User:Kounelaki.27/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

$\nabla ^{2}\mathbf {v} =\mathbf {0}$

and thus 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.

Potential of Laplacian field

Since the curl of $\mathbf {v}$ is zero, it follows that (when the domain of definition is simply connected) $\mathbf {v}$ can be expressed as the gradient of a scalar potential (see irrotational field) which we define as $\phi$:

$\mathbf {v} =\nabla \phi \qquad \qquad (1)$

since it is always true that $\nabla \times \nabla \phi =0$.

Other forms of {} can be expressed as

{}.

When the field is incompressible, then

{}.

And substituting equation 3 into the above equation yields

{}.

Therefore, the potential of a Laplacian field satisfies Laplace's equation.

Cauchy-Reimann equations

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

Potential flow theory

The Laplacian vector field has an impactful application in fluid dynamics. Consider the Laplacian vector field to be the velocity field v which is both irrotational and incompressible.

Irrotational flow is defined as a flow in which the vorticity, $\omega$, is zero, and since

See also

• Potential flow
• Harmonic function