Tensor calculus independent study questions?

[brandon078902]

I'm a mathematics major and up until now I've taken Calc 1,2,3 (so single + multivariable) a combined course in Elementary Linear Algebra + Differential Equations and PDE's. My school doesn't offer any tensor calculus classes, but I was interested in learning some of it on my own. Do I have enough of a math background to understand tensor calc, and if so, are there any textbook suggestions that you could offer? I'd like something accessible to somebody at my level, because I've seen other threads say you need to know topology/ other math concepts I haven't learned yet to understand most textbooks on tensor calc. Any information would be appreciated because I'm kinda clueless!

 

[vanhees71]

Have a look at some good physics book on general relativity on this subject. My favorites are Landau/Lifshitz vol. 2 and S. Weinberg, Graviation and Cosmology. For the more modern approach with differential forms (Cartan calculus) have a look at Misner, Thorne, Wheeler, Gravitation (aka "the phone book"). There you should have enough mathematical background.

 

[Fredrik]

Tensors are often taught in courses on differential geometry, and most if not all of those books require you to know topology. But it should be possible to ignore everything that involves topology, and just study the tensors. That's what I did, a long time ago.

The best place I know to start is chapter 3 of A first course in general relativity by Schutz. You may have to read some stuff from chapters 1-2 just to understand the notation, but that shouldn't slow you down much. Schutz defines the dual space V* of an arbitrary finite-dimensional vector space V. Then he defines a tensor as a multilinear map $T:V^*\times\cdots\times V^*\times V\times\cdots\times V\to\mathbb R$. The components of a tensor are the numbers you get when you plug in basis vectors as input. The tensor transformation law is the relationship between the components of a tensor associated with two different bases.

All of this is still relevant when you study tensors in the context of differential geometry. What changes is really just that the space V is now the tangent space $T_pM$ at a point $p$ of a manifold $M$. There's a lot of topology involved in the definition of a manifold, and in the basic theorems, but you can just ignore all that and take this as your starting point:

• An $n$-dimensional manifold $M$ is some kind of set (details not important), together with a bunch of functions $x:U_\alpha\to\mathbb R^n$ such that $M=\bigcup_\alpha U_\alpha$. These functions are called coordinate systems or charts.
• A coordinate system $x:U\to\mathbb R^n$ is used to define the partial derivative functionals $\frac{\partial}{\partial x^i}\big|_p$ associated with a point $p\in U\subseteq M$ by $$\frac{\partial}{\partial x^i}\bigg|_p f=(f\circ x^{-1})_{,i}(x(p))$$ for all nice enough functions $f:M\to\mathbb R$. The notation $_{,i}$ denotes the usual kind of partial differentiation with respect to the $i$th variable. (Note that $f\circ x^{-1}$ maps a subset of $\mathbb R^n$ to a subset of $\mathbb R$. That's why the usual kind of partial differentiation works).
  
• The tangent space at $T_pM$ is defined as the vector space spanned by those partial derivative functionals. (There are more elegant definitions, but they require much more work, and you end up with the same thing anyway). Note that this makes the $n$-tuple $\big(\frac{\partial}{\partial x^1}\big|_p,\dots,\frac{\partial}{\partial x^n}\big|_p\big)$ an ordered basis for $T_pM$. That's why a change of coordinate system $x\to y$ induces a change of basis $\frac{\partial}{\partial x^i}\big|_p\to \frac{\partial}{\partial y^i}\big|_p$, which induces a change of tensor components.

 

[brandon078902]

Thanks for the replies.
So the best way to learn tensor calculus is through a physics text? I should mention that I've had a freshman calc-based mechanics course, but that's the extent of my physics knowledge

 

[Fredrik]

The physics text I mentioned is a great place to start, but once you've read that chapter, you should continue with a real math book. (I can't comment on the approach vanhees71 suggested, since I haven't read any of those books). For me it was Spivak, but Lee ("Introduction to smooth manifolds") is probably a better option now. It's a better book, but I haven't read the part of it that introduces tensor fields, because I already understood those quite well.

Even if you start at the beginning of Schutz and read chapters 1-3, you wouldn't need any more physics than you already know. Chapter 1 is an awesome presentation of special relativity, so if that interests you at all, maybe you should read chapters 1-3. If not, then skip chapter 1, skim chapter 2 to see what notation he's using, and then study chapter 3.

 

[vanhees71]

Schutz is pretty much equivalent to the physics books I mentioned; it's a good choice in any case too. Of course, if you aim at an understanding in a mathematically rigorous way, you have to read math books. To understand them, to have a naive idea about the object's meaning nevertheless also helps a lot to understand the rigorous maths better.

 

[brandon078902]

What other math would you advise I learn before attempting tensor calc then? @Fredrik you said many texts require a knowledge of topology. I'm wondering how advanced a knowledge I'd need to grasp an introductory text on tensors. I'd be willing to study the prerequisite subjects required, and if there are any texts that would suit that purpose, it'd be great.

 

[Fredrik]

It's pretty difficult stuff. If you try to study the topology first, I think you would have to spend several times as much time on the topology as on the tensors. That stuff in Schutz will only take you a few days, and will really help your understanding of tensors. But if you insist on topology first so that you can read Lee's introduction to smooth manifolds from page 1, you could end up spending 3 months studying topology, and then 3 weeks studying the definitions and basics theorems about manifolds and tangent spaces, before you even begin to study tensors.

So if it's tensors that you're interested in, I strongly recommend that you skip the topology and study the tensors first. Then when you understand tensors, you can think about filling in the gaps in your knowledge about manifolds, by studying topology and then studying Lee.

Munkres is a good standard text on topology, but if you're studying topology specifically to prepare for Lee, then Lee might be the best option. He has another book called "Introduction to topological manifolds". Topology is however a difficult enough subject that one book may not be enough. Get a few of them so that you can check out another book when you get stuck on a proof, or find it difficult to understand a concept.

 

[brandon078902]

Thanks for all the replies!
I think I'll probably study the tensors first, then topology, and maybe go back and fill in any gaps afterwards