Functional relation between different functions of(x,y,z)

[souviktor]

Homework Statement

I have two scalar functions u(x,y,z) and v(x,y,z) which are differentiable..Now it is required to prove that a necessary and sufficient condition for these two to be functionally related by equation F(u,v)=0 is [$$\nabla$$u] $$\times$$ [$$\nabla$$v]=0

The Attempt at a Solution

clearly the cross products of the gradients are zero that means they point in he same direction.But what about the tangent planes?and how to approach this problem?

 

[lurflurf]

0=F
0=grad(F)
0=(Fu)grad(u)+(Fv)grad(v)
so clearly we need
0=grad(u)xgrad(v)