Partial derivatives question?

Question

I have to find $dw$ if $w=f(u,v,z)$ where $u=x^2+y^2,v=x^2-y^2,z=2xy$.

Now,I know that $dw= ( ∂w/ ∂z)*dz + (∂w/ ∂u)*du + (∂w/ ∂v)*dv$ The problem is that for example,if I want to find $∂w/ ∂z$ I don't know how to relate $w$ to $z$?

Answer 1

(I withdraw my earlier approach.)

We can "solve" the $u$ and $v$ equations for $x$ and $y$ as

$$2x^2 \ = \ u + v \ \ \ , \ 2y^2 \ = \ u - v \ ,$$

which will allow us to write $ \ z \ $ as

$$ \ z \ = \ 2xy \ = \ (4x^2y^2)^{1/2} \ = \ [ \ (u+v)(u-v) \ ]^{1/2} \ = \ [ \ (u^2 -v^2) \ ]^{1/2}, $$

from which we can now extract

$$dz \ = \ \frac{\partial z}{\partial u} \cdot du \ + \ \frac{\partial z}{\partial v} \cdot dv \ , $$

to replace in your chain-rule equation for $dw$ .