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**1.a. Show that ∇F[u(x,y,z),v(x,y,z)] = F**

_{u}∇v + F_{v}∇u**1.b. Show that a necessary and sufficient condition that u and v are functionally related by the equation F(u,v) = 0 is ∇u x ∇v = 0**

## Homework Equations

∇ = [itex]\frac{\partial}{\partial x}[/itex][itex]\widehat{i}[/itex] + [itex]\frac{\partial}{\partial y}[/itex][itex]\widehat{j}[/itex] + [itex]\frac{\partial}{\partial z}[/itex][itex]\widehat{k}[/itex]

**3. The Attempt at a Solution 1.a**

∇F[u(x,y,z),v(x,y,z)] = (F

_{u}u

_{x}+ F

_{v}v

_{x})[itex]\widehat{i}[/itex] + (F

_{u}u

_{y}+ F

_{v}v

_{y})[itex]\widehat{j}[/itex] + (F

_{u}u

_{z}+ F

_{v}v

_{z})[itex]\widehat{k}[/itex] = F

_{u}∇u + F

_{v}∇v

**4. The attempt at solution 1.b**

I'm honestly stuck. The necessary and sufficient condition throws me. If I work from the assumption that F[u,v] = 0, I can get:

∇F x ∇v = F

_{u}∇u x ∇v = 0

∇F x ∇u = F

_{v}∇v x ∇u = 0

Either of which lead to ∇u x ∇v = 0

But this seems to show neither necessity nor sufficiency.

I know that this leads to developing the Jacobian and I have an inkling that the delta function may help, but can't get anywhere with that. Any pointers would be greatly appreciated.