PlanetPhysics/Lamellar Field

A vector field \,$$\vec{F} = \vec{F}(x,\,y,\,z)$$,\, defined in an open set $$D$$ of $$\mathbb{R}^3$$, is\, lamellar \, if the condition $$\nabla\!\times\!\vec{F} = \vec{0}$$ is satisfied in every point \,$$(x,\,y,\,z)$$\, of $$D$$.

Here, $$\nabla\!\times\!\vec{F}$$ is the curl or rotor of $$\vec{F}$$.\, The condition is equivalent with both of the following:

The scalar potential has the expression $$u = \int_{P_0}^P\vec{F}\cdot d\vec{s},$$ where the point $$P_0$$ may be chosen freely,\, $$P = (x,\,y,\,z)$$.\\
 * The line integrals $$\oint_s \vec{F}\cdot d\vec{s}$$ taken around any closed contractible curve $$s$$ vanish.
 * The vector field has a scalar potential \, $$u = u(x,\,y,\,z)$$\, which has continuous partial derivatives and which is up to a constant term unique in a simply connected domain; the scalar potential means that $$\vec{F} = \nabla u.$$

Note. \, In physics, $$u$$ is in general replaced with\, $$V = -u$$.\, If the $$\vec{F}$$ is interpreted as a force, then the potential $$V$$ is equal to the work made by the force when its point of application is displaced from $$P_0$$ to infinity.