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solenoidal field (Definition)

A solenoidal vector field is one that satisfies

$\displaystyle \nabla \cdot\mathbf{B}= 0$
at every point where the vector field $ \mathbf{B}$ is defined. Here $ \nabla \cdot\mathbf{B}$ is the divergence.

This condition actually implies that there exists a vector $ \mathbf{A}$, such that

$\displaystyle \mathbf{B}= \nabla \times\mathbf{A}.$

For a function $ f$ satisfying Laplace's equation

$\displaystyle \nabla ^2f = 0,$
it follows that $ \nabla f$ is solenoidal.



"solenoidal field" is owned by giri.
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Other names:  solenoidal

Attachments:
vector potential (Definition) by pahio
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Cross-references: Laplace's equation, function, vector, implies, divergence, point, vector field
There are 6 references to this entry.

This is version 6 of solenoidal field, born on 2002-11-13, modified 2004-10-18.
Object id is 3590, canonical name is SolenoidalField.
Accessed 4767 times total.

Classification:
AMS MSC26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

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