Vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a vector potential is a vector field A such thatConsequence
If a vector field v admits a vector potential A, then from the equality
one obtains
which implies that v must be a solenoidal vector field.Theorem
Let
be a solenoidal vector field which is twice continuously differentiable. Assume that v decreases sufficiently fast as ||x||→∞. Define
Then, A is a vector potential for v, that is,
A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.Nonuniqueness
The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is
where f is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.
This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.
