The Laplacian Operator, $\nabla ^{2}$

The Laplacian of a scalar function $f$ is

$$\nabla^{2}f = \nabla \cdot \left (\nabla f\right ) = {\pa... ...\over \partial y^{2}} + {\partial^{2}f\over \partial z^{2}},$$

and so is a natural generalisation of $d^{2}f/dx^{2}$ for the one dimensional function $f(x)$ to the thre dimensional function $f(x,y,z)$. Formulae involving combinations of functions or vector products can be worked out by remembering that $\nabla$ is both an operator and a vector. Thus,

$${d\over dx}(fg) = f{dg\over dx} + {df\over dx}g,$$

and so

$$\nabla (fg) = f \nabla g + g \nabla f,$$ (1.24)

and

$$\nabla \cdot (f{\bf A}) = f \nabla \cdot {\bf A} + \nabla f \cdot {\bf A}.$$ (1.25)