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 v can be expressed as the gradient of a scalar potential φ :
Then, since the divergence of v is also zero, it follows from equation that
which is equivalent to
Therefore, the potential of a Laplacian field satisfies Laplace's equation.
