How to compute the following divergence

Question

$div(\nabla\nabla u\cdot\nabla u)$, where $u$ is a scalar function. I was told $\nabla\nabla u$ is a matrix. It is not clear properly.

Answer 1

$u$ is a scalar function. $\nabla u$, the gradient of $u$, is a vector field. $\nabla\nabla u$ is the matrix "field" obtained from applying the gradient to each component of $\nabla u$ separately.

For example, if $u$ is a function of $x,y,z$ then $\nabla u=\langle u_x,\ u_y,\ u_z\rangle$.

And $$\nabla\nabla u=\begin{bmatrix}u_{xx}&u_{xy}&u_{xz}\\u_{yx}&u_{yy}&u_{yz}\\u_{zx}&u_{zy}&u_{zz}\end{bmatrix}$$

Now to compute $\nabla\nabla u\cdot \nabla u$, this is just matrix multiplication and you will end up with a vector field. And I'm assuming you know how to compute the divergence of a vector field.