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 $C^{2}$ vector field A such that

$\mathbf {v} =\nabla \times \mathbf {A} .$

Consequence
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If a vector field v admits a vector potential A , then from the equality

$\nabla \cdot (\nabla \times \mathbf {A} )=0$

(divergence of the curl is zero) one obtains
$\nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0,$

which implies that v must be a solenoidal vector field .
Theorem
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Let

$\mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}$

be a solenoidal vector field which is twice continuously differentiable . Assume that v (x ) decreases at least as fast as $1/\|\mathbf {x} \|$ for $\|\mathbf {x} \|\to \infty$ .
Define
$\mathbf {A} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .$

Then, A is a vector potential for v , that is,

$\nabla \times \mathbf {A} =\mathbf {v} .$

Here, $\nabla _{y}\times$ is curl for variable y .
Substituting curl[v] for the current density j of the retarded potential , you will get this formula. In other words, v corresponds to the H-field .
You can restrict the integral domain to any single-connected region Ω . That is, A' below is also a vector potential of v ;

$\mathbf {A'} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\Omega }{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .$

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 ${\boldsymbol {A''}}({\textbf {x}})$ is also qualify as a vector potential for v .

${\boldsymbol {A''}}({\textbf {x}})=\int _{\Omega }{\frac {{\boldsymbol {v}}({\boldsymbol {y}})\times ({\boldsymbol {x}}-{\boldsymbol {y}})}{4\pi |{\boldsymbol {x}}-{\boldsymbol {y}}|^{3}}}d^{3}{\boldsymbol {y}}$ Substitute j (current density ) for v and H (H-field )for A , we will find the Biot-Savart law.

Let ${\textbf {p}}\in \mathbb {R}$ 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 following ${\boldsymbol {A'''}}({\boldsymbol {x}})$ is also a vector potential for the ${\boldsymbol {v}}$

${\boldsymbol {A'''}}({\boldsymbol {x}})=\int _{0}^{1}s(({\boldsymbol {x}}-{\boldsymbol {p}})\times ({\boldsymbol {v}}(s{\boldsymbol {x}}+(1-s){\boldsymbol {p}}))\ ds$

Nonuniqueness
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The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v , then so is

$\mathbf {A} +\nabla f,$

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 .

See also
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References
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Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.