Understanding $a\vec b * \nabla c$

Question

I have some difficulties when working with nabla-operators and I might need advice.

Is it correct to say that $$a\vec b \cdot \nabla c = a\nabla \cdot (\vec bc) + ac\nabla \cdot \vec b~~~?$$

If so, why?

Thanks in advance!

Answer 1

Given two functions $c:\mathbb{R}^n\longrightarrow \mathbb{R}$ and $b:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ you can compute $\nabla \cdot (b(x)c(x))$ to be \begin{align*} \nabla \cdot (b(x)c(x)) &= \sum_{i=1}^n\frac{\partial }{\partial x_i}(b_i(x)c(x)) = \sum_{i=1}^n\left(\frac{\partial }{\partial x_i}(b_i(x))c(x)+b_i(x)\frac{\partial }{\partial x_i}c(x)\right)\\ &= c(x)\left(\nabla \cdot b(x)\right)+b(x)\cdot \nabla c(x) \end{align*} Thus your formula should be $$a b \cdot (\nabla c) = a\nabla \cdot (bc)-ac \nabla \cdot b$$ with $b,c$ as above and $a:\mathbb{R}\longrightarrow \mathbb{R}$ arbitrary.