How can we prove the triple product rule from the implicit function theorem, two variables?

Question

frostbite · Answer

Triple product rule:
$$\frac{ \partial x }{ \partial y }\frac{ \partial y }{ \partial z }\frac{ \partial z }{ \partial x }=-1$$
 implicit function theorem, two variables: see attachment

frostbite · Answer

@amistre64

amistre64 · Answer

does triple product refer to scalar or vector

$$scalar:a\cdot(b\times c)$$$$vector:a\times(b\times c)$$

amistre64 · Answer

i assume since its 1 that its scalar

frostbite · Answer

Well perhaps, I have no idea where to start, I was just told to give it a shot, but no idea where to start. And i sepose you mean -1?

amistre64 · Answer

yeah, -1 ;) http://en.wikipedia.org/wiki/Triple_product_rule i get this as a start from the wiki, and i am reading up on some other finds; but at the moment i got no idea :)

frostbite · Answer

Welll i can try uploade the full pdf I have. Don't think it is going to be any better however.

frostbite · Answer

p. 9 - 17.

amistre64 · Answer

yeah, i cant make much sense of it; either with the google or with the pdf :/

amistre64 · Answer

are we able to rewrite the partials in terms of df/dx, df/dy, df/dz ?  cause i tend to see alot of those being tossed about in there

amistre64 · Answer

and it might have something to do with a Jacobian ....

frostbite · Answer

As far I have been told we may use any known differential notation and calculus.

amistre64 · Answer

http://answers.yahoo.com/question/index?qid=20090319011005AAa0Djq can you make this link out? it seems to address alot of the stuff

amistre64 · Answer

(dF/dx) / (dF/dy)

dF  dy     
--  -- = dy/dx
dx  dF

amistre64 · Answer

dropped a negative again

frostbite · Answer

Well i don't see how he/she get the "dy/dx = -(∂F/∂x)/(∂F/∂y)....." part is that a calculus rule?

amistre64 · Answer

i think thats the implicit function thrm, or at least a consequence of it

frostbite · Answer

Yea was just thinking the same. and then you do that for all veriables and just multiply them?

amistre64 · Answer

yep, thats what it looks like to me; granted i cant verify it, but it reads good to me

frostbite · Answer

Hmm if it looks fine to you i trust your jodgement. Thanks a lot, seems alot more easy, just wondring why they did not write that consequence in the pdf.

amistre64 · Answer

http://www.youtube.com/watch?v=xtgTckGMuWE oh, and patrick has the best explanations