## Laplacian of a Product

Suppose
$\phi$
and
$\psi$
are differentiable functions, then
$\nabla^2(\phi \psi) =\nabla^2 \phi + 2( \mathbf{\nabla} \phi) \cdot ( \mathbf{\nabla} psi) + \nabla^2 \psi$
.
\begin{aligned} \nabla^2 (\phi \psi) &= \frac{\partial^2 (\phi \psi)}{\partial x^2} + \frac{\partial^2 (\phi \psi)}{\partial y^2} + \frac{\partial^2 (\phi \psi)}{\partial z^2} \\ &= \frac{\partial}{\partial x} (\frac {\partial (\phi \psi)}{\partial x} ) + \frac{\partial}{\partial y} (\frac{\partial (\phi \psi)}{\partial y}) + \frac{\partial}{\partial z} \frac{\partial (\phi \psi)}{\partial z} \\ &= \frac{\partial}{\partial x} (\phi \frac{\partial \psi}{\partial x} + \frac{\partial \phi}{\partial x} ) \psi )+ \frac{\partial}{\partial y} (\phi \frac{\partial \psi}{\partial y} + \frac{\partial \phi}{\partial y} \psi )+ \frac{\partial}{\partial y} (\phi \frac{\partial \psi}{\partial y} + \frac{\partial \phi}{\partial y} \psi ) \\ &= \frac{\partial \phi}{\partial x} \frac{\partial \psi}{\partial x} +\phi \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial \psi}{\partial x} \frac{\partial \phi}{\partial x} +\psi \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial \phi}{\partial y} \frac{\partial \psi}{\partial y} +\phi \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial \psi}{\partial y} \frac{\partial \phi}{\partial y} +\psi \frac{\partial^2 \phi}{\partial y^2} \\ & + \frac{\partial \phi}{\partial z} \frac{\partial \psi}{\partial z} +\phi \frac{\partial^2 \psi}{\partial z^2} + \frac{\partial \psi}{\partial z} \frac{\partial \phi}{\partial z} +\psi \frac{\partial^2 \phi}{\partial z^2} \\ &= \phi (\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} ) + \psi (\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} )+ 2 (\frac{\partial \phi}{\partial x}\frac{\partial \psi}{\partial x} + \frac{\partial \phi}{\partial y}\frac{\partial \psi}{\partial y} + \frac{\partial \phi}{\partial z}\frac{\partial \psi}{\partial z} ) \\ &= \nabla^2 \phi + 2( \mathbf{\nabla} \phi) \cdot ( \mathbf{\nabla} \psi) + \nabla^2 \psi \end{aligned}