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:
formula_1
Laplace's equation.
From the vector calculus identity formula_2 it follows that
formula_3
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 formula_4 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 formula_5 is zero, it follows that (when the domain of definition is simply connected) formula_5 can be expressed as the gradient of a scalar potential (see irrotational field) which we define as formula_7:
formula_8
since it is always true that formula_9.
Other forms of formula_10 can be expressed as
formula_11.
When the field is incompressible, then
formula_12.
And substituting equation 1 into the equation above yields 
formula_13
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" formula_7 "which is both irrotational and incompressible.
Irrotational flow is a flow where the vorticity, formula_15, is zero, and since formula_16, it follows that the condition formula_17 is satisfied by defining a quantity called the velocity potential formula_7, such that formula_19, since formula_20 always holds true.
Irrotational flow is also called potential flow.
If the fluid is incompressible, then conservation of mass requires that 
formula_21.
And substituting the previous equation into the above equation yields formula_22 which satisfies the Laplace equation.
In planar flow, the stream function formula_23 can be defined with the following equations for incompressible planar flow in the xy-plane:
formula_24.
When we also take into consideration formula_25, 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 formula_23) everywhere.
Further reading.
The Laplacian vector field theory is being used in research in mathematics and medicine. Math researchers study the boundary values for Laplacian vector fields and investigate an innovative approach where they assume the surface is fractal and then must utilize methods for calculating a well-defined integration over the boundary. 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.
References.
 ;templatestyles src="Reflist/styles.css" / ;