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Math Help - Differentiation Equation Curl, Divergence

  1. #1
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    Differentiation Equation Curl, Divergence

    How do I solve these:

    \nabla \cdot \mathbf{A} = 0

    \nabla \times \mathbf{A} = 0

    Where \mathbf{A} =\left(<br />
\begin{array}{c}<br />
 A_x \\<br />
 A_y \\<br />
 A_z<br />
\end{array}<br />
\right)

    From the first equation:

    \dfrac{\partial A_x}{\partial x}+\dfrac{\partial A_y}{\partial y}+\dfrac{\partial A_z}{\partial z}=0

    From the second equation:

    \dfrac{\partial A_x}{\partial y}=\dfrac{\partial A_y}{\partial x}

    \dfrac{\partial A_x}{\partial z}=\dfrac{\partial A_z}{\partial x}

    \dfrac{\partial A_y}{\partial z}=\dfrac{\partial A_z}{\partial y}

    What next?
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  2. #2
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    Well, you have two equations in three unknown functions. In general, that is not enough.
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  3. #3
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    Found what I was looking for. Helmholtz's Theorem -- from Wolfram MathWorld
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  4. #4
    MHF Contributor

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    You realize, I hope, that that does NOT answer your original question. That simply asserts that your v can be written as v= -\nabla \phi+ \nabla\times A but the formulas for \phi and A are given in terms of v.
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  5. #5
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    \phi and A are given in terms of the divergence and the curl of v, which I know.
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