Posted: Nov 13, 2021 2:14 pm
by Calilasseia
If you're looking for an elementary course on tensors, then the book:

Vector Analysis, With an Introduction To Tensor analysis by Murray R. Spiegel, Schaum's Outline Series, McGraw-Hill Publishing, ISBN-10 07 084378 3, is one I would recommend. Might be somewhat on the terse side, but it contains numerous solved problems illustrating the thinking underlying tensors.

However, there is one BIG issue to raise here, namely that this book concentrates upon the coordinate based approach to tensors. If you want a book that covers the coordinate free treatment of tensors, then you need to look elsewhere.

Likewise, if you're thinking of pursuing tensors all the way to the Ricci Calculus, you'll need to look for other texts.

However, the good news about the Spiegel book, is that it teaches you quickly that a tensor representation isn't just valid in one choice of coordinate system, but in all possible coordinate systems, which is what makes tensors so massively powerful.

Quite simply, you can choose whichever coordinate system is most convenient for you, the moment you have a tensor form of an equation, because that tensor form represents the quantities in question in all possible coordinate systems, in one compact, dense notation. Though it's precisely because that notation is compact and dense, that it'll take some time learning to understand said notation.

Head off to Chapter 8 in the Spiegel book, and have fun. It'll take you to the stage where you can, at the end of the chapter, begin to understand how general relativity works, though it's a fairly steep learning curve if you don't already have a fair amount of vector analysis under your belt first.