Analysis (implicit and inverse func thrms)

[steviet]

Homework Statement

The equations:

uz - 2e^(vz) = 0
u - x^2 - y^2 = 0
v^2 - xy*log(v) - 1 = 0

define z (implicitly) as a function of (u,v) and (u,v) as a function of (x,y),
thus z as a function f (x,y)

Describe the role of the inverse and implicit function theorems in the
above statement and compute

$$\partial$$z/$$\partial$$x(0,e).

(Note that when x=0 and y=e, u=e^2, v=1 and z=2)

Homework Equations

Implicit and inverse function theorems

The Attempt at a Solution

I'm finally starting to get a grasp on the 2 theorems and their
respective proofs (I hope), but as far as explaining their role in this
concrete example I'm a little lost.
If anyone can put me on the right path, it would be greatly appreciated.
(This is for an undergrad analysis course)

 

[HallsofIvy]

Perhaps it would be a good idea to check on exactly what the "implicit function theorem" and ""inverse function theorem" say!

In particular you might want to determine where, if ever, the Jacobian is zero.