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Homelamellar field
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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, $P=(x,\,y,\,z)$.
Defines:
scalar potential, potential, rotor
Related:
CurlFreeField, PoincareLemma, VectorPotential, GradientTheorem
Synonym:
lamellar, irrotational, conservative, laminar
Type of Math Object:
Definition
Major Section:
Reference
Parent:
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