# 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:

•  $\oint_{s}\vec{F}\cdot d\vec{s}$

taken around any contractible curve $s$ vanish.

• The vector field has a   $u=u(x,\,y,\,z)$  which has continuous partial derivatives and which is up to a 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 , then the potential $V$ is equal to the work made by the when its point of application is displaced from $P_{0}$ to infinity.

 Title lamellar field Canonical name LamellarField Date of creation 2013-03-22 14:43:44 Last modified on 2013-03-22 14:43:44 Owner pahio (2872) Last modified by pahio (2872) Numerical id 18 Author pahio (2872) Entry type Definition Classification msc 26B12 Synonym lamellar Synonym irrotational Synonym conservative Synonym laminar Related topic CurlFreeField Related topic PoincareLemma Related topic VectorPotential Related topic GradientTheorem Defines scalar potential Defines potential Defines rotor