Timeline for How should one present curl and divergence in an undergraduate multivariable calculus class?
Current License: CC BY-SA 2.5
11 events
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| Mar 15, 2014 at 19:17 | comment | added | John McVirgooo | I've noticed this attitude towards tensors also; where the invariance of the product of a tensor with its dual isn't emphasised in most text books. Instead, the authors just state that the components of a tensor transform in a way which defines it as a tensor! | |
| Apr 5, 2011 at 10:08 | comment | added | Jesse Madnick | To be pedantic, I think technically it's $\text{curl} = \sharp \circ \ast \circ d \circ \flat$, but no matter. | |
| May 4, 2010 at 13:41 | vote | accept | Kevin H. Lin | ||
| Apr 21, 2010 at 15:35 | comment | added | Tim Perutz | Kevin, it seems that this is often phrased in the following form: "the only isotropic rank 3 tensor in 3 dimensions is the alternating symbol". In that form, it's covered in Temple's "Cartesian tensors: an introduction" - inspired, I think, by Weyl's "The classical groups" - and (apparently) also in Goldstein's "Classical mechanics". Of course, these are not references at the level of Stewart. | |
| Apr 20, 2010 at 22:14 | comment | added | Kevin H. Lin | @Willie: Yeah, I had noted that formula for curl above. @Tim: Do you know of any reference or book where this is written up nicely? I would like to show it to my students, but I don't have the time/inclination right now to do a nice write-up of it myself. | |
| Apr 20, 2010 at 18:33 | comment | added | Willie Wong | I am sure you can work this out yourself, but just to note: div,grad, and curl are not diffeomorphism invariant, just invariant under isometries. This is of course due to the fact that $grad = \sharp\circ d$, $div = \delta \circ\flat$ and $curl = \sharp \circ d \circ * \circ \flat$, so the metric is involved heavily. This is also why it is somewhat more natural to work with d and forms instead of those operations on vectors in the context of differential topology. | |
| Apr 20, 2010 at 8:45 | comment | added | Kevin H. Lin | ... up by, for instance, an introduction to some ideas from special relativity. (Ok, ok, ok, that's probably way too much to ask. But a mention would still be nice.) | |
| Apr 20, 2010 at 8:42 | comment | added | Kevin H. Lin | Hmm, well, it's possible that someone showed this to me at some point, but I must have not been paying attention ;). Anyway, I feel that this explanation is so nice that it should be at least mentioned in these courses and books; it doesn't have to necessarily be worked out in explicit detail. And yeah, I agree with your feelings about Stewart, though to be fair I feel similarly about a lot of other calculus books as well. Some mention of more modern work would be helpful to counteract such perceptions. For instance, an introduction to the Galilean principle could be nicely followed ... | |
| Apr 20, 2010 at 1:56 | comment | added | Tim Perutz | Kevin, I'm glad if this is new to you. I admit that the argument itself may a bit fiddly for a mass-market calculus text (it reduces to showing that the cross product is characterized by bilinearity, rotation-invariance and scale), but the statement is clear enough. In its own terms, Stewart's text is solid enough, but he sometimes seems to me to present mathematics as a subject that fossilized some time in the Late Triassic...;) | |
| Apr 19, 2010 at 23:39 | comment | added | Kevin H. Lin | Dear Tim: I'll have to check that argument for myself, but argh, that's beautiful!!! I feel frustrated that they don't seem teach this in undergrad multivariable calculus courses. This is certainly not in Stewart's book, and certainly this is the first time I've seen this myself. I hope that at least this is commonly taught in undergrad classical mechanics courses, but I never took a proper classical mechanics course myself. | |
| Apr 19, 2010 at 23:09 | history | answered | Tim Perutz | CC BY-SA 2.5 |