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Homework Help: Vector Calc Problem

  1. Sep 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that[tex]\int (\nabla X A) d\tau[/tex] = -[tex]\int A X da[/tex]
    Where A is an arbitrary Vector Field and S is the surface bounding the volume V (The left hand side is supposed to be an integral over a volume and the right side an integral over a surface, but I didn't know how to put it in)
    Hint: Consider the dot product of the left hand side with an arbitrary constant vector and use (V1xV2)[tex]\bullet[/tex]V3=V1[tex]\bullet[/tex](V2x V3) and [tex]\nabla\bullet[/tex](V1x V2) = V2[tex]\bullet[/tex]([tex]\nabla[/tex] x V1)-V1[tex]\bullet[/tex]([tex]\nabla[/tex] x V2)
    where V1, V2, and V3 are any vector fields.


    2. Relevant equations
    I'm thinking either the divergence theorem or the curl theorem will come into play here, but i'm not sure which one, if not both.


    3. The attempt at a solution
    I dotted the left side with [tex]\stackrel{B\rightarrow}{}[/tex], my arbitrary constant vector, then used a circular permutation to write it as [tex]\nabla[/tex][tex]\bullet[/tex]([tex]\stackrel{A\rightarrow}{}[/tex] X [tex]\stackrel{B\rightarrow}{}[/tex]) d[tex]\tau[/tex], then used the second formula in the hint to expand it. I'm thinking ([tex]\nabla[/tex] X [tex]\stackrel{B\rightarrow}{}[/tex]) will go to 0 because [tex]\stackrel{B\rightarrow}{}[/tex] is a constant, but I don't know where to go from here.

    4. Note on notation
    Those fat dots are supposed to indicate a dot product, and the letters with arrows are supposed to be vector fields. I'm not too sure on how to work this yet.
     
  2. jcsd
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