Can All Vector Fields Be Represented by the Vector Laplacian?

[LucasGB]

Can all vector fields be described as the vector Laplacian of another vector field?

 

[LucasGB]

Perhaps I should elaborate a little bit. The vector Laplacian is an operator which allows us to obtain a vector field B from a vector field A (B is the vector Laplacian of A). My question is: is it correct to say that ALL vector fields D can be though of as being the vector Laplacian of another vector field C?

 

[Mosis]

why do you think this should be true?

 

[LucasGB]

Because I saw a proof of Helmholtz's Theorem where the guy assumed this was true.

 

[Marin]

Hi!

It's an intriguing problem you're posting over here :)

So, if I got it correctly, the question is, if for any given field F there exists a field B such that \vec F=\Delta\vec B

Well, this is a vector equation, so it has (assuming it really holds) to hold for every component, which implies:

F_i=\partial^2_l B_i for all i = 1,2,3, or put another way:

F_i=\Delta B_i which is the Poisson equation.

So you have to find out if the Poisson equation always has a solution. I checked in Wikipedia - it was not clearly stated, but it looks like the equation is indeed analytically solvable via Green's functions.

 

[LucasGB]

That's a very interesting breakdown of the problem. In fact, I think, and I could be wrong, that if we don't specify boundary conditions, there are infinitely many solutions to Poisson's equation.