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

The Laplacian of a scalar function $f$ is

\nabla^{2}f = \nabla \cdot \left (\nabla f\right ) =
...\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,
\end{displaymath} (1.24)

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

Prof. Alan Hood