# Looking for a good introductory Tensor Analysis Textbook

• Relativity
• astroboulders
In summary, a physicist would recommend reading the referenced works from the document, then moving on to other sources if desired.
astroboulders
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!

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).

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

jbergman and vanhees71
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.

weirdoguy
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.

weirdoguy
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.

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.

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.

Last edited:
vanhees71

## What is a good introductory textbook for learning Tensor Analysis?

One highly recommended introductory textbook for learning Tensor Analysis is "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" by Pavel Grinfeld. It provides a clear and accessible introduction to the subject, with plenty of examples and exercises.

## Is "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" suitable for beginners?

Yes, "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" by Pavel Grinfeld is suitable for beginners. It starts with the basics and gradually progresses to more advanced topics, making it a good choice for those new to the subject.

## Are there any prerequisites for understanding Tensor Analysis?

Yes, a solid understanding of linear algebra and calculus is essential before diving into Tensor Analysis. Familiarity with differential geometry can also be helpful but is not strictly necessary for beginners.

## Does the textbook include exercises and solutions?

Yes, "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" includes numerous exercises at the end of each chapter. While it does not provide solutions to all exercises, it offers enough guidance to help students practice and understand the material.

## Can this textbook be used for self-study?

Absolutely, the textbook is well-structured and written in a way that makes it suitable for self-study. The clear explanations and examples help readers to grasp the concepts without needing additional guidance.

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