Tensor Calculus: Modern Books with Physics Examples

[Amok]

Hello, could someone recommend a good book on tensor calculus? I'd like it to be relatively modern (I have an old book) and maybe contain some examples drawn from physics. Chapters on related subjects such as differential forms and calculus of variations would be a plus.

Cheers.

 

[CJ2116]

It's kind of old (Written in the late 60's), but I have used Tensor Analysis on Manifolds by Bishop and Goldberg as a reference for a while now with good results. It can be a little concise, but for the price I'm more than satisfied!

I have only read the first few chapters in it, but The Geometry of Physics is an excellent book, I use it as an occasional supplement to Wald's General Relativity. It has a lot of worked examples, illustrations and exercises. Covers most of the standard tensor calculus stuff, but with an emphasis (obviously) on physics. To be honest, though, I found the notation and presentation kind of hard to follow.

 

[Amok]

Well, I was looking for something relatively modern, or at least with modern editions. Thanks for replying. Anyone read this one:

https://www.amazon.com/dp/1842654489/?tag=pfamazon01-20

 

[George Jones]

I, too, like Bishop and Goldberg (despite its vintage, it has a relatively modern style) and Frankel. Another possibility is Fecko.

As n!kofeyn has stated, contents of differential geometry references vary widely. Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,

https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20.

This book is not as rigorous as the books by Lee and Tu, but it more rigorous and comprehensive than the book by Schutz. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge firld theories than traditionally is presented in physics courses.

Fecko has an unusual format. From its Preface,

A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).

The book is reviewed at the Canadian Association of Physicists website,

http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.

From the review

There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...

A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.

Personal observations based on my limited experience with my copy of the book:

1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in, say, Nakahara;
3) the simple examples are often effective.

 

[Daverz]

Bishop & Goldberg is certainly "modern" enough.

If you want something with a newer publish date, you could try

Frankel, Geometry of Physics
Szekeres, A Course in Modern Mathematical Physics