Timeline for How to make Calculus II seem motivated, interesting, and useful?
Current License: CC BY-SA 3.0
18 events
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| Jan 29, 2016 at 23:12 | comment | added | David Roberts | "The most probable reason that freshman calc is one-size-fits-all is that other departments want it as a "weeder" course." -- and possibly the fear that if an 'engineer-only' course is offered, the engineering department might decide it can teach that course instead of the maths department, and take back the funding that service teaching brings to the latter. | |
| May 31, 2014 at 19:21 | comment | added | nomen | @Ben: en.wikipedia.org/wiki/…. It's not a complex or controversial idea. | |
| May 31, 2014 at 18:34 | comment | added | nomen | @Ben: pick up any book on differential forms. Understand what a volume element is. An exact differential form is a linear functional to $\mathbf{R}$ and can therefore that vector space can inherit the ring structure. Remember $(f/g)(x) = f(x)/g(x)$? Start here: en.wikipedia.org/wiki/Volume_element | |
| May 31, 2014 at 18:08 | comment | added | user507 | @nomen: I would be happy to see a link supporting your claims, which seem incorrect to me, or explaining your terminology, which seems nonstandard to me. "Cromulent"...I'll take this as humorous rather than snarky, but I don't get the joke. | |
| May 30, 2014 at 4:03 | comment | added | nomen | @Ben: Exact differential forms are invertible. And those are the ones we want to turn into fractions, to do the things physicists like. And your example is a perfectly cromulent form. | |
| May 30, 2014 at 1:08 | comment | added | user507 | @nomen: I'd be interested if you could point me to some information on what you mean by fractional techniques. Differential forms are not invertible. Since differential forms are nilpotent, you can't use them to express ideas like $ds^2=dx^2+dy^2$. That would be an example of what I mean when I say they're not sufficient to do what people have traditionally done with infinitesimals. There's a good discussion here: mathoverflow.net/q/25089/21349 | |
| May 29, 2014 at 19:55 | comment | added | nomen | @Ben: I don't mean "infinitesimal". The infinitesimal techniques are a subset of what can be done by doing manipulations of fractions of differential forms. Differential forms are sufficient. Indeed, just the FTC is enough to justify the fractional techniques. | |
| May 29, 2014 at 19:40 | comment | added | user507 | @nomen: Sorry, I don't understand what you mean by "fractional" techniques. Do you mean "infinitesimal?" Yes, scientists and engineers take a sophomore vector calculus course that discusses multivariate integration. No, differential forms are not normally covered in such a course. Differential forms are neither necessary nor sufficient to cover all of the traditional-style things that scientists and engineers do with infinitesimals. | |
| May 28, 2014 at 23:27 | comment | added | vonbrand | Funny, a while back I used $\sum_{n \ge 0} n! z^n$ in earnest... (as a formal series, mind you) | |
| May 28, 2014 at 23:24 | comment | added | vonbrand | @BenCrowell, I was thinking more of the $\tan \theta / 2$ substitutions and such. Integration by parts is a basic manipulation technique, as is knowing something like partial fractions. | |
| May 27, 2014 at 20:16 | comment | added | nomen | @Ben: Don't those scientists take a course in multivariate integration? Differential forms are the only "justification" the "fractional" techniques need. Just the FTC is enough. | |
| May 27, 2014 at 17:06 | history | edited | user507 | CC BY-SA 3.0 |
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| May 27, 2014 at 16:44 | comment | added | user507 | @MattF.: Good point about singling out Robinson. Fixed. Math and engineering majors will learn partial derivatives in a vector calculus course, so I don't see the harm in omitting them from the second semester of a freshman survey. The trouble with pretending that NSA and SIA never happened is that we perpetuate a culture in which scientists and engineers universally use infinitesimals (as they've done for 300 years), but are not prepared properly by their freshman calc course to use them. Ignoring developments since 1965 also makes it harder for students to decode the Leibniz notation. | |
| May 27, 2014 at 16:40 | comment | added | user507 | @vonbrand: I added a specific example. | |
| May 27, 2014 at 16:40 | history | edited | user507 | CC BY-SA 3.0 |
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| May 27, 2014 at 2:30 | comment | added | user173 | You've elevated Robinson pretty high among the many (e.g. Bishop, Conway, Lawvere) who refounded the calculus. I'm happy leaving all of them out: even for math majors, probabilities and partial derivatives and numerical computations are the serious omissions for me. | |
| May 27, 2014 at 2:28 | comment | added | vonbrand | I'm yet to come across a nasty integral in a proof where the "how to evaluate it" aspect i is more relevant than its value. Care to exemplify? | |
| May 27, 2014 at 0:09 | history | answered | user507 | CC BY-SA 3.0 |