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, a vector potential is a vector field such that
Consequence
If a vector field admits a vector potential, then from the equalityone obtains
which implies that must be a solenoidal vector field.
Theorem
Letbe a solenoidal vector field which is twice continuously differentiable. Assume that decreases at least as fast as for. Define
where denotes curl with respect to variable. Then is a vector potential for. That is,
The integral domain can be restricted to any simply connected region. That is, also is a vector potential of, where
A generalization of this theorem is the Helmholtz decomposition theorem, which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.
By analogy with the Biot-Savart law, also qualifies as a vector potential for, where
Substituting for and for, yields the Biot-Savart law.
Let be a star domain centered at the point, where. Applying Poincaré's lemma for differential forms to vector fields, then also is a vector potential for, where
Nonuniqueness
The vector potential admitted by a solenoidal field is not unique. If is a vector potential for, then so iswhere 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.