Learn about tensors for physics

Question

Can I have suggestions for some good book regarding tensors for physics.

Answer 1

I suggest that you read what is said about tensors in three General Relativity books:

1. Spacetime and Geometry - An Introduction to General Relativity - Sean Carroll - Chapter 1, specially sections 1.4, 1.5, 1.6 and 1.7, but it is good to read all of it. If you want to understand tensor fields, and not just tensors, then you should read the whole Chapter 2;

2. Gravitation - Misner, Thorne, Wheeler - Chapters 2 and 3. Specifically about tensors in general there is section 3.2, but I suggest reading the two chapters to actually understand the point. This one gives quite a nice geometric intuition, specially about differential forms (completely skew-symmetric tensors);

3. General Relativity - Robert Wald - Chapter 2 and specially Section 2.3 which gives a quite self-contained description of what are tensors in general and why the definition makes sense for physics. Again, if you want to understand tensor field, though, you should read the whole chapter;

Those books are certainly for physicists, as they cover standard General Relativity, and all they have inside then discussion about what are tensors and how to manipulate them in a modern way. Of course, it is not because they are GR books that what they say about tensors is specific to GR.

In other words: it is good to understand tensors as multilinear maps instead of as sets of components plus a transformation property. Furthermore, the definition as multilinear maps shows why, when working totally in coordinates, the transformation definition makes sense. Without it, the transformation definition seems like it came out of thin air.

Finally, if you want a more advanced treatment, I suggest you look at

4. Linear Algebra and Geometry - Kostrikin and Manin - Chapter 4.

But I would suggest you only do so after you feel comfortable with the usual approach.

This book is quite though with a quite abstract approach. For starters, in physics, the first approach is already quite good. There are advantages, though. In particular, it discusses the tensor product that appears in QM.

By the way, if you ever want to learn this other approach and find the definition of the tensor product confusing, I suggest to check out these questions on Math.SE:

• Kostrikin's Definition of Tensor Product
• Why is the tensor product constructed in this way?

Answer 2

In my opinion, there are two major ways to learn tensors for physicist, depending on the desired depth of understanding. The first, learning it through a physics text, in which vector fields, one-forms, and tensor fields are introduced in a quite hand-wavy way using tons of pictures and illustrations. The benefit of this approach, is that you can learn it fast and you can do ton of calculations in GR without truly understanding what you are doing. To achieve this, i recommend Sean Carroll's text, as it is probably the more mathematically incline physics text.

The second approach, is to learn tensor fields through differential geometry, where it actually originated. The upside is that you will truly understand the geometrical aspect of general relativity and can further study to more theoretical topics later on. For this approach, i recommend Semi-Riemannian Geometry: With Applications to Relativity by Barrett O'Neill (chap 1-3) if you are in a rush. The pre-requisite is a working knowledge of basic general topology and multivariable analysis (Jacobian matrix etc).

If you have time, you can approach more elaborate texts such as John Lee's Smooth Manifolds (chap 1-3,8,12-13) + Riemannian Manifold's (as much as you need). John Lee's text are known to be verbose and friendly to readers, intutive but rigorous. The downside of his text's are, they are quite long. I believe his text's tend to attract theoretical physicist, given his style. Another good mathematical text for physicist is Chris Isham's Modern Differential Geometry for Physicist text (chapter 1-3). Chris Isham was a leading theoretician in the UK, and his writing style is nice for physicist, where he focuses on intuition rather than rigirous proof's. Although he covers precisely the required structures. His text does'nt cover Riemannian geomtry though. If you would like to know where the inspiration of tensor products came from, the best book i've come across that elaborates completely this notion is Steve Roman's Advanced Linear Algebra (chap 3 & 14). I recommend learning all your linear algebra from Steve Roman honestly, its a wonderful exposition.

Lastly, if you want to learn tensors for special relativity purposes only, look no further, check out Éric Gourgoulhon Special Relativity in General Frames. Probably the best book to introduce some tensor notation in general relativity as well.

Answer 3

An Introduction to Tensors and Group Theory for Physicists is a good book, recommended to me by some professors in my institution. It doesn't go into too much of abstract algebra, other than what's absolutely necessary.

Another really good book would be A Student's Guide to Vectors and Tensors by Daniel A. Fleisch.

Some other recommendations for physicist approach of tensors may be found here. I'm not quoting from there, because I've already mentioned the best books.

Answer 4

I learned about tensors from Schutz's book about general relativity ("A first course in general relativity") and from Wald's general relativity book ("General relativity").

These are not strictly books about tensors, but since they form a lot of the mathematical apparatus of GR, they are explained well there (and with physical interpretations and applications).

Answer 5

See Nuclear Shell Theory by Amos de Shalit and Igal Talmi. The book is constructed from two parts and one for tensors a total of 586 pages. My thought is he dived into some confusing details which should be read carefully. The following parts consider applications that makes a good deal with mathematics.