PlanetPhysics/Lamellar Field

A [[../NeutrinoRestMass/|vector field]] \,$\overrightarrow{F}=\overrightarrow{F}(x,y,z)$,\, defined in an open set $D$ of ${ℝ}^3$, is\, lamellar \, if the condition $\mathrm{\nabla }\times \overrightarrow{F}=\overrightarrow{0}$ is satisfied in every point \,$(x,y,z)$\, of $D$.

Here, $\mathrm{\nabla }\times \overrightarrow{F}$ is the [[../Curl/|curl]] or rotor of $\overrightarrow{F}$.\, The condition is equivalent with both of the following:

• The line integrals ${\oint }_s\overrightarrow{F}\cdot d\overrightarrow{s}$ 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 [[../Bijective/|domain]]; the [[../Vectors/|scalar]] potential means that $\overrightarrow{F}=\mathrm{\nabla }u.$

The scalar potential has the expression $u=\underset{P_0}{\overset{P}{\int }}\overrightarrow{F}\cdot d\overrightarrow{s},$ 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 $\overrightarrow{F}$ is interpreted as a force, then the potential $V$ is equal to the [[../Work/|work]] made by the force when its point of application is displaced from $P_0$ to infinity.