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Comment by Koshkin | Hacker News Reader

I have always hated when tensors are defined as something whose coordinates are transformed in a certain way. I just find it inherently unfriendly and un-geometric. No matter how much talk is given about simpler cases such as scalars, vectors, covectors, etc., the final defining formula would still look to me as daunting as always. (There is nothing “gentle” about the formulas on page 14.) Surprisingly or not, the whole thing clicked for me when I eventually learned about tensors in a more abstract algebraic setting, where they are defined as multilinear forms. The coordinate transformation laws were very easy to understand and remember after that.

But if you want to learn about tensors from how their coordinates transform, here’s a treat:

https://www.youtube.com/watch?v=CliW7kSxxWU

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I taught a module on tensors in my undergrad Mathematical Methods for Physics course last semester. The students had a weak mathematical background, so I had to explain better than any of the books and videos do.

The main problem with most introductions to the topic are that they deal with coordinate systems where the basis vectors are orthogonal, so the covariant and contravariant components are the same. You need to deal with non-orthogonal basis vectors, because then you realize that naturally there are two ways of defining basis vectors at a given point in space. And naturally these two different sets of basis vectors have different transformation properties under geometric transformations. This book [https://danfleisch.com/sgvt/] does a decent job of getting you started on this approach - of course, it has no rigor.

Yes! As a former Physics major who was puzzled by tensors for years (despite being relatively comfortable using and manipulating them), I actually had a similar breakthrough reading Fleisch's book and finally "getting" covariant and contravariant components via thinking about non-orthogonal basis vectors. Definitely one of those "why couldn't they have just explained it this way back when I was an undergrad?" moments. I got so excited when I was reading it that I literally got up, ran to the living room and subjected my partner (not a math or physics person at all) to an impromptu fifteen minute lecture with diagrams in multiple colors of ink. She wasn't as impressed.

(Fleisch's "Student's Guide to Waves" is also highly recommended as a book I wish I'd had when I was a student).

> The main problem with most introductions to the topic are that they deal with coordinate systems where the basis vectors are orthogonal, so the covariant and contravariant components are the same. You need to deal with non-orthogonal basis vectors, because then you realize that naturally there are two ways of defining basis vectors at a given point in space.

Or, preferably, not with basis vectors at all, but that notion seems to do violence to a physicist's way of thinking about linear algebra—I've never quite understood why the physical approach to the subject is so bound up in co-ordinates, when it seems like physicists would be one of the groups most likely to benefit from fully grokking a co-ordinate-free approach.

Because most physicists are ultimately interested in predicting the results of experiments with actual numbers. And that demands the use of coordinates.

> Because most physicists are ultimately interested in predicting the results of experiments with actual numbers. And that demands the use of coordinates.

Of course, at some point, you need numbers! But there's no reason that those numbers need to infest the whole computation; you can de-coordinatise as soon as possible and re-coordinatise as late as possible, and in between you not only can but must think in a coordinate-free fashion.

(I am a mathematician, not a physicist, and am not presuming to tell physicists how to do their job—this isn't an argument about whether they should do things this way. I'm just pointing out that they could, and that it seems not only possible but advantageous. But of course long-established knowledge of what actually works in physics beats this tyro's guess at what could.)

I can also recommend this great series on Tensor Calculus: https://www.youtube.com/watch?v=kGXr1SF3WmA&list=PLJHszsWbB6...

What is also confusing that for physics to be a tensor/vector only the transformation behavior under the orthogonal group is considered, so acceleration is a vector. Math and GR (ok, this is also physics) then uses the coordinate transforms on a manifold, so these are different things.

Could you recommend a text/paper/blog article which develops tensor theory from an algebraic setting?

Chapter 12 of this book https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth...

Hm, that’s a good question! Pretty much any modern book on, or with a chapter on, multilinear algebra, or even a text with a modern treatment of differential geometry (smooth manifolds) would, I imagine, do a decent job. Also, searching the web for “tensors as multilinear forms” turns up quite a few promising links. For a video lecture that could give you a flavor of what tensors are from the linear-algebraic perspective, try

https://www.youtube.com/watch?v=4l-qzZOZt50

(You might get more mileage if you also watch the preceding lecture(s) of this excellent series.)

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