Which book should I buy for Tensor Analysis?

[Mmmm]

I'm just starting out with learning a bit of general relativity and have read the first 3 chapters of Schutz's 'A first course in General Relativity' (up to and including the Tensor analysis chapter). I have managed to do about 90% of the exercises but I don't really feel confident with it. I've decided that I need to do some studying of tensor analysis so that I can feel at ease with this subject but I'm not sure which book to buy.

I have also read the first couple of chapters of Schutz's 'Geometrical methods of mathematical physics' in order to get some more insight into this, but I think that the problem is that Schutz never gives any examples and so when I tackle a problem I'm never sure if I did it in a 'good' way or not.

I have a copy of Heinbockel's 'Introduction to tensor calculus and continuum mechanics' which I have heard is nice and slow and gives lots of examples, but this book uses the old index notation and I'm not sure if it is worth my while reading this or not. Will one method help understanding with the other or should I avoid the index notation altogether?

I have seen mentioned in this forum Goldber's 'Tensor Analysis on Manifolds'. does this have lots of examples? and the Schaum book I know has lots of examples but is this another index notation book?

 

[Fredrik]

Do you have a copy of Wald's book too? I felt that the two of them were sufficient to get me to understand tensor fields. Schutz explains tensors extremely well in my opinion (in the SR part of the book) but he's just talking about some arbitrary vector space and some basis of that vector space, so it's not easy to see how this relates to manifolds. Wald defines the tangent space and explains how a coordinate system can be used to construct a basis for the tangent space. (Schutz didn't explain this well if I remember correctly). Now you can apply everything you've learned from Schutz, with the vector space being T_pM (the tangent space of M at p) and the basis vectors being the derivative operators \partial/\partial x^\mu|_p. After that it's pretty easy to understand e.g. vector fields as local sections of the tangent bundle.

It might be a good idea to also read (the interesting parts of) some book that uses the index free notation, but unfortunately I don't know what the best recommendation is. Wald uses the abstract index notation, which must be the best one by far when you have to construct new tensors from old ones using the operation of contraction, but is (in my opinion) a little awkward in definitions of e.g. a connection or the curvature tensor.

 

[Mmmm]

I was lead to believe that Wald's book was a fair bit more difficult than the Schutz and so planned on reading it after I had finished Schutz. Do you suggest dipping into both simultaneously then? I really want to get to grips with this before I continue as it is pretty important stuff.