3 ms·

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-geomet

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

tmearnest 5y ago
Could you recommend a text/paper/blog article which develops tensor theory from an algebraic setting?
- hrwvjigegby 5y ago
  Chapter 12 of this book https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth_manifolds.pdf https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth...
- Koshkin 5y ago
  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 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.)
abdullahkhalids 5y ago
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/ https://danfleisch.com/sgvt/] does a decent job of getting you started on this approach - of course, it has no rigor.
- thraxil 5y ago
  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).
- JadeNB 5y ago
  > 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.
maze-le 5y ago
I can also recommend this great series on Tensor Calculus: https://www.youtube.com/watch?v=kGXr1SF3WmA&list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx https://www.youtube.com/watch?v=kGXr1SF3WmA&list=PLJHszsWbB6...
andi999 5y ago
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.

tmearnest 5y ago

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

hrwvjigegby 5y ago
Chapter 12 of this book https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth_manifolds.pdf https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth...
Koshkin 5y ago
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 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.)

abdullahkhalids 5y ago

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/ https://danfleisch.com/sgvt/] does a decent job of getting you started on this approach - of course, it has no rigor.

thraxil 5y ago
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).
JadeNB 5y ago
> 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.

maze-le 5y ago

I can also recommend this great series on Tensor Calculus: https://www.youtube.com/watch?v=kGXr1SF3WmA&list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx https://www.youtube.com/watch?v=kGXr1SF3WmA&list=PLJHszsWbB6...

andi999 5y ago

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.