## 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 any one provide a pointer to a proof?

Thanks.

Bob Kolker

## The Attempt at a Solution

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Twigg
Gold Member
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)

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

Twigg
Gold Member
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.