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 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 ${\vec {F}}=\nabla u.$

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,\, $P=(x,\,y,\,z)$.\\

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.