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

LucasGB

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

LucasGB

Re: Can all vector fields be described as the vector Laplacian of another vector fiel

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

Re: Can all vector fields be described as the vector Laplacian of another vector fiel

why do you think this should be true?

LucasGB

Re: Can all vector fields be described as the vector Laplacian of another vector fiel

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

Marin

Re: Can all vector fields be described as the vector Laplacian of another vector fiel

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

Re: Can all vector fields be described as the vector Laplacian of another vector fiel

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.

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