## Proof that the Gradient of a Harmonic Function is Both Sinusoidal and Irrotational

A function
$f$
is said to be harmonic if
$( \mathbf{\nabla} \cdot \mathbf{\nabla}) f = \nabla^2 f=0$
A vector
$\mathbf{v}$
solenoidal if
$\mathbf{\nabla} \cdot \mathbf{v} =0$

$( \mathbf{\nabla} \cdot \mathbf{\nabla}) f = \mathbf{\nabla} \cdot (\mathbf{\nabla} f)=0$
so
$\mathbf{\nabla} f$
is solenoidal.
A vector
$\mathbf{v}$
irrotational if
$\mathbf{\nabla} \times \mathbf{v} =0$

\begin{aligned} \mathbf{\nabla} \times (\mathbf{\nabla} f) &= (\frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \mathbf{k}) \times (\frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z} \mathbf{k})\mathbf{j}) \\ &= (\frac{\partial^2 f}{\partial y \partial z} - \frac{\partial^2 f}{\partial z \partial y}) \mathbf{i} + (\frac{\partial^2 f}{\partial z \partial x} - \frac{\partial^2 f}{\partial x \partial z}) \mathbf{j} + (\frac{\partial^2 f}{\partial x \partial y} - \frac{\partial^2 f}{\partial y \partial x}) \mathbf{k} =0 \end{aligned}

Since all partial derivatives commute if
$f$
is twice differentiable so
$\mathbf{\nabla} f$
is irrotational.