Laplace's Equation

The scalar form of Laplace's equation is the partial differential equation,

$$\nabla^2 \phi = 0$$ (1)

where $\nabla^2$ is the second-order spatial derivative operator, and $\phi$ is the sought PF. Laplace's equation is a special case of the Helmholtz differential equation,

$$\nabla^2 \phi + k^2 \phi = 0,$$ (2)

when wavenumber k=0. A physical interpretation of this observation is that PF $\phi$ is the zero-frequency solution of the frequency-domain wave-equation, and is independent of the velocity field and thereby solely a geometric construct. A harmonic PF satisfying Laplace's equation has a number of important properties that are valid either on the boundary of, or entirely within, the defining domain. Most relevant to this discussion are that a PF:

• is uniquely determined by either the values, or normal derivatives thereof, on the domain boundary;
• has an average value over a spherical neighborhood equal to the value at its center; hence, PFs do not have local maxima or minima in the domain;
• is curl-free (i.e., $\nabla \times \phi =0$) ensuring non-overlapping, and at most simply connected, equipotential surfaces;
• gradient field is uniquely defined, locally orthogonal to the equipotential surfaces, and related to the PF through,  

  $$\phi (b) -\phi (a) = \int_{a}^b {\rm d}\phi = \int_{a}^b \... ...y}{\rm d}y + \frac{\partial \phi}{\partial z}{\rm d}z \right).$$ (3)

Each of these properties make PF solutions of Laplace's equation an important tool for generating orthogonal coordinate systems.