Purchase information on CreateSpace.

Purchase information on Amazon.

What We Are Doing about the High Cost of Textbooks

There is an interesting article in the current issue of the Notices of the American Mathematics Society (August 2013)  from the AIM (American Institute of Mathematics) Editorial Board.   The first paragraph:

What We Are Doing about the High Cost of Textbooks

 Let’s begin with the obvious: The price of textbooks
has risen much faster than the cost of living over
the last thirty years, but there has not been a
significant increase in their quality. We don’t
propose to analyze the economic and educational
factors that underlie this phenomenon. Instead,
we will describe our efforts to help lower the
cost of textbooks for standard undergraduate
mathematics courses in North American colleges
and universities.
Here is a link to the entire article:

http://www.ams.org/notices/201307/rnoti-p927.pdf

In our own smaller way we have done the same thing since 2008 with our mathematics textbooks.  All of our texts are available as FREE PDF files as well as inexpensive paperbacks from either Amazon or CreateSpace.  I expect that this is the future of publishing for academic mathematicians if not for all disciplines. 

Our experience with publishing in the traditional way with a well-known publisher, who should remain nameless (Pearson!), was not at all rewarding.  The texts were outrageously expensive for the students, they offered minimal assistance in preparing the manuscript, scarcely any promotion, modest royalties, and (worst of all) they simply dropped the title from their list (without informing us) when they thought they could make more money with another one. Creating our web site classicalrealanalysis.com and distributing our texts there has been much more rewarding.

We have no connection (as yet) with the AIM people who are a part of a
larger NSF project1 to develop open source software and curriculum materials for undergraduate mathematics education.  But we agree that this is the future.




 1 Information about Project UTMOST (Undergraduate Teach-
ing in Mathematics with Open Software and Texts) can be
found at http://utmost.aimath.org.

Updated files (June 2013) for THEORY OF THE INTEGRAL

I have updated the FREE PDF file for the text

     THEORY OF THE INTEGRAL

          


Click here for the PDF.

Click here to visit the web page on the site.


Thanks go to one of our more alert readers (Tom Savage) for spotting that the correction factor in the integration by parts formula for the Stieltjes integral was misstated (i..e, wrong!) in Section 5.7.

This version of the file has this error corrected.  I also made a small addition to Section 1.2.  Adam Besenyei supplied me with a reference to an early paper of the American mathematician Osgood who addressed the problem discussed in that section.

Review of "The Calculus Integral"

Our textbook THE CALCULUS INTEGRAL has just been reviewed in the most recent issue of the American Math Monthly:

Table of Contents June/July 2013 Math Monthly

The review, by David Bressoud,  covers

• Calculus by Michael Spivak; 

• Calculus Deconstructed: A Second Course in First-Year Calculus by Zbigniew Nitecki;  

• Approximately Calculus by Shahriar Sharhriari;  

• A Guide to Cauchy's Calculus: A Translation and Analysis of Calcul Infinitesimal by Dennis M. Cates; 

and  

• The Calculus Integral by Brian S. Thomson

 I cannot provide a link to the entire review (which should be of interest to anyone following the teaching of integration theory).  But maybe fair use allows me to quote two of the paragraphs of interest to readers of our text:

...``The final book is the most challenging of all, Brian Thomson’s The Calculus Integral. The author introduces it as appropriate for an honors course in calculus, but it would require some very special students. I could see it used for a senior seminar. Every proof, every example, and every counterexample is left as an exercise. The proofs
are scaffolded and answers are provided in the second half of the book, but if approached as intended, working through this text would be daunting. Like Nitecki and Spivak, Thomson focuses on the theorems of calculus. He begins with sequences, the careful derivation of properties of continuous functions, and the definition and basic
properties of differentiation. He takes an abrupt turn when he gets to integration. Like the other four authors, Thomson rejects Riemann’s definition of the definite integral, but he takes it a step further. He returns to Newton’s definition of the integral as antiderivative, what he calls “the calculus integral.” Thomson defines the definite integral
of f as the change in a uniformly continuous function that is an antiderivative of f at all but at most finitely many points. A function is integrable if and only if such an antiderivative exists. The fundamental theorem of integral calculus is now the observation, via the mean value theorem, that for any integrable function f and any partition
of the interval over which we integrate, for each i there exists a value xi in the ith subinterval such that the definite integral is exactly the sum over i of f (xi ) times the length of the ith subinterval. From here, he explores Riemann sums as approximations to definite integrals. Thomson’s handling of integration has the advantage that it clarifies the point of the fundamental theorem of integral calculus, which is the general equivalence of two very different approaches to integration.
 

There is much more to Thomson’s book. He continues on to the monotone convergence theorem, then to sets of measure zero, to absolute continuity, and on to define the Lebesgue integral of f as the change in a function F that is an antiderivative of f almost everywhere and is absolutely continuous in the Vitali sense. He concludes with
the Henstock–Kurzweil integral. His book is magnificent, but even if we stop before preparing for the monotone convergence theorem, it would be supremely demanding for an honors course in the first year.''

Students and teachers of integration theory should be familiar with David Bressoud  whose two "radical" books are must reads for anyone with a serious interest in this subject.