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
If a vector field
admits a vector potential
, then from the equality
(divergence of the curl is zero) one obtains
must be a solenoidal vector field.
be a solenoidal vector field which is twice continuously differentiable.
decreases at least as fast as
denotes curl with respect to variable
is a vector potential for
The integral domain can be restricted to any simply connected region
also is a vector potential of
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
(current density) for
, yields the Biot-Savart law.
be a star domain centered at the point
Applying Poincaré's lemma for differential forms to vector fields, then
also is a vector potential for
{\displaystyle \mathbf {A'''} (\mathbf {x} )=\int _{0}^{1}s((\mathbf {x} -\mathbf {p} )\times (\mathbf {v} (s\mathbf {x} +(1-s)\mathbf {p} ))\ ds}
The vector potential admitted by a solenoidal field is not unique.
is a vector potential for
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.