Divergence and rotational equal to zero - solutions?

Tosh5457
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Hi, I'd like to know the solutions for these equations, and how to arrive at them. Is it possible to derive the general form of F(x,y,z) analytically? I'm still studying linear differential equations so I have no clue on what to do with partial differential equations...

div F = 0
curl F = 0

Thanks :)
 
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Doesn't this lead to Laplace equation ?
 
dextercioby said:
Doesn't this lead to Laplace equation ?

Yes, it leads to the vectorial Laplace equation (each component's laplacian is 0).
 
Vectorial ? No, scalar, take curl F=0. Then F = grad phi. Phi is a scalar. Phi will be involved in a scalar equation.
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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