Timeline for How should one present curl and divergence in an undergraduate multivariable calculus class?
Current License: CC BY-SA 2.5
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| Sep 8, 2021 at 18:41 | answer | added | Marco Farinati | timeline score: 1 | |
| Aug 31, 2019 at 2:10 | history | edited | Gerry Myerson |
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| Aug 30, 2019 at 14:27 | answer | added | Tom Copeland | timeline score: 0 | |
| Aug 23, 2018 at 0:16 | answer | added | David Roberts♦ | timeline score: 2 | |
| Jun 12, 2018 at 10:24 | comment | added | Ali Taghavi | @KevinH.Lin This is not about the method of presenting "curl" in Multivariable calculus but it is just a comment on a possible reinterpretation of the Curle not as a vector field but as a differential form: mathoverflow.net/questions/291099/… | |
| Jun 10, 2018 at 15:28 | answer | added | isomorphismes | timeline score: 1 | |
| Jan 15, 2013 at 17:04 | comment | added | Zsbán Ambrus | Have you tried to start with the two-dimensional analogs first? Once the students understand the theorems in two dimensions (both the differential and the integral forms), you can motivate curl by saying that it's what makes it possible to have analogous theorems in three dimensions. | |
| Jan 15, 2013 at 16:49 | answer | added | Scott Taylor | timeline score: 5 | |
| Apr 20, 2012 at 0:54 | answer | added | Jeff | timeline score: 3 | |
| Feb 16, 2011 at 15:21 | answer | added | John Sidles | timeline score: 9 | |
| Oct 22, 2010 at 19:02 | answer | added | Tobias Hagge | timeline score: 6 | |
| Jul 14, 2010 at 0:29 | answer | added | David Corwin | timeline score: 9 | |
| Jul 14, 2010 at 0:24 | comment | added | David Jordan | More importantly, for the one or two students in the class who get excited about this sort of thing, and want to know more, it's a great way to encourage them to take an analysis or differential geometry course. I would say, rather than claiming that curl and div and grad are just the deRham differential, that the deRham differential is really just the next grad, curl and div. Granted, I am making a contentless point here, and I didn't really figure this out until I started TA'ing, but still...=] | |
| Jul 14, 2010 at 0:19 | comment | added | David Jordan | I would like to add a comment which doesn't constitute an answer. I have always explained the definition of divergence and curl just as Qiaochu suggests, by starting a proof of Gauss's and Stoke's theorems, computing the flux or divergence integrals on small boxes and deriving the formulas for divergence and curl as a limit. This has the advantage that these two theorems, which are rarely explained or motivated in a calculus class, are essentially self-evident, if one is comfortable with some heuristics about the integral as summing up small contributions. | |
| Jul 14, 2010 at 0:08 | answer | added | Jesse Madnick | timeline score: 27 | |
| Jul 7, 2010 at 1:59 | history | edited | Yemon Choi |
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| Jul 7, 2010 at 1:41 | history | edited | Jose Brox |
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| May 4, 2010 at 13:41 | vote | accept | Kevin H. Lin | ||
| Apr 21, 2010 at 19:12 | history | rollback | Kevin H. Lin |
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| Apr 21, 2010 at 19:07 | history | rollback | Kevin H. Lin |
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| Apr 21, 2010 at 18:43 | answer | added | Willie Wong | timeline score: 3 | |
| Apr 21, 2010 at 18:08 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Apr 21, 2010 at 17:01 | answer | added | Donu Arapura | timeline score: 10 | |
| Apr 21, 2010 at 15:38 | answer | added | Allen Knutson | timeline score: 4 | |
| Apr 21, 2010 at 12:42 | answer | added | muad | timeline score: 8 | |
| Apr 20, 2010 at 17:56 | answer | added | B. Bischof | timeline score: 6 | |
| Apr 20, 2010 at 4:17 | answer | added | Pete L. Clark | timeline score: 20 | |
| Apr 20, 2010 at 3:35 | answer | added | Christopher Olah | timeline score: 2 | |
| Apr 20, 2010 at 1:55 | comment | added | Kevin H. Lin | @sigfpe: I mentioned that picture already. I'm still curious, though, how one can prove/show/explain that that "intuitive" picture is equivalent to the formula $\nabla \times \textbf{F}$. I'm thinking that something along the lines of Tim Perutz's answer would work. | |
| Apr 20, 2010 at 1:52 | comment | added | Dan Piponi | This has always been my intuitive picture of curl: en.wikipedia.org/wiki/… | |
| Apr 20, 2010 at 1:47 | comment | added | Kevin H. Lin | Or, if you prefer, you can interpret my question as, "if a student comes to me outside of class and asks for more insight, what should I tell him/her?" | |
| Apr 19, 2010 at 23:09 | answer | added | Tim Perutz | timeline score: 52 | |
| Apr 19, 2010 at 21:09 | comment | added | Kevin H. Lin | What I am really hoping for is some way to convey to them that curl is in fact as natural as gradient and divergence, despite initial appearances. | |
| Apr 19, 2010 at 21:08 | comment | added | Kevin H. Lin | @Gerald: Thank you for your advice. I should say that I am certainly not going to try to teach my students about differential forms and the Hodge star. My main issue is just that every time curl comes up, there are inevitably some students who ask about where the "unnatural-looking" formula comes from. On the other hand, they usually don't ask such questions about gradient and divergence, because their formulas "look" natural to them. I don't want to tell them that the curl formula is just some magic formula that has these magical properties. | |
| Apr 19, 2010 at 21:07 | answer | added | some guy on the street | timeline score: 11 | |
| Apr 19, 2010 at 20:58 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Apr 19, 2010 at 20:54 | comment | added | Gerald Edgar | My advice ... at this level, strictly stick to the textbook. If a student comes to you outside of class and asks for more insight, go ahead. But any deviation from the text will likely cause far more confusion that it prevents! | |
| Apr 19, 2010 at 20:50 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Apr 19, 2010 at 20:49 | answer | added | Justin Hilburn | timeline score: 5 | |
| Apr 19, 2010 at 20:44 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Apr 19, 2010 at 20:36 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Apr 19, 2010 at 20:15 | answer | added | Qiaochu Yuan | timeline score: 27 | |
| Apr 19, 2010 at 20:13 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Apr 19, 2010 at 19:58 | history | asked | Kevin H. Lin | CC BY-SA 2.5 |