# Thread: Differentiation Equation Curl, Divergence

1. ## Differentiation Equation Curl, Divergence

How do I solve these:

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

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

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

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?

2. Well, you have two equations in three unknown functions. In general, that is not enough.

3. Found what I was looking for. Helmholtz's Theorem -- from Wolfram MathWorld

4. 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.

5. $\phi$ and $A$ are given in terms of the divergence and the curl of $v$, which I know.