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Homework Help: Vector decomposition (Helmholtz)

  1. Feb 18, 2010 #1
    I have to show that a generic vector can be decomposed into an irrotational and solenoidal component:

    V(r) = -Grad[phi(r)] + Curl[A(r)]

    I'm not sure how to start. Do I need to take the curl or div of V and use a vector identity? Any help would be greatly appreciated!
  2. jcsd
  3. Feb 18, 2010 #2
    Helmholtz' Theorem starts with the two components in my original post and defines the divergence and curl as:

    div[V] = s(r)
    curl[V] = c(r), where div[c(r)] = 0

    But I can't find anything about how we can define a generic vector as two components:

    V = -grad[phi] + curl[A], where "phi" is the scalar potential and "A" is the vector potential. I need to do this before I can show that s(r) and c(r) uniquely specify the vector.

    I hope that makes my problem a little more clear.
  4. Feb 18, 2010 #3


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    Taking the divergence/curl of both sides of this equation seems like a good place to start. What do you get when you do that?

    P.S. You may wish to use boldface font to denote vectors, to make things clear.
  5. Feb 18, 2010 #4
    I actually just found an easy way of showing it using projection operators. Thanks for the reply.

    Consider this question solved.
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