What is nabla scalar (a.u) where a is a scalar field and u a vector field?

Question

We have a domain D of say R² and a function a from D to R and a function u from R² to R² what is Nabla dot (au) ? If u were from R² to R we could have simply used the product rule

Answer 1

As you have the question phrased, both $\vec{a}$ and $\vec{u}$ are vector valued functions, where the domain of $\vec{a}$ is $D$ and the domain of $\vec{u}$ is all of $\mathbb{R}^2$, so I will answer the question under that assumption. So I believe in that case you are looking for a formula for, $$ \nabla(\vec{a} \cdot \vec{u}) $$

Try to use calculus you already know on the components. So say, $\vec{u}=\langle f_1(x,y), g_1(x,y) \rangle$ and $\vec{a}=\langle f_2(x,y),g_2(x,y) \rangle $. Then,

$$\nabla(\vec{a}\cdot \vec{u})=\nabla(f_1f_2+g_1g_2)=\nabla(f_1f_2)+\nabla(g_1g_2) $$

Now just calculate the above gradients to get a general formula.