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:
:<math>\begin{align}
\nabla \times \mathbf{v} &= \mathbf{0}, \\
\nabla \cdot \mathbf{v} &= 0.
\end{align}</math>
Laplace's equation
From the vector calculus identity <math>\nabla^2 \mathbf{v} \equiv \nabla (\nabla\cdot \mathbf{v}) - \nabla\times (\nabla\times \mathbf{v})</math> it follows that
:<math>\nabla^2 \mathbf{v} = \mathbf{0}</math>
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 <math>{\bf v} = (xy, yz, zx)</math> 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 <math>\mathbf{u}</math> is zero, it follows that (when the domain of definition is simply connected) <math>\mathbf{u}</math> can be expressed as the gradient of a scalar potential (see irrotational field) which we define as <math>\phi</math>:
:<math> \mathbf{u} = \nabla \phi \qquad \qquad (1) </math>
since it is always true that <math> \nabla \times \nabla \phi = 0 </math>.
Other forms of <math> \mathbf{u} = \nabla \phi </math> can be expressed as
<math> 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} </math>.
And substituting the previous equation into the above equation yields <math>\nabla ^2 \phi = 0</math> which satisfies the Laplace equation. Medical researchers proposed a method to obtain high resolution in vivo measurements of fascicle arrangements in skeletal muscle, where the Laplacian vector field behavior reflects observed characteristics of fascicle trajectories.
See also
- Potential flow
- Harmonic function