Partial Differentials Identity

[sebb1e]

Homework Statement

Prove that if z=z(x,y) is invertible that:

(dz/dx)(dy/dz)(dx/dy)=-1 where the d's represent partial differentiation not total differentiation

Homework Equations

The Attempt at a Solution

I guess you start with the 6 total derivatives and substitute them into each other in someway. Beyond this I have no idea.

 

[lanedance]

so what is the definition of invertible here?

 

[sebb1e]

I think that you can write it as x=x(y,z) and y=y(x,z)?

 

[lanedance]

try starting with the total differential
$$dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y} dy$$

 

[sebb1e]

I've reduced it to:

dz=2(dz/dx)(dx/dy)(dy/dz)dz+((dz/dy)(dy/dz)+(dy/dx)(dx/dy))dz

So I need to show that (dz/dy)(dy/dz)+(dy/dx)(dx/dy)=2 where the d's are partial derivatives. If total derivatives this is obvious, am I missing something obvious here?

Also, is using this differential notation perfectly rigorous, I was expecting to have to do it using total derivatives or does that just create a lot more work.

 

[lanedance]

so try 2 differentials
$$dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y} dy$$
$$dx = \frac{\partial x}{\partial z}dz + \frac{\partial x}{\partial y} dy$$

now try subtstituting in for dx

you'll have to make some assumptions (may need to prove) but due to the invertibility you're probably safe to assume
$$\frac{\partial z}{\partial x} = \frac{1}{\frac{\partial x}{\partial z}}$$

though i must say I haven't dealt with a function specifically defined as invertible like this

as for the differential v total derivatives, they can be derived from each other
http://en.wikipedia.org/wiki/Total_derivative