# 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:
    ${\begin{aligned}\nabla \times \mathbf {v} &=\mathbf {0} ,\\\nabla \cdot \mathbf {v} &=0.\end{aligned}}$

## Laplace's equation

From the vector calculus identity $\nabla ^{2}\mathbf {v} \equiv \nabla (\nabla \cdot \mathbf {v} )-\nabla \times (\nabla \times \mathbf {v} )$ it follows that
    $\nabla ^{2}\mathbf {v} =\mathbf {0}$
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 ${\bf {v}}=(xy,yz,zx)$ satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a Laplacian vector field.

## Cauchy-Riemann equations

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

## Potential of Laplacian field

Suppose the curl of $\mathbf {u}$ is zero, it follows that (when the domain of definition is simply connected) $\mathbf {u}$ can be expressed as the gradient of a scalar potential (see irrotational field) which we define as $\phi$:
    $\mathbf {u} =\nabla \phi \qquad \qquad (1)$
since it is always true that $\nabla \times \nabla \phi =0$.
Other forms of $\mathbf {u} =\nabla \phi$ can be expressed as
$u_{i}={\frac {\partial \phi }{\partial x_{1}}}\quad ;\quad u={\frac {\partial \phi }{\partial x}},v={\frac {\partial \phi }{\partial y}},w={\frac {\partial \phi }{\partial z}}$.
When the field is incompressible, then
$\nabla \cdot u=0\quad {\textrm {or}}\quad {\frac {\partial u_{j}}{\partial x_{j}}}=0\quad {\textrm {or}}\quad {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0$.
And substituting equation 1 into the equation above yields
    $\nabla ^{2}\phi =0.$
Therefore, the potential of a Laplacian field satisfies Laplace's equation.

## Potential flow theory

The Laplacian vector field has an impactful application in fluid dynamics. Consider the Laplacian vector field to be the velocity potential $\phi$ which is both irrotational and incompressible.
Irrotational flow is a flow where the vorticity, $\omega$, is zero, and since $\omega =\nabla \times u$, it follows that the condition $\omega =0$ is satisfied by defining a quantity called the velocity potential $\phi$, such that $u=\nabla \phi$, since $\nabla \times \nabla \phi =0$ always holds true.
Irrotational flow is also called potential flow.
If the fluid is incompressible, then conservation of mass requires that
∇⋅u=0or ∂u$_j$/∂x$_j$=0or ∂u/∂x+∂v/∂y+∂w/∂z=0{\textstyle \nabla \cdot u=0\quad {\textrm {or}}\quad {\frac {\partial u_{j}}{\partial x_{j}}}=0\quad {\textrm {or}}\quad {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0}.
And substituting the previous equation into the above equation yields $\nabla ^{2}\phi =0$ which satisfies the Laplace equation.
In planar flow, the stream function $\psi$ can be defined with the following equations for incompressible planar flow in the xy-plane:
$u={\frac {\partial \psi }{\partial y}}\quad {\textrm {and}}\quad v=-{\frac {\partial \psi }{\partial x}}$.
When we also take into consideration $u={\frac {\partial \phi }{\partial x}}\quad {\textrm {and}}\quad v={\frac {\partial \phi }{\partial y}}$, we are looking at the Cauchy-Reimann equations.
These equations imply several characteristics of an incompressible planar potential flow. The lines of constant velocity potential are perpendicular to the streamlines (lines of constant $\psi$) everywhere.