[BUG] partial derivative product rule

[Tevian Dray]

>>>>> Suda Kunyosying writes:

    SK> Problem 54 of Stewart's section 15.5 can be proved easily enough. But
    SK> is there a physical situation we can use to explain why the product:
    SK> of partial derivatives (z wrt x, x wrt y, and y wrt z)= -1 ?

One way to look at this involves the iterated coordinate transformations:
	(x,y) |--> (x,z) |--> (y,z) |--> (x,z)
and the stated identity follows from the fact that multiplying Jacobian
matrices of 2 (or more) transformations yields the Jacobian matrix of the
composition of the transformations; in this case, the product of the three
matrices must be the identity.  (I think it is also necessary to use the
composition property on two transformations in order to complete the proof.)

A more geometric interpretation follows from the realization that the normal
vector to the surface is just the (3-d) gradient of the function of
3-variables which defines the surface, which can also be expressed in terms of
the (2-d) gradients of the "graph" functions, so that the following vectors
must be parallel:
	grad(f)-k, grad(g)-j, grad(h)-i
This again yields the desired result after a bit of algebra, but I don't see
an obvious way to finish the argument geometrically.

I see no physical context here -- and it's easy to get lost in the notation,
since different variables are being held constant in each differentiation.

Tevian