Laplacian vector field

In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:

<math> \nabla \times \mathbf{v} = 0, <math>

<math> \nabla \cdot \mathbf{v} = 0. <math>

Since the curl of v is zero, it follows that v can be expressed as the gradient of a scalar potential (see irrotational field) φ :

<math> \mathbf{v} = \nabla \phi \qquad \qquad (1) <math>.

Then, since the divergence of v is also zero, it follows from equation (1) that

<math> \nabla \cdot \nabla \mathbf{v} = 0 <math>

which is equivalent to

<math> \nabla^2 \phi = 0 <math>.

Therefore, the potential of a Laplacian field satisfies Laplace's equation.

See also: potential flow, harmonic function