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Math Help - Volume integral to a surface integral ... divergence theorem

  1. #1
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    Volume integral to a surface integral ... divergence theorem

    Suppose A is a vector satisfying curlA = aA, a a function of position.

    Is it possible to turn the volume integral of A^2 into a surface integral? I know I have to write A^2 as the divergence of something, but what?
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  2. #2
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    A couple of questions for clarification.

    1. What quantities here are vector quantities, and which ones are scalar?

    2. Which multiplication do you mean when you write \mathbf{A}^{\!2}? Is that \mathbf{A}\cdot\mathbf{A}? Or is it \mathbf{A}\times\mathbf{A}?

    3. I'm also curious as to what physical quantity \mathbf{A} represents. Is that the vector magnetic potential?
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  3. #3
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    Hi Ackbeet,

    1/. a is scalar, A is a vector
    2/. A^2 = A dot A
    3/. Yes, it is a vector magnetic potential
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  4. #4
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    Ok, so re-writing your problem statement, I have the following:

    Assume \nabla\times\mathbf{A}=a\mathbf{A}, where \mathbf{A} is the vector magnetic potential defined by \mathbf{B}=\nabla\times\mathbf{A}, where \mathbf{B} is the magnetic field, and the scalar function a depends only on the coordinates.

    We wish to show that we can convert

    \displaystyle \iiint_{V}(\mathbf{A}\cdot\mathbf{A})\,dV into a surface integral via the divergence theorem.

    The divergence theorem states that if \mathbf{F} is a continuously differentiable vector field defined on a neighborhood of V, then

    \displaystyle\iiint_{V}(\nabla\cdot \mathbf{F})\,dV=\iint_{S}\mathbf{F}\cdot\mathbf{n}  \,dS.

    Here's a question: do you know if a is ever zero? Because, if not, you might be able to manipulate the equation

    \displaystyle \frac{\nabla\times\mathbf{A}}{a}=\mathbf{A} to simplify

    \displaystyle\mathbf{A}\cdot\mathbf{A}=\left(\frac  {\nabla\times\mathbf{A}}{a}\right)\cdot\left(\frac  {\nabla\times\mathbf{A}}{a}\right)=\frac{1}{a^{2}}  \left(\nabla\times\mathbf{A}\right)\cdot\left(\nab  la\times\mathbf{A}\right).

    I have another question: are you told that you can perform this transformation? Or are you merely trying to simplify a particular volume integral? If the second, can you provide a little more context?
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