Here's the first one. I thought it was as simple as multiplying f by the Kronecker Delta (which gives the identity matrix with f's instead of 1's), but now I'm not sure if the δ term is actually supposed to be the Dirac delta? I know nothing about the Dirac delta, as it's not covered in my book or my lectures. What would be the best way to prove this?
∇ ⋅ (f δ) = ∇f
For this final one, I'm having a similar issue with the last one. I don't know if the δ term is Kronecker or Dirac, nor do I know what to do with it. I'm also not familiar with using the : rule, although I've been familiarizing myself with it online.
δ : ∇a = ∇ ⋅ a
Indeed $\delta$ is the identity matrix, its entries given by the Kronecker delta. With implicit summation over repeated indices your equations are $\partial_i (f\delta_{ij})=\partial_j f,\,\delta_{ij}\partial_i a_j =\partial_i a_i$. Note $:$ takes the trace of a matrix product.