Properties
The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as:
automatically results in the identity (as can be shown, for example, using Cartesian coordinates):
The converse also holds: for any solenoidal v there exists a vector potential A such that (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)
The divergence theorem, gives the equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero:
- ,
where is the outward normal to each surface element.
Read more about this topic: Solenoidal Vector Field
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