PlanetMath: vector potential

PlanetMath

(more info)

Math for the people, by the people.

Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS

Login

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Owner confidence rating: Very high

Entry average rating: No information on entry rating

[parent]

vector potential

(Definition)

Let $\vec{U} = \vec{U}(x,\,y,\,z)$ be a vector field in $\mathbb{R}^3$ with continuous partial derivatives. Then the following three conditions are equivalent:

The surface integrals of $\vec{U}$ over all contractible closed surfaces vanish:
$\displaystyle \oint_S\vec{U}\cdot d\vec{S} = 0$
The divergence of $\vec{U}$ vanishes everywhere in the field:
$\displaystyle \nabla\!\cdot\!\vec{U} = 0$
There exists the vector potential $\vec{A} = \vec{A}(x,\,y,\,z)$ of $\vec{U}$ :
$\displaystyle \nabla\!\times\!\vec{A} = \vec{U}$

Bibliography

1: K. V¨AISÄLÄ: Vektorianalyysi. Werner Söderström Osakeyhtiö, Helsinki (1961).

"vector potential" is owned by pahio.

(view preamble)

See Also: surface integration with respect to area, lamellar field, Kalle Väisälä

This object's parent.

Attachments:: example of vector potential (Example) by pahio

Log in to rate this entry.
(view current ratings)

Cross-references: divergence, vanish, contractible, integrals, surface, partial derivatives, continuous, vector field
There are 3 references to this entry.

This is version 5 of vector potential, born on 2006-02-28, modified 2008-03-03.
Object id is 7662, canonical name is VectorPotential.
Accessed 1450 times total.

Classification:

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

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact

post | correct | update request | add derivation | add example | add (any)