lamellar field

A vector fieldF=F(x,y,z),  defined in an open set D of 3, is  lamellar  if the condition


is satisfied in every point  (x,y,z)  of D.

Here, ×F is the curl or rotor of F.  The condition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath with both of the following:

  • The line integrals


    taken around any contractible curve s vanish.

  • The vector field has a   u=u(x,y,z)  which has continuousMathworldPlanetmathPlanetmath partial derivativesMathworldPlanetmath and which is up to a unique in a simply connected domain; the scalar potential means that


The scalar potential has the expression


where the point P0 may be chosen freely,  P=(x,y,z).

Note.  In physics, u is in general replaced with  V=-u.  If the 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 P0 to infinityMathworldPlanetmath.

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