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Vector Potential




Formally, given a vector field v, a ''vector potential'' is a vector field '''A''' such that
: \mathbf{v} =
abla imes \mathbf{A}.

If a vector field v admits a vector potential '''A''', then from the equality
:
abla \cdot (
abla imes \mathbf{A}) = 0
( Divergence of the Curl is zero) one obtains
:
abla \cdot \mathbf{v} =
abla \cdot (
abla imes \mathbf{A}) = 0,
which implies that v must be a Solenoidal Vector Field .

An interesting question is then if any solenoidal vector field admits a vector potential. The answer is affirmative, if the vector potential satisfies certain conditions.


THEOREM


Let
:\mathbf{v} : \mathbb R^3 o \mathbb R^3