I haven’t figured out a good way to make Hermitian matrices seem inevitable and natural (which is the way I want these posts to read). Given their definition, you can show why Hermitian matrices have the properties quantum mechanics needs, but that’s not the point of this series of posts. In the meantime, if you’re long out of college , like math, and aren’t in a rush, here are some math books which I’ve found excellent for self-study.

The best book on calculus is fairly old — first written in 1967 and now in its third (and last) edition of 1994– Calculus by Michael Spivak. It’s light years from Thomas, but just as user friendly. Mathematicians are fond of saying that it isn’t a spectator sport. Thomas’ exercises are plug and chug, Spivak’s are blink and think. Unless you’re a mathematician who missed your true calling, you’ll need the answer book.

Spivak really explains what’s going on under the hood. Nothing is left out. He starts with some known properties of numbers and goes on from there. He even constructs the real numbers from the rationals.

Spivak’s book has been described as an analysis book rather than a book on calculus. It is, in the sense that everything is built from the ground up. No smoke and mirrors. However, there’s lots of analysis he doesn’t cover. If you feel you know calculus pretty well, then get Stephen Abbott’s book “Understanding Analysis”. It’s well written, and rather than present the results of analysis as a fait accompli, each chapter begins with a a motivational section which makes you see why the section requires, well, analysis. For instance, the chapter on limits begins the infinite sequence

((-1)^n)/n

which allows you to group terms to get the associated series to converge to anything you want. That certainly gets you thinking about convergence and what a limit really is.

Spivak’s Calculus book is resolutely one dimensional. For advanced calculus in 3 or more dimensions you can’t beat the book by the Hubbards’ “Vector Calculus, Linear Algebra and Differential forms.” It’s extremely user friendly, probably because one of the authors (Barbara Hubbard) is not a mathematician but an excellent writer who wrote an extremely accessible book on wavelets. So her husband had to explain it all to her, with the result that the book is full of sidebars with explanations of all the traps, misunderstandings and ambiguities which arise when learning the stuff. The discussion of manifolds is pretty concrete, but OK. If you really want to know what’s going on, John Lee’s books “Introduction to Manifolds” and “Introduction to Topological Manifolds” are better and user friendly — but you had better know a fair amount of math before attacking them, as they are written at the graduate level.

For complex number theory, there is simply nothing like “Visual Complex Analysis” by Tristam Needham. Again, extremely user friendly, if idiosyncratic, with a diagram on nearly every page. Definitely not dry.

I learned my linear algebra (this time around) using “Linear Algebra Done Right” by Sheldon Axler. It’s extremely clear, but there is no motivation, no physical applications. The logical bones are laid bare (and left there).

Finally, if you are starting out with these or any recently written books, do not hesitate to Email questions to the authors about things you don’t understand (or what you think is a misprint or an error). This is a huge advance for the self-studier, and one undreamt of until the internet. My experience is that they always respond. One responded so quickly, that my wife said “he must not have much to do”.

Do not be surprised if you find errors. Math books must be the world’s hardest to proofread. I’m a very slow math reader, since I’m reading for understanding, so I grind to a halt with anything I don’t understand. Most of the time it’s me but sometimes it’s an error, which authors love to hear about (they really do) so they can put them on the errata page (usually posted on the web — another great advance).

For the vast majority of humanity, reading math for fun is like having root canals for fun, but for the aficionado it’s meat and drink. Enjoy

## Comments

Is this Spivak’s “Calculus on Manifolds”? I would also very strongly recommend Robert Bartle’s “Introduction to Real Analysis”. He extensively establishes all the foundations of the subject; the derivative is not introduced until the 6th chapter. The book is simple but rigorous and I spent a lot of time understanding the proofs and working out the problems. Another fantastic book which I spent a huge amount of productive time understanding and which I am again trying to read now is “Introduction to Topology and Modern Analysis” by George Simmons. Again, simple but rigorous, and I remember spending hours mulling over a proof or two with a professor. I wonder if they use Apostol these days.

No, It’s just Spivak’s “Calculus” pure and simple. He also wrote a 5 volume monster series on differential geometry, apparently pitched at some (incredibly advanced) undergraduates at Brandeis. You have to know an awful lot of math to start reading it. I didn’t get very far. Thanks for the other tips.

The one I am talking about is the following one. It’s a very unusual book; I think you will love it

http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219/ref=ntt_at_ep_dpt_2

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