A vector field \,
,\, defined in an open set
of
, is\, lamellar \, if the condition
is satisfied in every point \,
\, of
.
Here,
is the curl or rotor of
.\, The condition is equivalent with both of the following:
- The line integrals
taken around any closed contractible curve
vanish.
- The vector field has a scalar potential \,
\, which has continuous partial derivatives and which is up to a constant term unique in a simply connected domain; the scalar potential means that ![{\displaystyle {\vec {F}}=\nabla u.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/181260c21fb3036151db88ff7e95e952983d9bd8)
The scalar potential has the expression
where the point
may be chosen freely,\,
.\\
Note. \, In physics,
is in general replaced with\,
.\, If the
is interpreted as a force, then the potential
is equal to the work made by the force when its point of application is displaced from
to infinity.