Oh, sorry—for some silly reason I thought you were using “intuition” in the derogatory sense, like “lack of rigor”. Of course one wants a rigorous proof that also provides a good intuition for why the result is true!

]]>I too yearn for an intuitive proof and I regret that I don’t have one, as I tried to indicate in my comment.

]]>Thanks. Neither my proof nor Jules Jacobs’ nor the argument in the video used ‘intuition’, though my quick proof sketch didn’t fill in the details of every step, e.g. justifying that you can work to first order at a point.

]]>Wedge the first two equations with each other and use the third to replace in the right-hand side. Taking into account and , we find

But and , so we get , whence . ]]>

There is a not-so-well-known book by Michael Spivak that is worth mentioning:

*Physics for Mathematicians: Mechanics I*.

Concerning that book, Spivak’s friend, mentor, and colleague Richard S. Palais wrote:

I would like to give a strong second to this recommendation. I do have a copy and have been reading and liking it. And let me add that it contains tons of fascinating historical etc. material. While not everything that I know about is done just as I would have presented it, as anyone who knows Spivak’s other famous books would expect, the presentation is exceptionally clear and readable. I saw Spivak [born 1940] at the New Orleans Joint Annual Meeting this weekend [in 2011] and he told me that electricity and magnetism is next—but don’t hold your breath, he said the Mechanics volume took 6 yrs of very hard work.

After extensive and thorough coverage of hidden assumptions in the subject, Spivak moves forward to a treatment of Lagrangian and Hamiltonian mechanics, relying on his books on DG.

]]>Like John, I also took multivariable calculus at my local university while I was in high school. I don't remember who the author of the textbook was, but I don't think that it was substantially different from the textbooks in common use at state universities in the USA today. I really learnt the subject the next year, reading Spivak's Calculus on Manifolds in the university's library. (First I read his regular calculus textbook, then Calculus on Manifolds, then his 5-volume magnum opus on differential geometry, although I didn't finish the last or understand all of what I did read of it.)

As for what I use, I teach this course at my local community college, from an ordinary textbook that we get from our contracted publisher, along with course notes that I've written that look a bit more under the hood. You can find the latter at http://tobybartels.name/MATH-2080/2021FA/ (or start at http://tobybartels.name/MATH-2080/ for a more up-to-date version if you're reading this in 2022 or later).

]]>Well, that’s certainly interesting. Thanks for explaining that. I had wondered why you so often discussed learning math from physics texts.

BTW, texts at one time popular for some approximation to that course were:

1957 Apostol, *Mathematical Analysis*

1959 Nickerson, Spencer, and Steenrod, *Advanced Calculus* (Princeton Math 303-304)

1965 Fleming, *Functions of Several Variables*

(1965 Spivak, *Calculus on Manifolds* (very concise notes))

1968 Loomis & Sternberg, *Advanced Calculus* (Harvard Math 55)

1968 Lang, *Analysis I*, 1983, 1997 *Undergraduate Analysis*

Just thought I’d mention those texts in case anyone cares to opine on them.

(I wonder what text Toby uses.)

Re Feynman, let me put in a plug for a most excellent, and free!!, on-line version of his lectures:

https://www.feynmanlectures.caltech.edu/.

I took that course at George Washington University when I was a senior in high school, and apparently the textbook was unmemorable. In practice, I got good at this stuff from taking physics courses, and reading Feynman’s *Lectures on Physics*.