Mathematical physics text with plenty of applications

Question

I'm looking for texts on mathematical physics. I've seen various other threads, but the texts recommended in those threads were mathematical methods of theoretical physics texts, that is to say those texts focused on the math methods useful for physicists and assumed the applications of those methods would be covered in some other text. What I'm looking for is a book which introduces some math and then looks at its applications (back and forth) rather than a text which just goes over some math. (Example of books which are NOT what I'm looking for: Mathematics for physicists by Dennery, Math methods for physicists by Weber and Arfken etc)

I've taken courses in multi-variable calc, linear algebra, real analysis, and I'm currently taking a course in abstract algebra. But so far I've only seen applications of multi-variable to physics. I'm looking for books which look at the applications of the other subjects mentioned above to physics, of course books which cover subjects beyond those mentioned are also welcome.

So far I'm liking the flavor of the text "A Course in Modern Mathematical Physics" by Szekeres, but I'd prefer a text which got to the physics side of things more quickly (in the Szekeres text the first eight chapters are strictly on pure math).

After some amazon searching it looks like Theoretical Physics by Joos, Mathematical Physics by Henzel, and Physical Mathematics by Cahill may be good bets?

Answer 1

For an introductory book on the topic, consider "Numerical and Analytical Methods for Scientists and Engineers, Using Mathematica" by Daniel Dubin (ISBN-13: 978-0471266105).

Whats sets this book apart from the rest is that it combines theoretical physics, teaches the math, and solves practical physics problems both by hand and by using Mathematica. This book is at the advanced undergraduate level (junior or senior level).

Some examples include solutions to the heat equation, wave equation, electrostatics. Non-linear phenomena are also briefly discussed. On the math side, there's a lot of emphasis on Sturm-Liouville theory and methods for solving such problems. It covers both analytical techniques and numerical methods such as some discretization techniques and finite-element analysis using Mathematica in the context of the above-mentioned physics problems.

Answer 2

I think you should have a look at Reed&Simons classic book on mathematical physics ("Methods of Modern Mathematical Physics", 4 volumes). Excellent and clear writing style, many further references and it covers most of the important analytic methods which are used in physics.

For geometry stuff, I recommend Bishops "Tensor analysis on manifolds". Very clear writing style, many applications, and very low cost. However, it does not cover all of the important topics.

If you are looking for algebra stuff, there is a good book by McWeeny on discrete symmetries (Symmetry: An Introduction to Group Theory and Its Applications), really wonderfully written. Sadly, it does not cover continuous groups. For these, I would recommend the classic book by Gilmore - "Lie Groups, Lie Algebras, and Some of Their Applications", which also covers many applications, as the title already implies.

Good luck out there!