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Vector potential

Mathematical concept in vector calculus

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated February 17, 2026

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 \mathbf {v}, a vector potential is a C^{2} vector field \mathbf {A} such that \mathbf {v} =\nabla \times \mathbf {A} .

01Consequence

If a vector field \mathbf {v} admits a vector potential \mathbf {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 \mathbf {v} must be a solenoidal vector field.

02Theorem

Let \mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3} be a solenoidal vector field which is twice continuously differentiable. Assume that \mathbf {v} (\mathbf {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 _{\mathbf {s} }\times \mathbf {v} (\mathbf {s} )}{\left\|\mathbf {x} -\mathbf {s} \right\|}}\,d^{3}\mathbf {s} where \nabla _{\mathbf {s} }\times denotes curl with respect to variable \mathbf {s}. Then \mathbf {A} is a vector potential for \mathbf {v}. That is, \nabla \times \mathbf {A} =\mathbf {v} .

The integral domain can be restricted to any simply connected region \Omega. That is, \mathbf {A'} also is a vector potential of \mathbf {v}, where \mathbf {A'} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\Omega }{\frac {\nabla _{\mathbf {s} }\times \mathbf {v} (\mathbf {s} )}{\left\|\mathbf {x} -\mathbf {s} \right\|}}\,d^{3}\mathbf {s} .

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, \mathbf {A''} (\mathbf {x} ) also qualifies as a vector potential for \mathbf {v}, where

\mathbf {A''} (\mathbf {x} )=\int _{\Omega }{\frac {\mathbf {v} (\mathbf {s} )\times (\mathbf {x} -\mathbf {s} )}{4\pi \left|\mathbf {x} -\mathbf {s} \right|^{3}}}d^{3}\mathbf {s}

Substituting \mathbf {j} (current density) for \mathbf {v} and \mathbf {H} (H-field) for \mathbf {A}, yields the Biot-Savart law.

Let \Omega be a star domain centered at the point \mathbf {p}, where \mathbf {p} \in \mathbb {R} ^{3}. Applying Poincaré's lemma for differential forms to vector fields, then \mathbf {A'''} (\mathbf {x} ) also is a vector potential for \mathbf {v}, where

\mathbf {A'''} (\mathbf {x} )=\int _{0}^{1}s\left[(\mathbf {x} -\mathbf {p} )\times \mathbf {v} (s\mathbf {x} +(1{-}s)\mathbf {p} )\right]ds

03Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If \mathbf {A} is a vector potential for \mathbf {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.

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Sources and credits

This article is adapted from the Wikipedia article “Vector potential”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Helmholtz decomposition

In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational vector field and a solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed.

Magnetic vector potential

In classical electromagnetism, magnetic vector potential is the vector quantity defined so that its curl is equal to the magnetic field, B: ∇ × A = B {\textstyle \nabla \times \mathbf {A} =\mathbf {B} } . Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well.

Solenoidal vector field

In vector calculus and differential equations, a divergence free vector field is a vector field v with divergence zero at all points in the field: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.} In electromagnetic theory, a divergence-free vector field is commonly called a solenoidal vector field, while in hydrodynamics it is commonly called an incompressible vector field, and in mathematical physics it is sometimes called an transverse vector field).