Divergence and rotational equal to zero - solutions?

[Tosh5457]

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 :)

 

[dextercioby]

Doesn't this lead to Laplace equation ?

 

[Tosh5457]

Doesn't this lead to Laplace equation ?

Yes, it leads to the vectorial Laplace equation (each component's laplacian is 0).

 

[dextercioby]

Vectorial ? No, scalar, take curl F=0. Then F = grad phi. Phi is a scalar. Phi will be involved in a scalar equation.