# Tensor Analysis: Best Books for Beginners to Experts

• ForMyThunder
In summary, a good book to learn Tensor Analysis from the basics to the advanced material is "Tensor Analysis on Manifolds" by Bishop and Goldberg.
ForMyThunder
Could anyone tell me what a good book is that describes Tensor Analysis from the basics to the advanced material? It would be truly helpful.

What is the context?
What is your current preparation and your long-term goals for learning this?

Well, I've been through Calculus, Vector Calculus, Ordinary and Partial Differential Equations, and Complex Variables. I guess I just want to learn it because I was planning on going on towards Relativistic Physics and Quantum Mechanics and Field Theory.

Oh yeah, Schaum's Outlines. I almost forgot about them. I've read the "Advanced Calculus" from them. Thanks!

Geometrical Methods of Mathematical Physics - Schutz
Tensor Analysis on Manifolds - Bishop & Goldberg (Dover = cheap)
General Relativity - Wald
,etc.

All under \$40 each.

I really liked

Tensor Analysis on Manifolds - Bishop & Goldberg

it covers a lot of the basic things, and tensor analysis in general. It also have some small chapters covering the next things to read, such as riemannian geometry

I second Bishop and Goldber's Tensor Analysis on Manifolds. I'm reading through this book right now, actually, and it has been quite a pleasurable experience. The notation is a bit awkward (he writes f(x) as fx without parentheses, for example) sometimes, but for the most part this is a thoroughly modern book.

I will say, however, that to get the most out of this book you need some basic background in topology. You could make it through this book without knowing much about topology, but I think you'd miss out on a lot of good material concerning the topological peculiarities of various structures studied in the book. You also need to be familiar with some topics from advanced calculus such as the jacobian, the implicit function theorem, the inverse function theorem, and integration on arbitrary-dimensional Euclidean spaces.

Previously I was grappling with Edwards' Advanced Calculus: A Differential Forms Approach (which isn't really about tensors in general but differential forms specifically). This book took too pragmatic an approach for my taste. Maybe I'm insane, but I actually find the modern, abstract definitions easier to understand and use than the old, often physics-based explanations. Eventually I got tired of trying to translate the practical explanations into the abstract currency of modern mathematics, and I got myself a copy Tensor Analysis on Manifolds, which cured all my tensor-analytic ills.

Last edited:
Thanks. I've already have "Tensor Analysis on Manifolds" and I am reading it now. Thanks anyway. It was a big help.

## 1. What is Tensor Analysis?

Tensor analysis is a branch of mathematics that deals with the study of tensors, which are mathematical objects that describe the geometric relationships between different coordinate systems. Tensors are used to represent physical quantities such as forces, velocities, and stresses in a way that is independent of the coordinate system used to describe them.

## 2. Who should study Tensor Analysis?

Tensor analysis is an important tool in many scientific fields including physics, engineering, and computer science. It is particularly useful in fields that involve the study of continuous media, such as fluid mechanics, electromagnetism, and solid mechanics. Scientists and researchers who work in these fields can greatly benefit from a solid understanding of tensor analysis.

## 3. What are some recommended books for beginners in Tensor Analysis?

Some popular books for beginners in tensor analysis include "Tensor Calculus for Physics" by Dwight E. Neuenschwander, "A Student's Guide to Vectors and Tensors" by Daniel Fleisch, and "Tensor Analysis: Theory and Applications" by Zair Ibragimov. These books provide a solid foundation in the basics of tensor analysis and are written in an accessible and easy-to-understand manner.

## 4. Are there any books specifically for experts in Tensor Analysis?

Yes, there are several advanced books on tensor analysis for experts in the field. Some recommended titles include "Tensor Analysis: Spectral Theory and Special Tensors" by Mikhail Botvinnik, "An Introduction to Tensor Calculus and Continuum Mechanics" by J. H. Heinbockel, and "The Geometry of Tensor Calculus" by Tevian Dray. These books delve deeper into the mathematical aspects of tensor analysis and are geared towards readers with a strong background in mathematics.

## 5. Is it necessary to have a strong background in mathematics to understand Tensor Analysis?

While a strong mathematical background is helpful, it is not absolutely necessary to understand tensor analysis. Many introductory books on tensor analysis assume only a basic understanding of calculus and linear algebra. However, to fully grasp the mathematical intricacies of tensor analysis, a solid understanding of multivariable calculus, linear algebra, and differential geometry is recommended.

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