One second derivative, the divergence of the gradient, occurs so often it has its own name and notation. It is called the Laplacian of the function $V$, and is written in any of the forms \begin{eqnarray*} \triangle V = \nabla^2 V = \grad\cdot\grad V \end{eqnarray*} In rectangular coordinates, it is easy to compute \begin{eqnarray*} \triangle V = \grad\cdot\grad V = \PARTIAL{V}{x} + \PARTIAL{V}{y} + \PARTIAL{V}{z} \end{eqnarray*}

The simplest homogeneous partial differential equation involving the Laplacian \begin{equation} \nabla^2 V=0 \end{equation} is called Laplace's equation. The inhomgeneous version \begin{equation} \nabla^2 V=f(\rr) \end{equation} is known as Poisson's equation. There are many important techniques for solving these equations that are beyond the scope of this text.