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Question about vector fields, div, curl grad

  1. Mar 10, 2016 #1
    1. The problem statement, all variables and given/known data

    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?


    Bob Kolker
    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 10, 2016 #2


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    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)
  4. Mar 20, 2016 #3
    I have no doubt the equations are true. I am looking for a reference to a proof. Can you help me out? :Thanks.
  5. Mar 20, 2016 #4


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    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.
  6. Mar 20, 2016 #5

    Charles Link

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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.
    Last edited: Mar 20, 2016
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