Books/resources for exercises on tensors and multilinear algebra

In summary, the conversation discusses the need for a book with more exercises on tensor and index manipulation, specifically for the purpose of being able to easily do calculations in physics, such as special and general relativity. Several books are mentioned as potential resources, including Schaums Outlines and A General Relativity Workbook by Thomas Moore. The speaker also mentions finding coordinate-free notation useful for conceptual understanding but acknowledges the usefulness of indices for calculations.
  • #1
Shirish
244
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I'm self-studying tensors from a book that doesn't have exercises. The book is Semi-Riemannian Geometry by Newman. To get a better feel for index manipulation, tensor results and calculations, I'm looking for a book that has many exercises in these topics.

I'd be grateful if those knowledgeable in the subject can point me to such books. The end use case is to get comfortable enough with tensor and index machinery that I can easily do calculations in Physics (e.g. special/general relativity, etc.)



I've studied till chapter 6, ToC here : https://onlinelibrary.wiley.com/doi/book/10.1002/9781119517566

More specifically, the topics I've covered so far are: vectors, covectors, basic linalg results, linear transformations, matrix representations of transformations/vectors/covectors, change of basis, tensors, basis/components of tensor spaces, sums/direct sums, subspace annihilator, pullback of covectors/covariant tensors by a linear transformation, ordinary and metric contraction of tensors, bilinear functions, inner product space, adjoints, orthonormal bases, linear isometries, perp, time cones, Lorentz vector spaces, flat/sharp maps and index raising/lowering (for vectors/covectors only so far).
 
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  • #3
I think Schaum's is underrated.
 
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  • #4
My issue with Schaum's is that while good, the books never covered exactly what my course covered. I attributed it to the books being written a few years earlier and that my prof was somewhat eccentric in the topics chosen. I recall trying to learn Lagrangian physics from them for a course using Marion's book and finding the problems didn't match at all what we were doing.

In contrast, they worked great for self-study. I learned Calculus, Vector, and Tensor Calculus that way before taking the courses.
 
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  • #5
jedishrfu said:
My issue with Schaum's is that while good, the books never covered exactly what my course covered. I attributed it to the books being written a few years earlier and that my prof was somewhat eccentric in the topics chosen. I recall trying to learn Lagrangian physics from them for a course using Marion's book and finding the problems didn't match at all what we were doing.

In contrast, they worked great for self-study. I learned Calculus, Vector, and Tensor Calculus that way before taking the courses.
I think they're good as a complement , for independent study and/or as a prep for the class.
 
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  • #6
Shirish said:
I'm self-studying tensors from a book that doesn't have exercises. The book is Semi-Riemannian Geometry by Newman. To get a better feel for index manipulation, tensor results and calculations, I'm looking for a book that has many exercises in these topics.

I'd be grateful if those knowledgeable in the subject can point me to such books. The end use case is to get comfortable enough with tensor and index machinery that I can easily do calculations in Physics (e.g. special/general relativity, etc.)
A nice complement to the book your studying is the book "A General Relativity Workbook" by Thomas Moore. There is nary a (coordinate-free) multilinear map to be seen, but there is nice coverage of "index gymnastics".

I personally find that coordinates and indices can be useful for calculation, and that coordinate-free notation often is useful conceptually, although these aren't completely general statements.
 
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  • #7
George Jones said:
A nice complement to the book your studying is the book "A General Relativity Workbook" by Thomas Moore. There is nary a (coordinate-free) multilinear map to be seen, but there is nice coverage of "index gymnastics".

I personally find that coordinates and indices can be useful for calculation, and that coordinate-free notation often is useful conceptually, although these aren't completely general statements.
I had a look at this book - it seems nice! I'll update more in this thread as and when I find more resources with exercises.
 
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1. What are tensors and multilinear algebra?

Tensors and multilinear algebra are mathematical concepts used in the study of linear transformations and their properties. Tensors are multidimensional arrays that can represent geometric or physical quantities, while multilinear algebra deals with the algebraic manipulation of these tensors.

2. Why are tensors and multilinear algebra important?

Tensors and multilinear algebra are important in many fields of science and engineering, including physics, computer science, and machine learning. They provide a powerful framework for understanding and solving complex problems involving multiple dimensions and transformations.

3. Are there any recommended books or resources for learning about tensors and multilinear algebra?

Yes, there are several books and online resources available for learning about tensors and multilinear algebra. Some popular options include "Tensor Calculus for Physics" by Dwight E. Neuenschwander and "Multilinear Algebra" by Werner H. Greub.

4. How can I practice and improve my understanding of tensors and multilinear algebra?

One way to practice and improve your understanding of tensors and multilinear algebra is to work through exercises and problems. Many textbooks and online resources provide exercises and solutions for practice. You can also try implementing algorithms and applications using tensors and multilinear algebra in a programming language.

5. Is it necessary to have a strong background in mathematics to learn about tensors and multilinear algebra?

While a strong foundation in mathematics can certainly be helpful, it is not necessary to have a deep understanding of advanced math concepts to learn about tensors and multilinear algebra. However, some familiarity with linear algebra and calculus is recommended for a better understanding of these topics.

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