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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

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.
- abdullahkhalids 5y ago
  Because most physicists are ultimately interested in predicting the results of experiments with actual numbers. And that demands the use of coordinates.
  - JadeNB 5y ago
    > 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.)

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.

abdullahkhalids 5y ago
Because most physicists are ultimately interested in predicting the results of experiments with actual numbers. And that demands the use of coordinates.
- JadeNB 5y ago
  > 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.)