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.
However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a Laplacian vector field.
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.