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Homework Help
Calculus and Beyond Homework Help
Showing scalar functions u(x,y,z) and v(x,y,z) are related
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[QUOTE="bla1089, post: 4417825, member: 440957"] [b]1.a. Show that ∇F[u(x,y,z),v(x,y,z)] = F[SUB]u[/SUB]∇v + F[SUB]v[/SUB]∇u[/b] [b]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[/b] [h2]Homework Equations[/h2] ∇ = [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] [b]3. The Attempt at a Solution 1.a[/b] ∇F[u(x,y,z),v(x,y,z)] = (F[SUB]u[/SUB]u[SUB]x[/SUB] + F[SUB]v[/SUB]v[SUB]x[/SUB])[itex]\widehat{i}[/itex] + (F[SUB]u[/SUB]u[SUB]y[/SUB] + F[SUB]v[/SUB]v[SUB]y[/SUB])[itex]\widehat{j}[/itex] + (F[SUB]u[/SUB]u[SUB]z[/SUB] + F[SUB]v[/SUB]v[SUB]z[/SUB])[itex]\widehat{k}[/itex] = F[SUB]u[/SUB]∇u + F[SUB]v[/SUB]∇v[b]4. The attempt at solution 1.b[/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[SUB]u[/SUB]∇u x ∇v = 0 ∇F x ∇u = F[SUB]v[/SUB]∇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. [/QUOTE]
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Calculus and Beyond Homework Help
Showing scalar functions u(x,y,z) and v(x,y,z) are related
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