# Irrotational vector field

In fluid mechanics, an irrotational vector field is a vector field whose curl is zero. If the field is denoted as v, then

[itex] \nabla \times \mathbf{v} = 0 [itex].

Since there is an identity of vector calculus which states that the curl of any gradient is zero:

[itex] \nabla \times \nabla \phi = 0 [itex]

where φ is a scalar field, it follows that any irrotational field can be expressed as the gradient of a scalar potential:

[itex] \mathbf{v} = \nabla \phi [itex].

If, in addition to being irrotational, a field is also incompressible, then the field is called a Laplacian field.

An irrotational field is practically synonymous with a lamellar field. The adjective "irrotational" implies that irrotational fluid flow (whose velocity field is irrotational) has no rotational component: the fluid does not move in circular or helical motions; it does not form vortices.

From the zero curl definition of an irrotational field, it can be deduced, by means of Stokes' theorem, that the circulation of any closed loop in the field is zero:

[itex] \oint_S \mathbf{v} \cdot \, d\mathbf{s} = \int\!\!\!\int_A \nabla \times \mathbf{v} \cdot d\mathbf{A} = 0 [itex]

where A is the area enclosed by loop S. This lack of circulation means that irrotational field lines (streamlines of irrotational flow) do not form loops (or helices).

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