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: \operatorname{curl} \, \mathbf{v} =
abla imes \mathbf{v} = 0 .

There is an identity of Vector Calculus which states that the curl of any Gradient is zero:
: \operatorname{curl} \,
abla \phi =
abla imes
abla \phi = 0
where \phi is a scalar field.

Conversely, any irrotational field can be expressed as the gradient of a Scalar Potential :
: \mathbf{v} =
abla \phi .

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

In .

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:
: \oint_S \mathbf{v} \cdot \, d\mathbf{s} = \int\!\!\!\int_A
abla imes \mathbf{v} \cdot d\mathbf{A} = 0
where ''A'' is the area enclosed by loop ''S''. This lack of circulation means that irrotational field lines ( Streamline s of irrotational flow) do not form loops (or Helices ).