Using $\epsilon_{ijk}$ notation, evaluate $$\vec{\nabla} \cdot[(\vec{a} \times \vec{r}) \times \vec{a}] \quad \text{ and} \quad \vec{\nabla} \times \vec{r}$$ where $\vec{a}$ is and arbitrary constant vector and $\vec{r}$ is the position vector.
Our professor asked us this last week and i've been scrambling around the hildebrand book to no avail, can anyone help me understand what goes on with nabla cross and dot products?
You can naively consider nabla a vector (technically, it's an operator and a co-vector, but for college maths, it's not important). $$\vec\nabla = \left(\frac\partial{\partial x},\frac\partial{\partial y},\frac\partial{\partial z}\right)^\intercal$$
When $\partial/\partial x$ is left-“multiplied” with a function, one needs to take a derivative: $(\partial/\partial x) f(x) = \partial f/\partial x = f'(x)$
Then, divergence can be written as a dot product and curl as a cross product: $$ \mathop{\mathrm{div}} \vec V = \frac{\partial}{\partial x}V_x + \frac{\partial}{\partial y}V_y+\frac{\partial}{\partial z}V_z = \vec\nabla \cdot \vec V,\\ \mathop{\mathrm{curl}} \vec V = \begin{pmatrix} \frac{\partial}{\partial y}V_z - \frac{\partial}{\partial z}V_y\\ \frac{\partial}{\partial z}V_x - \frac{\partial}{\partial x}V_z\\ \frac{\partial}{\partial x}V_y - \frac{\partial}{\partial y}V_x \end{pmatrix} = \vec\nabla \times \vec V $$
Although there is a proper math framework behind this, you can safely consider it a notation abuse.