Physics Library
 An open source physics library
Encyclopedia | Forums | Docs | Random | Template Test |  
Login
create new user
Username:
Password:
forget your password?
Main Menu
Sections

Talkback

Downloads

Information
[parent] lamellar field (Topic)

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

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

    $\displaystyle \oint_s \vec{F}\cdot d\vec{s}$
    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

    $\displaystyle \vec{F} = \nabla u.$
The scalar potential has the expression

$\displaystyle u = \int_{P_0}^P\vec{F}\cdot d\vec{s},$
where the point $P_0$ 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 force, then the potential $V$ is equal to the work made by the force when its point of application is displaced from $P_0$ to infinity.



Anyone with an account can edit this entry. Please help improve it!

"lamellar field" is owned by pahio.

View style:


This object's parent.

Attachments:
examples of lamellar field (Example) by pahio

Cross-references: work, scalar, domain, curl, vector field
There is 1 reference to this object.

This is version 1 of lamellar field, born on 2008-08-27.
Object id is 292, canonical name is LamellarField.
Accessed 449 times total.

Classification:
Physics Classification: 02.30.Em (Potential theory)
 45.20.Dd (Newtonian mechanics)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:

No messages.

Testing some escape charachters for html category with a generator has an injective cogenerator" now escape ” with "