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It seems to me that some hand-waving (by which I mean some arguments that aim at giving some form of intuition on the problem even at expenses of complete rigour [and not mnemonics for high-schoolers or totally bogus oversimplistic smoke curtains]) may be really useful at times to get some insight on a problem.

For example, in Levi's Mathematical Mechanic there are many intelligent examples of problems where some physical intuition (while not perfectly rigorous) may help yield some result and even converted into a formal argument.

So, I would like to collect a "big list" of "useful" (and possibly somewhat sophisticated), insightful and interesting heuristics and hand-waving arguments (which may also include some reference to physical principles e.g. see this)

Community
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Dal
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    Does "handwawing" mean hand-waving, hand-drawing, or handwriting? –  Dec 17 '14 at 15:12
  • Hard to answer. I feel like most heuristics go along the lines of "this A looks like B, and for B we know that P(B), so probably P(A)". Now the examples which I'd be brave enough to post are ones which can be formalized anyway. If I write down a heuristic with a wrong conclusion, people will not like it either. – Nikolaj-K Dec 24 '14 at 18:34

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Answers by Christian Blatter often contain A little bit of Physics. This one is a wonderful example:

A parabola is the trajectory $\,\vec{r}\,$ of a particle with initial position $\vec{s}$ , initial velocity $\,\vec{v}\,$ and constant acceleration $\,\vec{a}$ . This leads to the representation $\,\vec{r}(t) = \vec{s} + \vec{v}\, t + \vec{a}\, t^2$ , as has been employed in:

The following answer is inspired by the physical - what is "slimness" - and the physics / mechanics of solid bodies - Moments of inertia . With respect to the latter, any - "slim" or "fat" - 2-D body can be thought as an ellipse (of inertia). Then there is a wonderful relationship between the physical, the physics, and the mathematics of Steiner ellipses :

Couldn't really distinguish between physics intuition and a mathematical argument at this place:

Han de Bruijn
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  • $+1$ Thank you. This is the kind of things I was looking for.
    I would really appreciate if you (or Prof. Blatter himself) could compile a collection of such answers. :)     – Dal Dec 25 '14 at 16:33
  • @Dal: I would be glad to do it, but it's Xmas time and I shall be with my family soon. I'd suggest to check the Answers given by Dr. Blatter and myself (you can find them in our profiles) and I'm sure you will find some. A little bit of Physics in Mathematics is my personal bias. – Han de Bruijn Dec 25 '14 at 19:41
  • Okay, I'll have a look. Do you happen to have written some mathematical papers in which you use physical arguments? – Dal Dec 29 '14 at 00:55
  • @Dal: Could not find explicit examples of the kind. Rather the other way around: physics papers in which mathematical arguments are employed, as is commonly the case. – Han de Bruijn Dec 29 '14 at 20:18
  • @Dal: Think I've found some more. See my update of the answer. – Han de Bruijn Dec 30 '14 at 10:11
  • Thank you very much. If you find something else, please, don't hesitate to add. – Dal Dec 30 '14 at 13:10
  • @Dal: Didn't hesitate to add another update. – Han de Bruijn Jan 08 '15 at 12:36
  • @Dal: added another update. Hope it's relevant. – Han de Bruijn Feb 21 '15 at 14:00
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I use the following in my calculus class to find limits of various quotients when the variable approach zero. They are:

$\begin{eqnarray}(BIG + small)^r &=& BIG^r + r\ BIG^{r-1} small + \cdots\\ \sin(small) &=& small + \cdots\\ \cos(small) &=& 1 - {1 \over 2}small^2 + \cdots\\ \tan(small) &=& small + \cdots\\ e^{small} &=& 1 + small + \cdots\\ \ln(1+small) = &=& small + \cdots \end{eqnarray}$

abel
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  • Sorry, but a semi-rigorous version of asymptotic expansions is not what I'm looking for. – Dal Dec 24 '14 at 18:03
  • @Dal, that is alright. you asked for handwaving that is useful and i gave you one. – abel Dec 24 '14 at 18:05
  • How is this not rigorous? – Andrés E. Caicedo Dec 24 '14 at 18:49
  • @AndresCaicedo, it is rigouros. but, sometimes i feel the reasoning to be circular. kind of using taylor series to find the limits while you are supposed to do limits first. my students like and they can compute complex limits with ease. so i will continue teach this. – abel Dec 24 '14 at 19:15
  • @AndresCaicedo What I meant is that these are only "expansions" with "small" instead of x for x ->0, which could be useful only as mnemonics. – Dal Dec 25 '14 at 16:30