Question

How do you compute the first partial derivatives of w with respect to u and v given w=f(x,y)=e(^x^2+y^2) and g(u,v)={x(u,v),y(u,v)}={u cos v,u sin v} ???

Answer 1

This is an example of the chain rule for partial derivatives.
First you notice that w is a function f of x and y. But x and y themselves are functions of u and v.
So w is a function of u and v, and we want to know how w changes when u and v change as independent variables.

By the chain rule:
partial w / partial u = (partial f / partial x) * (partial x / partial u) + (partial f / partial y) * (partial y / partial u)

Similarly,
partial w / partial v = (partial f / partial x) * (partial x / partial v) + (partial f / partial y) * (partial y / partial v).

f(x, y) = e^(x^2 + y^2). So partial f / partial x = 2x * e^(x^2 + y^2).
Similarly, partial f / partial y = 2y * e^(x^2 + y^2).

Also, we see that x(u, v) = u cos v, so partial x / partial u = cos v
and partial x / partial v = -u sin v.

We also have y(u, v) = u sin v, so partial y / partial u = sin v and finally
partial y / partial v = u cos v.

Then applying the chain rule gives you the nasty final answer.