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orthonormal comments on The Best Textbooks on Every Subject - Less Wrong

167 Post author: lukeprog 16 January 2011 08:30AM

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Comment author: orthonormal 24 April 2013 01:05:09AM 7 points [-]

Calculus: Spivak's Calculus over Thomas' Calculus and Stewart's Calculus. This is a bit of an unfair fight, because Spivak is an introduction to proof, rigor, and mathematical reasoning disguised as a calculus textbook; but unlike the other two, reading it is actually exciting and meaningful.

Analysis in R^n (not to be confused with Real Analysis and Measure Theory): Strichartz's The Way of Analysis over Rudin's Principles of Mathematical Analysis, Kolmogorov and Fomin's Introduction to Real Analysis (yes, they used the wrong title; they wrote it decades ago). Rudin is a lot of fun if you already know analysis, but Strichartz is a much more intuitive way to learn it in the first place. And after more than a decade, I still have trouble reading Kolmogorov and Fomin.

Real Analysis and Measure Theory (not to be confused with Analysis in R^n): Stein and Shakarchi's Measure Theory, Integration, and Hilbert Spaces over Royden's Real Analysis and Rudin's Real and Complex Analysis. Again, I prefer the one that engages with heuristics and intuitions rather than just proofs.

Partial Differential Equations: Strauss' Partial Differential Equations over Evans' Partial Differential Equations and Hormander's Analysis of Partial Differential Operators. Do not read the Hormander book until you've had a full course in differential equations, and want to suffer; the proofs are of the form "Apply Theorem 3.5.1 to Equations (2.4.17) and (5.2.16)". Evans is better, but has a zealot's disdain of useful tools like the Fourier transform for reasons of intellectual purity, and eschews examples. By contrast, Strauss is all about learning tools, examining examples, and connecting to real-world intuitions.

Comment author: TimS 24 April 2013 01:15:05AM 1 point [-]

Spivak was a lot of fun - and very readable. Amusing footnotes, too. (I still remember the rant against Newtonian notation for derivatives).

Comment author: [deleted] 24 April 2013 01:17:28AM 0 points [-]

If you like Spivak, they've reprinted his five volume epic on differential geometry. It's pretty glorious.

Comment author: [deleted] 24 April 2013 01:15:30AM 0 points [-]

I'm confused. Did you mean the entire 4-volume set of Hormander -- in which case, it's not remotely comparable to Evans or Strauss -- or the first volume that you linked -- in which case, it's not even really about PDEs?

In terms of introductory PDE books, I'd favor Folland over all three.

Comment author: komponisto 25 April 2013 02:21:16AM 0 points [-]

And after more than a decade, I still have trouble reading Kolmogorov and Fomin.

Huh. I've always liked Kolmogorov and Fomin. (And shouldn't it be under "Real Analysis and Measure Theory"?)

Have you looked at Jost's Postmodern Analysis, by chance? (I found the title irresistibly curiosity-provoking, and the book itself rather good, at least if memory serves.)

Comment author: lukeprog 30 October 2014 07:54:41PM 1 point [-]

Thanks! Added.

Comment author: Princess_Stargirl 04 November 2014 01:22:35AM *  3 points [-]

In my opinion the "good stuff" in evans is in chapters 5-12. Evans is a pretty good into book on the modern "theory" of Linear and Non-linear PDEs. Strauss by comparison is a much less demanding book that is concerned with concrete examples and applications to physics. (less demanding is a good thing if the material covered is similar, but in this case its not).

Possibly Strass is overall the better book. And I really dislike Evan's chapter 1-4 (he does not use Fourier theory when it helps, his discussion of the underlying physics of some equations is very lacking, etc). But directly comparing Strauss and Evans seems odd to me. The books have very different goals and target audiences.

If the comparison is evans 1-4 vs strauss then I too would recommend Strauss. And this restricted comparison makes a ton of sense imo.

Comment author: orthonormal 09 November 2014 09:06:49PM 1 point [-]

I'll agree with that. Evans would be better for a second course on PDEs than a first course.