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bla1089
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1.a. Show that ∇F[u(x,y,z),v(x,y,z)] = Fu∇v + Fv∇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
∇ = [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)] = (Fuux + Fvvx)[itex]\widehat{i}[/itex] + (Fuuy + Fvvy)[itex]\widehat{j}[/itex] + (Fuuz + Fvvz)[itex]\widehat{k}[/itex] = Fu∇u + Fv∇v4. 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 = Fu∇u x ∇v = 0
∇F x ∇u = Fv∇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.
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)] = (Fuux + Fvvx)[itex]\widehat{i}[/itex] + (Fuuy + Fvvy)[itex]\widehat{j}[/itex] + (Fuuz + Fvvz)[itex]\widehat{k}[/itex] = Fu∇u + Fv∇v4. 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 = Fu∇u x ∇v = 0
∇F x ∇u = Fv∇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.