In vector calculus, an irrotational or conservative vector field is a vector field whose curl is zero. If the field is denoted as $\backslash mathbf\{v\}$, then

$\backslash operatorname\{curl\}\; \backslash ,\; \backslash mathbf\{v\}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{v\}\; =\; 0$.

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

$\backslash operatorname\{curl\}\; \backslash ,\; \backslash nabla\; \backslash phi\; =\; \backslash nabla\; \backslash times\; \backslash nabla\; \backslash phi\; =\; 0$

where $\backslash phi$ is a scalar field.

Conversely, any irrotational field can be expressed as the gradient of a scalar potential:

$\backslash mathbf\{v\}\; =\; \backslash nabla\; \backslash phi$.

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

In fluid mechanics, 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:

$\backslash oint\_S\; \backslash mathbf\{v\}\; \backslash cdot\; \backslash ,\; d\backslash mathbf\{s\}\; =\; \backslash int\backslash !\backslash !\backslash !\backslash int\_A\; \backslash nabla\; \backslash times\; \backslash mathbf\{v\}\; \backslash cdot\; d\backslash mathbf\{A\}\; =\; 0$

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).

Vector calculus