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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 $S$ 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 $\vec{U}$:

    $\displaystyle \nabla\!\times\!\vec{A} = \vec{U}$

Under those conditions, the vector field $\vec{U}$ is called solenoidal.

Bibliography

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



"vector potential" is owned by pahio.
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Other names:  solenoidal vector field
Also defines:  solenoidal

Attachments:
example of vector potential (Example) by pahio

Cross-references: divergence, vector field
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This is version 3 of vector potential, born on 2009-04-18, modified 2009-04-18.
Object id is 655, canonical name is VectorPotential.
Accessed 1417 times total.

Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
 02.40.Hw (Classical differential geometry)
Pending Errata and Addenda
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