lamellar field
A vector field $\overrightarrow{F}=\overrightarrow{F}(x,y,z)$, defined in an open set $D$ of ${\mathbb{R}}^{3}$, is lamellar if the condition
$$\nabla \times \overrightarrow{F}=\overrightarrow{0}$$ 
is satisfied in every point $(x,y,z)$ of $D$.
Here, $\nabla \times \overrightarrow{F}$ is the curl or rotor of $\overrightarrow{F}$. The condition is equivalent^{} with both of the following:

•
The line integrals
$${\oint}_{s}\overrightarrow{F}\cdot \mathit{d}\overrightarrow{s}$$ 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
$$\overrightarrow{F}=\nabla u.$$
The scalar potential has the expression
$$u={\int}_{{P}_{0}}^{P}\overrightarrow{F}\cdot \mathit{d}\overrightarrow{s},$$ 
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 , 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  20130322 14:43:44 
Last modified on  20130322 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 