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Vector analysis identity

  1. May 16, 2007 #1
    1. The problem statement, all variables and given/known data

    we are to show a=(1/2) closed loop integral over [r x dl]

    2. Relevant equations

    3. The attempt at a solution

    I suppose this can be done formally from the alternative form of Stokes' theorem that can be obtained by replacing the vector field in curl theorem by VxC where C is a constant vector

    The identity is :

    surface int [(da x grad) x V]=closed loop integral over [dl x V]

    The RHS matches.But how to show that LHS leads to the required value?
  2. jcsd
  3. May 16, 2007 #2

    Meir Achuz

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    There is an identity:
    [tex]\oint{\bf dr\times V}={\bf \int(\nabla V )\cdot dS
    - \int dS(\nabla\cdot V)}[/tex].
    This can be derived by dotting the left hand side by a constatn vector, and then applying Stokes' theorem.
    Applying this with V=r works.
    Last edited: May 16, 2007
  4. May 16, 2007 #3
    OK,thank you.Your method worked nicely...
    First I was sceptical about the grad V in your RHS...However,I started from the very beginning by dotting c with the required integral and it worked well.
  5. May 17, 2007 #4

    Meir Achuz

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    There is an easier way I overlooked. Just take
    [tex]{\vec k}\cdot\oint{\vec r}\times{\vec dr}[/tex]
    where k is a constant vector, and apply Stokes' theorem.
    Last edited: May 17, 2007
  6. May 17, 2007 #5
    I did just that...your dr reolaced by dl...
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