Where does this result come from?

Question

I'm sorry about the non-specific title, I wasn't sure where this question would fit in...

I'm reading through a few notes for my PDE course, and I'm struggling to see where the following result comes from;

For every $v \in C^{\infty}(\Omega)$, it holds; $$v \Delta v + |Dv|^2 = div({v \cdot Dv}) $$

I haven't done vector-y things in quite a while, and I've not encountered the $D$ notation before - does it simply refer to a differential operator??

Any insight into where this result comes from would also be much appreciated.

Answer 1

$D$ is just the gradient - you may be more familiar with it as $\nabla$. This identity follows from the product rule: in coordinates we have

$$ \text{div} (v \cdot Dv) = \sum_i \partial_i (v \partial_i v) = \sum_i v \partial_i \partial_i v + \sum_i (\partial_i v \partial_i v) = Dv \cdot Dv + v \Delta v.$$ This is a specific case of the general product rule $$\text{div}(f X) = Df \cdot X + f \text{div} X,$$ which holds for all functions $f$ and vector fields $X$.