What does $\text{div} (A \text{ grad }b)$ mean?

Question

I often see this term in my Applied Mathematics course.

If $\text{grad }b= \frac{\partial b}{\partial x}\hat{i}+\frac{\partial b}{\partial y}\hat{j}+\frac{\partial b}{\partial z}\hat{k}$, then would $\text{div}(A\text{ grad }b)$ mean $K(\frac{\partial^2 b}{\partial x^2}+\frac {\partial^2 b}{\partial y \partial x}+\frac {\partial^2 b}{\partial z \partial x})\hat{i}+...$?

Answer 1

Do you mean $\nabla\cdot\vec \nabla X$?

Assuming X is a scalar potential field like electrical potential [voltage unit], Gravitational potential [J/Kg] or fluid potential (I am from electrical engineering side, so not such good at mechanical examples),

Then, $\vec \nabla X$ gives you the vector field of that potential such as electrical field, gravity field or ...

and $\nabla\cdot\vec \nabla X$ gives you the source of that field such as density of electric charge at each point or mass density which causes Gravitational force or ...

Answer 2

$\nabla \cdot (\vec A \nabla \phi)$ is a vector that can be written

$$\nabla \cdot (\vec A \nabla \phi)=\left(\hat x_i\frac{\partial }{\partial x_i}\right)\cdot \left(\hat x_jA_j\hat x_k \frac{\partial \phi}{\partial x_k}\right)=(\nabla \cdot \vec A)\nabla \phi +(\vec A \cdot \nabla)(\nabla \phi)$$