Grasping Tensors: Math Resource for Physics Students

[Clever-Name]

So I tried asking this over in the math resource section but it won't let me post a thread there, so I figured here is the next best place to ask.

I am rather worried about Tensor mathematics. I have read a bit into the courses I'm going to be taking next year and I'm rather scared... First of all I'm entering 3rd year in an Honors Spec Physics and Minor in Advanced Physics (lots of mathematical physics courses *CFT, Elem. particles, upper quantum/classical mech/EM*) and I'm trying to find a decent Tensor Math resource so that I'm at least somewhat prepared.

I have looked at various Physics textbooks on things like Classical Mechanics, intro to GR, Field Theory, etc. and they all assume a prior knowledge of tensors, so I'm always completely lost right away.

Here is my math background so far:

4 semesters of Calculus (Differential calculus, vector calculus, *Green's Thm, Stoke's Thm etc.* everything offered in my university for calc classes)

2 semesters of Linear Algebra (covering: Vector space examples. Inner products, orthogonal sets including Legendre polynomials, trigonometric functions, wavelets. Projections, least squares, normal equations, Fourier approximations. Eigenvalue problems, diagonalization, defective matrices. Coupled difference and differential equations; applications such as predator-prey, business competition, coupled oscillators. Singular value decomposition, image approximations. Linear transformations, graphics.)

ODE's (covering: Introduction to first order differential equations, linear second and higher order differential equations with applications, complex numbers including Euler's formula, series solutions, Bessel and Legendre equations, existence and uniqueness, introduction to systems of linear differential equations.)

So I'm wondering: Should I really be worried at all? Will it all fall into place once used in the context of physics applications? Or should I really be looking ahead and studying in advance to prepare myself.

 

[khemist]

Many general relativity textbooks will briefly go over tensors. There are also many textbooks that deal specifically with tensors. It is always good to review a topic before learning it in a classroom. I have found the hard part about tensors (never actually taking a course, just some self study) is remembering the transformation equations (which define tensors) and how easy it is to accidentally switch the co variant and contra variant indices (I am semi dyslexic, which doesn't help...). Honestly though, the same difficulties that people have in any other type of mathematics applies here. From what I understand, using tensors is a part of differential geometry, so its kinda like a combination of vector calculus, linear algebra, and geometry.

 

[osnarf]

Aug13-04, 05:57 PM
There are several threads here in which people have recommended sets of notes on relativity in which tensors are taught.

In one of those sets of notes, that I particularly liked, the author Sean Carroll recommended the book by Frank Warner, on Differentiable manifolds and lie groups, as a "standard". I kind of like Michael Spivak's little book Calculus on manifolds, and his much longer series on Differential Geometry, say the first volume for starters.

This is a mathematician talking, so I recommend getting some opinions from the physics experts too. Of course Carroll is presumably a physicist.

Warnes book is nice because it also has an introduction to "Hodge theory" as I recall. Others here have recommended Tensor analysis on manifolds by Bishop and Goldberg, because it is not only a good classic text, but it is available in paper for a song.

Look on the threads "What is a tensor", and Differential geometry lecture notes, and Math "Newb" Wants to know what a Tensor is, and others , for some free sites with downloadable material on tensors.

I would warn you of one thing. I myself am primarily educated in the mathematics of manifolds and tensor bundles as in Spivak's calculus on manifolds. As you can see from numerous exchanges I have had with physicists on this forum I have great difficulty understanding what they are talking about. Thus i would suggest that it is not enough to understand only the mathematical concepts of manifolds and tensors, but one should go further and see these concepts in use either in differential geometry, or in physics.

Quote from Mathwonk, when asked about a good intro to tensors and manifolds.

I would second calc on manifolds, you can start reading in the 4th chapter if you already are well grounded. But from what he said in the last paragraph and from what your intended uses are, maybe it would be best to learn from a relativity book that gives an intro?

 

[Clever-Name]

Interesting. Alright I'll start looking for that Spivak book as well as some good GR intro textbooks (which actually go over tensors instead of just jumping right in). Thanks for the advice!

 

[lanedance]

I'm working through the start of this text and find it a reasonable hands on intro
http://www.math.odu.edu/~jhh/counter2.html

 

[hunt_mat]

You might want to have a look at Bernard shutz's beek entitled geometrical methods in physics or something like that.