Question about vector fields, div, curl grad

[bobkolker]

Homework Statement

I need a pointer to a proof of the following items:
if div X =0 then X = curl Y for some field Y.
if curl X = 0 then X = grad Y for some field Y.

Can anyone provide a pointer to a proof?

Thanks.

Bob Kolker

Homework Equations

The Attempt at a Solution

 

[Twigg]

Here are a couple of hints

First one: think about what X would look like on the boundary of a sphere
Second one: think about what X would look like on the interior of a closed plane curve (a loop that lies in a single plane)

 

[bobkolker]

I have no doubt the equations are true. I am looking for a reference to a proof. Can you help me out? :Thanks.

 

[Twigg]

Unfortunately I don't have a good reference for you, but I remember these two proofs. They both follow a similar procedure using the divergence theorem and Stokes theorem around arbitrary orientable closed surfaces and closed plane curves respectively. Start with the second one, it's a little simpler.

 

[Charles Link]

I think this is called Helmholtz's theorem in E&M (Electricity and Magnetism). The div(curl A)=0 in all cases and also curl (grad V)=0 in all cases, but the converse that there exists a field, etc. is Helmholtz's theorem.