# Differentiation Equation Curl, Divergence

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• Jul 11th 2009, 03:58 PM
fobos3
Differentiation Equation Curl, Divergence
How do I solve these:

$\displaystyle \nabla \cdot \mathbf{A} = 0$

$\displaystyle \nabla \times \mathbf{A} = 0$

Where $\displaystyle \mathbf{A} =\left( \begin{array}{c} A_x \\ A_y \\ A_z \end{array} \right)$

From the first equation:

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

From the second equation:

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

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

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

What next?
• Jul 11th 2009, 04:11 PM
HallsofIvy
Well, you have two equations in three unknown functions. In general, that is not enough.
• Jul 12th 2009, 04:18 AM
fobos3
Found what I was looking for. Helmholtz's Theorem -- from Wolfram MathWorld
• Jul 12th 2009, 08:30 AM
HallsofIvy
You realize, I hope, that that does NOT answer your original question. That simply asserts that your v can be written as $\displaystyle v= -\nabla \phi+ \nabla\times A$ but the formulas for $\displaystyle \phi$ and A are given in terms of v.
• Jul 12th 2009, 11:00 AM
fobos3
$\displaystyle \phi$ and $\displaystyle A$ are given in terms of the divergence and the curl of $\displaystyle v$, which I know.