What Are the Best Books for Mastering Tensor Calculus?

[Genericcoder]

Hi guys,

I am interested to learn tensor calculus but I can't find a good book that provide rigorous treatment to tensor calculus if anyone could recommend me to one I would be very pleased.

 

[micromass]

Why do you want to know tensor calculus? Probably for some physics subject like relativity, yes? The physics book will definitely do the necessary tensor calculus then, so there is no real need for a separate book.

 

[Genericcoder]

no actually I want to know it for math related area since I am undergrad student in math I am interested in it because it appears a lot when we deal with Hilbert space etc. I want though a book that gives a well defined definition for tensors.

 

[HakimPhilo]

I'm not qualified enough to recommend you such book but it seems that "A Student's Guide to Vectors and Tensors" by Daniel Fleisch is very loved, quoting one review:

But "A Student's Guide to Vectors and Tensors" (author Daniel Fleisch) is the first actual book I've found that (i) is pitched at a genuinely introductory level, assuming about first year university maths (ii) adopts an explanatory approach rather that the usual tedious 'endless succession of proofs' approach so beloved of many maths authors (think: Dover editions !) (iii) has many exercises, and worked solutions on the author's web site.

I like the way the author takes time to very slowly and carefully explain confusing (to me at least) yet not necessarily difficult areas such as Contraveriant vs Covariant representations; dual bases and more.

I should caution that [...] this book may well be too basic.

 

[micromass]

no actually I want to know it for math related area since I am undergrad student in math I am interested in it because it appears a lot when we deal with Hilbert space etc. I want though a book that gives a well defined definition for tensors.

When specifically do you see tensor calculus when you deal with Hilbert spaces? I'm starting to think we are talking about different tensors. Can you tell me what you think tensor calculus is?

 

[Genericcoder]

If we take for example the tensor product between two vectors each lives in Hilbert space I know that it has to satisfy certain properties but I can't find a good definition of what exactly is a tensor or a tensor product.

 

[micromass]

If we take for example the tensor product between two vectors each lives in Hilbert space I know that it has to satisfy certain properties but I can't find a good definition of what exactly is a tensor or a tensor product.

Where did you read about this? Can you give me some specific reference or citation? This would really help me to find out what exactly you need.

 

[Genericcoder]

For example this is a specific example of what I am talking about and that was also what my professor presented when he was explaining some stuff about C* algebra those aren't really well defined like it doesn't give what specifically what is a tensor ! http://www.quantiki.org/wiki/Tensor_product

 

[Daverz]

It may be too detailed, but I like the book by Wasserman. I have not seen the second edition, though. Everything is very carefully done, but at the same time it is down to earth.

https://www.amazon.com/dp/0195065611/?tag=pfamazon01-20

(First edition; there's a link to the newer edition there).

 

[micromass]

For example this is a specific example of what I am talking about and that was also what my professor presented when he was explaining some stuff about C* algebra those aren't really well defined like it doesn't give what specifically what is a tensor ! http://www.quantiki.org/wiki/Tensor_product

Thank you, that helps. So you don't want a book on tensor calculus! Tensor calculus is the name for a discipline that is used a lot in applied mathematics and it is related to your link, but it is not what you want. I was confused because you used this term.

Firstly, what is a tensor? A tensor on a $k$-vector space $V$ is just a multilinear map $V\times ... \times V\rightarrow k$. This is a covariant tensor. A contravariant tensor is a multilinear map $V^*\times ...\times V^*\rightarrow k$. Then there are also mixed tensors, which are less important for now, they are multilinear maps of the form $V\times ...\times V\times V^*\times ...\times V^*\rightarrow k$.

The above paragraph is the concrete picture of tensors and is the one used in physics. In mathematics however, we abstract the above picture and we form things called "tensor products of vector spaces". This is what is described in your link. Well, your link is about C*-algebras and Hilbert spaces which are more advanced.

The first thing to do is to understand the "easy" case of tensor products of vector spaces. All other forms of tensor products will build on that.

This is what I would do:
- First, I would take a look at the beautiful book "Linear Algebra Done Wrong", which is freely available here: http://www.math.brown.edu/~treil/papers/LADW/LADW.pdf Try to understand entire chapter 8

- Second, I would get the book "Advanced Linear algebra" by Roman. It has an entire chapter on tensor products of vector spaces (and a lot more good stuff). After reading this, you will know the theory of tensor products in vector spaces.

- You might be interested in tensor products on more general spaces such as modules (if you are not, skip this step). The book "Introduction to Commutative Algebra" by Atiyah and Macdonald does a great job. For the noncommutative case, check out the first two or three chapters of "An introduction to homological algebra" by Rotman.

- You are likely more interested in tensor products of hilbert spaces and C*-algebras. For this, I recommend the second chapter of Kadison & Ringrose "fundamentals of the theory of operator algebras", it is a chapter on Hilbert spaces. Tensor products on C*-algebras are much more subtle. As reference, you cannot find much better than the appendix of "K-Theory and C*-Algebras: A Friendly Approach" by Wegge-Olsen.

 

[Genericcoder]

Thanks a lot micro mass that's exactly what I need

 

[Orodruin]

Firstly, what is a tensor? A tensor on a $k$-vector space $V$ is just a multilinear map $V\times ... \times V\rightarrow k$. This is a covariant tensor. A contravariant tensor is a multilinear map $V^*\times ...\times V^*\rightarrow k$. Then there are also mixed tensors, which are less important for now, they are multilinear maps of the form $V\times ...\times V\times V^*\times ...\times V^*\rightarrow k$.

The above paragraph is the concrete picture of tensors and is the one used in physics. In mathematics however, we abstract the above picture and we form things called "tensor products of vector spaces". This is what is described in your link. Well, your link is about C*-algebras and Hilbert spaces which are more advanced.

I would not call it the one used in physics. The most common one perhaps, but tensor products of vector spaces also play a central role in physics and C* algebras and Hilbert spaces are at the very foundation of quantum mechanics.