How do I workout the Poisson equation?

Question

I have trouble with working out the Poisson equation. I don't know how to go from step $1$ to step $2$. Why is in $(2.)$ the $k$ in front of the nabla sign? How did the nabla sign come in front of the $k$ (in the bracketed part.)

$$\begin{array}{l} (1.) -\nabla \cdot(k \nabla u)=f \\ (2.) -k \nabla \cdot \nabla u-(\nabla k) \cdot(\nabla u)=f \end{array}$$

It would be great if someone can give me a hint.

Ter

Answer 1

It is a matter of notation. For three variables, we write

• $\nabla u = (u_x,u_y,u_z)$
• $\nabla = (\partial_x, \partial_y,\partial_z)$

Then, $$\begin{aligned}\nabla\cdot(k\nabla u)&= (\partial_x, \partial_y,\partial_z)\cdot(ku_x,ku_y,ku_z)\\ &=\partial_x(k u_x)+\partial_y(ku_y)+\partial_z(ku_z) \end{aligned}$$