PlanetMath: lamellar field

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lamellar field

(Definition)

A vector field $\vec{F} = \vec{F}(x,\,y,\,z)$ , defined in an open set of $\mathbb{R}^3$ , is lamellar if the condition

$\displaystyle \nabla\!\times\!\vec{F} = \vec{0}$

is satisfied in every point $(x,\,y,\,z)$ of

.

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
$\displaystyle \oint_s \vec{F}\cdot d\vec{s}$
taken around any closed contractible curve 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
$\displaystyle \vec{F} = \nabla u.$

The scalar potential has the expression

$\displaystyle u = \int_{P_0}^P\vec{F}\cdot d\vec{s},$

where the point

may be chosen freely, $P = (x,\,y,\,z)$ .

Note. In physics, is in general replaced with . If the $\vec{F}$ is interpreted as a force, then the potential is equal to the work made by the force when its point of application is displaced from to infinity.

"lamellar field" is owned by pahio.

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See Also: curl free field, Poincaré lemma, vector potential

Other names:

lamellar, irrotational, conservative, laminar

Also defines:

scalar potential, potential, rotor

This object's parent.

Attachments:: examples of lamellar field (Example) by pahio

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Cross-references: infinity, application, expression, domain, simply connected, partial derivatives, continuous, vanish, curve, contractible, line integrals, equivalent, point, open set, vector field
There are 50 references to this entry.

This is version 15 of lamellar field, born on 2004-10-11, modified 2008-08-27.
Object id is 6360, canonical name is LaminarField.
Accessed 7911 times total.

Classification:

26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

Pending Errata and Addenda

None.[ View all 4 ]

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