Divergence operating on a vector field ($\mathbf{u}=[u, v, w]\in\mathbb{R}^3$) outputs a scalar field:
$\nabla\cdot \mathbf{u} = \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\in \mathbb{R}$
If there is no dot product symbol $\cdot$, is it a gradient on a vector?
$\nabla \mathbf{u} = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x}\\\frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y}\\\frac{\partial u}{\partial z} & \frac{\partial v}{\partial z} & \frac{\partial w}{\partial z} \end{bmatrix}\in \mathbb{R}^{3 \times 3}$
It seems the dot product symbol is really really important - should not make a typo by mistake.