I believe this is simple to derive, but I can't seem to do it.
How does one get from the LHS to RHS
$\displaystyle f\nabla^2u=\nabla \cdot (f\nabla u)-\nabla u\cdot \nabla f$
where $\displaystyle f=f(x,y)$ and $\displaystyle u=u(x,y)$.
I believe this is simple to derive, but I can't seem to do it.
How does one get from the LHS to RHS
$\displaystyle f\nabla^2u=\nabla \cdot (f\nabla u)-\nabla u\cdot \nabla f$
where $\displaystyle f=f(x,y)$ and $\displaystyle u=u(x,y)$.
It is better to check it directly
$\displaystyle \nabla \cdot (f\nabla u)=div(f \nabla u)=div(f({ \frac{\partial u}{\partial x}},{ \frac{\partial u}{\partial y}}))=$
$\displaystyle ={ \frac{\partial }{\partial x}}( f{ \frac{\partial u}{\partial x}})+{ \frac{\partial }{\partial y}}( f{ \frac{\partial u}{\partial y}})=...$