I am tasked to prove that $\nabla (\phi {\bf a}) = \phi \nabla {\bf a} + \nabla \phi \cdot {\bf a}$, where $\phi$ and ${\bf a}$ are scalar and vector fields respectively. I have a few questions regarding this proof. The first is that on inspection, there seems to be some sort of dimensional inconsistency. On the LHS, we are given the gradient of a vector (which I am unsure of how to interpret, but I’ll leave that for another time), while the last term of the RHS is supposedly a scalar – am I interpreting the equation correctly?
I also tried to prove this using index notation, which gives me
$$ \partial_i (\phi a_j) = \phi \partial_i a_j + a_j \partial_i \phi$$
My logic for using different indices for the partial derivative and the vector field was because we were taking the gradient. Is this correct? Furthermore, how do i interpret $\partial_i a_j$ in the second term of the sum, and why are we allowed to compress the third term into a dot product without $\delta_ij$? Thank you very much!