In vector calculus, a Laplacian vector field is a vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

 which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:
A Laplacian vector field in the plane satisfies the Cauchy-Riemann equations

Cauchy-Riemann equations

In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable...

: it is holomorphic.

Since the curl of v is zero, it follows that v can be expressed as the gradient of a scalar potential

Scalar potential

A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...

 (see irrotational field) φ :.
Then, since the divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

 of v is also zero, it follows from equation (1) that
which is equivalent to.
Therefore, the potential of a Laplacian field satisfies Laplace's equation

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

.