# Math Help - Simple vector calculus derivation

1. ## Simple vector calculus derivation

I believe this is simple to derive, but I can't seem to do it.

How does one get from the LHS to RHS

$f\nabla^2u=\nabla \cdot (f\nabla u)-\nabla u\cdot \nabla f$

where $f=f(x,y)$ and $u=u(x,y)$.

2. Better not to use word "simple". Everyting is relative.
At first nabla acts as differential operator of two terms:
$\nabla \cdot (f\nabla u)=(\nabla f , \nabla u)+f(\nabla , \nabla u)$
and
$f(\nabla , \nabla u)=f \nabla^2 u$

3. It is better to check it directly
$\nabla \cdot (f\nabla u)=div(f \nabla u)=div(f({ \frac{\partial u}{\partial x}},{ \frac{\partial u}{\partial y}}))=$

$={ \frac{\partial }{\partial x}}( f{ \frac{\partial u}{\partial x}})+{ \frac{\partial }{\partial y}}( f{ \frac{\partial u}{\partial y}})=...$