Looking for a good introductory Tensor Analysis Textbook

astroboulders · Aug 3, 2023

Hello all,

I've taken math through differential equations and linear algebra, am in my senior year of physics curricula while conducting McNair research regarding General Relativity. I found a NASA document outlining Einstein's field equations, which suggests only preparative familiarity with tensor analysis (here's the NASA pdf). I'm wondering what textbooks would be recommended to begin undertaking tensor analysis study, given my mathematics skills haven't developed beyond differential equations and linear algebra.

Any help would be graciously received!

vanhees71 · Aug 4, 2023

For starting with GR I'd recommend the excellent volume 2 of Landau and Lifhitz. There you also get the minimum of tensor analysis needed (Ricci calculus and holonomous bases only).

haushofer · Aug 5, 2023

Tensor analysis for what? Many GR books like Carroll or Zee treat tensors in detail.

apostolosdt · Aug 5, 2023

The following is one of the simplest, yet quite accurate, intro to tensors.
https://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf

Can't replace a good textbook, though.

vanhees71 · Aug 6, 2023

A first glance at this source shows that it's not very accurate in its notation and in the concept. First of all, if you deal with general tensors on a vector space, i.e., objects that are invariant under the entire GL(n), you have to distinguish clearly between vectors and co-vectors, and this should be reflected in the usual notation with upper and lower indices to distinguish co- and contravariant bases and the corresponding components of tensors.

Second a linear form is not a scalar product. A linear form is a linear mapping from the vector space to the scalar field, while a scalar product is a postive definite bilinear form.

apostolosdt · Aug 6, 2023

vanhees71 said:

A first glance at this source shows that it's not very accurate in its notation and in the concept. First of all, if you deal with general tensors on a vector space, i.e., objects that are invariant under the entire GL(n), you have to distinguish clearly between vectors and co-vectors, and this should be reflected in the usual notation with upper and lower indices to distinguish co- and contravariant bases and the corresponding components of tensors.

Second a linear form is not a scalar product. A linear form is a linear mapping from the vector space to the scalar field, while a scalar product is a postive definite bilinear form.

Maybe so, but it exhibits a reasonable portion of rigour for the level of the OP, don’t you think? We are talking physics, rigour isn’t always on our side.

vanhees71 · Aug 6, 2023

No, for a physicists the clear distinction between co- and contravariant components is essential. As I said, then it's better to read the corresponding chapters in Landau and Lifshitz vol. 2. Also most other introductory textbooks on GR have better introductions to tensor calculus.

apostolosdt · Aug 6, 2023

You surely didn’t just take a cursory look and decided that no distinctions were made of contravariant and covariant stuff? Oh, never mind, hope the OP got some legible answers out of this thread.

robphy · Aug 6, 2023

Presumably, you've seen the referenced works on the last page of the document.
It might be good to start there (as well as any suggested introductory references that they make).

Be aware of conventions in signature and signs (e.g. ones regarding nabla vs semicolon), which vary among books...
and could pose unnecessary obstacles for a beginner.
So, I'd suggest that a beginner start with the references they give, then move on to other sources if desired.

From Misner-Thorne-Wheeler Gravitation (1973)...

Of course, one would have to extend this to include later references.

Looking for a good introductory Tensor Analysis Textbook

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