# On Learning Tensor Calculus

I am highly intrigued in knowing what tensors are, but I don't really know where to start with respect to initiative and looking for an appropriate textbook.

I have taken differential equations, multivariable calculus, linear algebra, and plan to take topology next semester. Since Summer break is right at the corner, I was wondering if anyone would tell me if my mathematical background is able to handle tensor calculus, or if I need to know other subjects in order to be ready to learn about tensor. Any and all help would be appreciated.

• I would have immediately suggested that linear algebra and multivariable calculus would be invaluable, and the other two certainly will be helpful. Have you already done a little research on what books you might use? – rschwieb May 8 '15 at 11:58
• I was thinking about "Introduction to Tensor Calculus, Relativity and Cosmology" by D.F. Lawden. I probably should have included this earlier, but I would like a physics-related approach to tensors and tensor calculus. – Hugo Bethancourt May 8 '15 at 12:01
• Which is stronger: your mathematics background or your physics background? – rschwieb May 8 '15 at 12:04
• I am planning a double major on both, but I would say that my physics background is a bit stronger than my mathematics one. – Hugo Bethancourt May 8 '15 at 12:06
• Hmm.. then your plan to pursue the physics route is probably the best for you. Personally, of the handful of books on the topic that I've read which were written by physicists, I found them to be the least readable and most confusing. I think it's just a discipline thing, though. Apparently physicists can follow them, and I imagine they say similar things about texts written by mathematicians. – rschwieb May 8 '15 at 12:08

What is a tensor? In short, a tensor is a generalization of a vector which is needed to express physical quantities which have more data than we can fit into a single vector field. However, it's more than that. We also need tensors of different transformation type. Ultimately, in physics, we wish to write equations which are independent of the choice of coordinates. Yet, we use coordinates. So, this brings you to the focus on components which transform inversely. With objects whose transformation properties are mirrored we are able to create scalars which are invariant. In math, a tensor product of vector spaces is a way of multiplying spaces. Or, for specific matrices, the tensor product is the Kronecker product which is pretty easy to understand calculationally; for $A \otimes B$ we just make a new matrix with blocks formed by $AB_{ij}$. A tensor is simply a multilinear mapping on a vector space and its dual. The tensor products of the basis and dual basis of the vector space are used to build a natural basis for the tensors over the given vector space. Of course, then from a manifold perspective, this is all just "at a point", we then wish to consider tensor fields... of course there is much to learn. I think Lawden is a good book, I used it in a General Relativity course, it was readable and I got a good amount out of it. As a beginning physics student, I used the big black book Gravitation by Misner Thorne and Wheeler. There's about 100-200 pages of plain old tensors and forms which are helpful, lots of exercises. From a physics perspective it was good. From a math perspective, it's not optimal. I thought the math in Sean Carrol's General Relativity text was also quite good, and a bit more modern than MTW.

All of this said, I suspect the book that you would enjoy is: Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists by Paul Rentein.

It's not just about tensors, and, I think that's a good thing. Tensors are part of a larger story and this book is written for someone with your general leaning.

which is Øyvind Grøn, Arne Næss: Einstein's Theory: A Rigorous Introduction for the Mathematically Untrained

This is a serious book! the second autho is (well, was ...) a retired philosopher which required from the first author that everything in the book should be understandable for him, without a need of pencil and paper. Still it reaches some graduate level. By the way, the philosopher author was very famous, at least in Norway.

The first review on Amazon says: "This is simply the clearest and gentlest introduction to the subject of relativity. This is NOT a popular reading, it does get into the math but the introduction and pace is what sets this text apart. The authors starts from baby steps and builds up the theory. The pace is easy and explanation are clear and the mathematical details are not left to the reader to figure out like in most text books.The physics is also explained very clearly From an electronics engineering background and not having dealt with tensors before, this text help me bridge the gap and allows me to fully appreciate the machinery underlying this wonderful theory.I have tried reading Schultz,Lawden etc to understand the subject on my own but this text is a much better intro to the subject." and other reviews are also very positive.

For a more mathematically flavored introduction, consider studying:

• Vector Calculus, Linear Algebra, and Differential Forms, by Hubbard and Hubbard

• Semi-Riemannian Geometry with an Introduction to Relativity, by O'Neill

• A good advanced linear algebra book. This should be one that focuses on abstract vector spaces and multilinear algebra, not Gaussian elimination. I hear the one by Roman is good.

The first one is an introduction to multilinear algebra and its relationship with calculus, pitched at undergraduate students. The second is a book on the geometry required for relativity, but the first two chapters are a relatively rigorous, if perhaps terse, introduction to the linear algebra required to do tensor analysis on a manifold.

In my own experience, when I was at your stage, I had seen some tensor analysis and had no idea what was going on. It wasn't until I read O'Neill's book pretty carefully that the subject "clicked" for me.