\[f\]
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 \[\mathbf{\nabla} \times \mathbf{v} =0\]
\[\begin{equation} \begin{aligned} \mathbf{\nabla} \times (\mathbf{\nabla} f) &= (\frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \mathbf{k}) \times (\frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z} \mathbf{k})\mathbf{j}) \\ &= (\frac{\partial^2 f}{\partial y \partial z} - \frac{\partial^2 f}{\partial z \partial y}) \mathbf{i} + (\frac{\partial^2 f}{\partial z \partial x} - \frac{\partial^2 f}{\partial x \partial z}) \mathbf{j} + (\frac{\partial^2 f}{\partial x \partial y} - \frac{\partial^2 f}{\partial y \partial x}) \mathbf{k} =0 \end{aligned} \end{equation}\]
Since all partial derivatives commute if
\[f\]
is twice differentiable so \[\mathbf{\nabla} f \]
is irrotational.