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Section 14.16 The Laplacian

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 \(V\text{,}\) and is written in any of the forms

\begin{gather*} \triangle V = \nabla^2 V = \grad\cdot\grad V . \end{gather*}

In rectangular coordinates, it is easy to compute

\begin{gather*} \triangle V = \grad\cdot\grad V = \PARTIAL{V}{x} + \PARTIAL{V}{y} + \PARTIAL{V}{z} . \end{gather*}

The simplest homogeneous partial differential equation involving the Laplacian

\begin{equation} \nabla^2 V=0\tag{14.16.1} \end{equation}

is called Laplace's equation. The inhomgeneous version

\begin{equation} \nabla^2 V=f(\rr)\tag{14.16.2} \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.