I often come across $\nabla \cdot \vec A$ but in an exercise I was asked to simplify $\vec A \cdot \nabla$, where $\vec A = xz \hat i - y^2 \hat j + 2x^2y \hat k$. I would simplify $\nabla \cdot \vec A$ as:
$$\nabla \cdot \vec A = \frac{\partial(xz)}{\partial x} - \frac{\partial(y^2)}{\partial y} + \frac{\partial(2x^2y)}{\partial z}$$ $$= z - 2y$$
And $\vec A \cdot \nabla$ as:
$$\vec A \cdot \nabla = xz\frac{\partial}{\partial x} - y^2\frac{\partial}{\partial y} + 2x^2y\frac{\partial}{\partial z}$$
Can this be simplified more?