Question about vector fields, div, curl grad

  • #1
14
3

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

Homework Equations




The Attempt at a Solution

 

Answers and Replies

  • #2
Twigg
Gold Member
205
36
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)
 
  • #3
14
3
I have no doubt the equations are true. I am looking for a reference to a proof. Can you help me out? :Thanks.
 
  • #4
Twigg
Gold Member
205
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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.
 
  • #5
Charles Link
Homework Helper
Insights Author
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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.
 
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