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June 13

Prove change of variables theorem for triple integrals

My book gives a proof for the case of double integrals, based on Green's theorem, but what about the corresponding theorem for triple integrals? What I'm trying to prove is that if F is a continuous function of x, y, and z on R, x=a(u,v,w), y=b(u,v,w), z=c(u,v,w), which form a transformation from R to S, and the Jacobian determinant of x, y, and z with respect to u, v, and w exists and doesn't change sign on R, then:

$\iiint \limits _{R}F(x,y,z)dxdydz=\iiint \limits _{S}F(a(u,v,w),b(u,v,w),c(u,v,w))\left\vert {\frac {\partial (x,y,z)}{\partial (u,v,w)}}\right\vert dudvdw$

where ${\frac {\partial (x,y,z)}{\partial (u,v,w)}}$ denotes the Jacobian determinant of x,y, and z with respect to u, v, and w. The integration by substitution article states this in terms of measure theory, but I'm sure the proof for this can't be that much more complicated than the case of double integrals.--Jasper Deng (talk) 00:20, 13 June 2013 (UTC)[reply]

The proof does generalize in a certain sense. The Jacobian of the transformation

(f,g,h)

can be written as the divergence of

f\nabla g\times \nabla h

. This can be integrated over a solid, the divergence theorem applied, and then a change of variables on the boundary surface followed by a second application of the divergence theorem gives you the change of variables for a region in 3 dimensions. However, this relies on change of variables on an arbitrary surface (smooth, say, but not necessarily flat). This is not usually how it's done, since it turns out to be most convenient to prove the change of variables formula directly, and then use this to define the integral over surfaces and higher dimensional manifolds. Sławomir Biały (talk) 03:10, 13 June 2013 (UTC)[reply]

I'm assuming the direct proof would then require measure theory. Something that's mysterious to me is why the Jacobian determinant can't change sign on R.--Jasper Deng (talk) 04:14, 13 June 2013 (UTC)[reply]

For the Jacobian determinant to change sign it would imply that at some point it was zero, i.e. the Jacobian drops rank and you have some singularity in the transformation from R to S.--Salix (talk): 08:41, 13 June 2013 (UTC)[reply]

No the direct proof doesn't require measure theory. You can find it, for instance, in Michael Spivak's entirely self-contained book Calculus on manifolds. Sławomir Biały (talk) 11:48, 13 June 2013 (UTC)[reply]

Name of a theorem

Is there any theorem which says:

If rain is falling on a ground you can always cross the ground in a path that no rain drop falls on you.

Note: My sir stated it vaguely while lecturing us on the derivation of a formula (number of paths between two points on a graph with integer co ordinates). Solomon7968 22:38, 13 June 2013 (UTC)[reply]

There couldn't possibly be any such theorem without additional assumptions regarding the distribution of raindrops. Looie496 (talk) 03:45, 14 June 2013 (UTC)[reply]

@User:Looie496 He used the term discrete in case of rain drop to distinguish it from continuous waterfall. Now wiki has pages discrete set, continuous set. Solomon7968 04:27, 14 June 2013 (UTC)[reply]

No set of dimension 0 can separate the plane. That could be what is being referred to. Although I'd guess you'd want a version for path-connected, which probably exists, but I don't know it.--80.109.106.49 (talk) 07:16, 14 June 2013 (UTC)[reply]

The worry is the use of "you". Zero-width humans are rare. Dbfirs 07:26, 14 June 2013 (UTC)[reply]

And statements of the Banach-Tarski paradox involving oranges ignore the atomic nature of matter. There's a grand tradition of constructing informal statements of theorems that don't stand up to scrutiny.--80.109.106.49 (talk) 07:33, 14 June 2013 (UTC)[reply]

Usually an informal statement of a theorem is intended to make the formal theorem easier to grasp, and to suggest it is either intuitively plausible or intuitively surprising. In this case, however, the informal statement only introduces complications and ambiguities. Do the person and the raindrops have non-zero size in any dimension ? What property characterises the places where raindrops fall ? Is the path of the raindrops assumed to be vertical ? How fast can the raindrops be falling ? Are they falling at constant speed ? How is time modelled in the scenario ? Does the person have to follow a continuous path ? a differentiable path ? Do they know where the raindrops will fall ? Do they have to define their path before starting ? etc. etc. Gandalf61 (talk) 08:08, 14 June 2013 (UTC)[reply]

Just mentioned above I have no clue except it happens because "Rain drop falls discretely". Hope it helps. Solomon7968 08:42, 14 June 2013 (UTC)[reply]

Sorry, I wasn't trying to answer your original question. I was simply pointing out the absurdity of your professor attempting to illustrate a theorem whose details you have not grasped and whose name you cannot remember with an example that introduces so many complications that it obscures whatever point he was originally trying to make. Gandalf61 (talk) 08:55, 14 June 2013 (UTC)[reply]

Actually not professor, a M.Stat guy from ISI teaching for a highly competetive examination. Solomon7968 09:05, 14 June 2013 (UTC)[reply]

Might be interested in Smart headlights see through rain and snow. Though of course that's the real world and so doesn't prove anything :) Dmcq (talk) 10:55, 14 June 2013 (UTC)[reply]

Searching math literature

It is easy to search for things like Fermat's last theorem, because everyone knows its name. But how would one go about looking for literature on a Diophantine equation like $x^{3}+2y^{3}=z^{3}$ if you don't know its name (and are not sure it has ever been given a name). The example given is just a random choice, but I'm wondering fairly generally if there are good tips for searching math literature when you know what the equations look like but are unsure of what they are called. I could just as well have used a random partial differential equation (rather than a Diophantine equation), and I'd be just as unsure how to search for literature on it. Any pointers? Dragons flight (talk) 23:03, 13 June 2013 (UTC)[reply]

See Wikipedia:Reference desk/Archives/Computing/2010 April 16#Internet searches for non-linear notations.

—Wavelength (talk) 23:30, 13 June 2013 (UTC)[reply]

Searching for a particular equation using a text based search engine is in general hard. One way to narrow the search is to figure out what type of equation you are dealing with and searching for that type. For instance, in your Diophantine equation, the highest power is 3, so this is a cubic Diophantine equation. There are three unknowns, so this is a cubic Diophantine equation in three variables. Searching Google for 'cubic diophantine equations three variables' leads to a couple of hits [1],[2] on the first page with equations closely related to yours and perhaps some clues to solving them. The same idea applies to PDEs: linear or nonlinear, autonomous or driven, order of the highest derivative, etc. --Mark viking (talk) 00:09, 14 June 2013 (UTC)[reply]

My Google search for equation formula search engine found some interesting results.

—Wavelength (talk) 04:11, 14 June 2013 (UTC)[reply]