# 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:

From the vector calculus identity it follows that

that is, that the field **v** satisfies Laplace's equation.

A Laplacian vector field in the plane satisfies the Cauchy–Riemann equations: it is holomorphic.

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

Then, since the divergence 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.