# Tensor Analysis Workbook

## Main Question or Discussion Point

Would anybody have some good recommendations for a workbook on tensor analysis?

I'm looking for the kind of book that would ask a ton of basic questions like:

"Convert the vector field $\vec{A}(x,y,z) \ = \ x^2\hat{i} \ + \ (2xz \ + \ y^3 \ + \ (xz)^4)\hat{j} \ + \ \sin(z)\hat{k}$ to spherical coordinates then compute it's covariant derivative. Re-express the Christoffel symbols for this field in terms of the metric tensor."

yada yada yada, basically anything with a lot of mechanical questions to practice the material. I know there's the schaums book but there aren't many questions like the above in it. If this were a question about calculus, that book would be like using the schaums calculus book whereas I'm looking for more along the lines of Demidovich, if such a thing exists. Since I've done a lot of searching I don't think such a thing will be obvious but there might be such a thing hidden away in an engineering book or a good math methods book, any ideas? Thanks!

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I bought the first edition of Analysis and Algebra on Differentiable Manifolds and got a lot out of it, for the most part. It has a mix of theory and applications.

My other suggestion would be to try and pick up a "mathematical methods in physics/engineering" book.

Thanks a lot for the suggestion, I was able to get my hands on it & it's really really great (!) for more advanced manifolds theory but unfortunately it's not what I was looking for as regards this thread.

I was hoping for something more along the lines of Borisenko, if you read those two pages you'll see what I'm talking about, just blindly differentiating some vector field using your intuition then whipping out the metric tensor & christoffel symbols to re-express it.

The problem is that I need a ton of questions, I'm not asking the right questions. For example one book will tell you how to convert between coordinate systems while another will ask you to describe all these geometric objects in different coordinate systems, while another will ask you to do the same thing using tensor notation, but that's three different books to answer the same question along practically the same lines. Then other ones will come up with many christoffel symbol identities but none will give any use for them, & none will even ask an example of finding the covariant derivative of a vector field or something. I'm finding myself spending so much time accumulating the right questions that I have no time to sit down & do them!

I'm really just hoping there are one or two books that ask just about every question one can imagine so I'm not finding myself confused with elementary questions, some book that really makes you appreciate the power of being able to set up integrals in oblate spheroidal coordinates for example & would even go so far as to compare this approach with another coordinate system etc...