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 that
If a vector field v admits a vector potential A, then from the equality
Then, A is a vector potential for v, that is,
You can restrict the integral domain to any single-connected region Ω. That is, A' below is also a vector potential of v;
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.
By analogy with Biot-Savart's law, the following is also qualify as a vector potential for v.
Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law.
Let and let the Ω be a star domain centered on the p then, translating Poincaré's lemma for differential forms into vector fields world, the followng is also a vector potential for the
The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is
This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.