Invector calculus, avector potential is avector field whosecurl is a given vector field. This is analogous to ascalar potential, which is a scalar field whosegradient is a given vector field.

Formally, given a vector field${\displaystyle \mathbf{v}}$, avector potential is a${\displaystyle C^2}$ vector field${\displaystyle \mathbf{A}}$ such that$${\displaystyle \mathbf{v}=\mathrm{\nabla }\times \mathbf{A}.}$$

If a vector field${\displaystyle \mathbf{v}}$ admits a vector potential${\displaystyle \mathbf{A}}$, then from the equality$${\displaystyle \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{A})=0}$$(divergence of thecurl is zero) one obtains$${\displaystyle \mathrm{\nabla }\cdot \mathbf{v}=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{A})=0,}$$which implies that${\displaystyle \mathbf{v}}$ must be asolenoidal vector field.

Let$${\displaystyle \mathbf{v}:{\mathbb{R}}^3\to {\mathbb{R}}^3}$$be asolenoidal vector field which is twicecontinuously differentiable. Assume that${\displaystyle \mathbf{v}(\mathbf{x})}$ decreases at least as fast as${\displaystyle 1/\Vert \mathbf{x}\Vert }$ for${\displaystyle \Vert \mathbf{x}\Vert \to \mathrm{\infty }}$. Define$${\displaystyle \mathbf{A}(\mathbf{x})=\frac{1}{4\pi }{\int }_{{\mathbb{R}}^3}\frac{{\mathrm{\nabla }}_y\times \mathbf{v}(\mathbf{y})}{\Vert \mathbf{x}-\mathbf{y}\Vert }{d}^3\mathbf{y}}$$where${\displaystyle {\mathrm{\nabla }}_y\times }$ denotes curl with respect to variable${\displaystyle \mathbf{y}}$. Then${\displaystyle \mathbf{A}}$ is a vector potential for${\displaystyle \mathbf{v}}$. That is,$${\displaystyle \mathrm{\nabla }\times \mathbf{A}=\mathbf{v}.}$$

The integral domain can be restricted to any simply connected region${\displaystyle \mathbf{\Omega }}$. That is,${\displaystyle {\mathbf{A}}^{\prime }}$ also is a vector potential of${\displaystyle \mathbf{v}}$, where$${\displaystyle {\mathbf{A}}^{\prime }(\mathbf{x})=\frac{1}{4\pi }{\int }_{\mathrm{\Omega }}\frac{{\mathrm{\nabla }}_y\times \mathbf{v}(\mathbf{y})}{\Vert \mathbf{x}-\mathbf{y}\Vert }{d}^3\mathbf{y}.}$$

A generalization of this theorem is theHelmholtz decomposition theorem, which states that any vector field can be decomposed as a sum of a solenoidal vector field and anirrotational vector field.

Byanalogy with theBiot-Savart law,${\displaystyle {\mathbf{A}}^{″}(\mathbf{x})}$ also qualifies as a vector potential for${\displaystyle \mathbf{v}}$, where

${\displaystyle {\mathbf{A}}^{″}(\mathbf{x})={\int }_{\mathrm{\Omega }}\frac{\mathbf{v}(\mathbf{y})\times (\mathbf{x}-\mathbf{y})}{4\pi |\mathbf{x}-\mathbf{y}{|}^3}{d}^3\mathbf{y}}$.

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

Let${\displaystyle \mathbf{\Omega }}$ be astar domain centered at the point${\displaystyle \mathbf{p}}$, where${\displaystyle \mathbf{p}\in {\mathbb{R}}^3}$. ApplyingPoincaré's lemma fordifferential forms to vector fields, then${\displaystyle {\mathbf{A}}^{‴}(\mathbf{x})}$ also is a vector potential for${\displaystyle \mathbf{v}}$, where

${\displaystyle {\mathbf{A}}^{‴}(\mathbf{x})={\int }_0^{1}s((\mathbf{x}-\mathbf{p})\times (\mathbf{v}(s\mathbf{x}+(1-s)\mathbf{p}))ds}$

The vector potential admitted by a solenoidal field is not unique. If${\displaystyle \mathbf{A}}$ is a vector potential for${\displaystyle \mathbf{v}}$, then so is$${\displaystyle \mathbf{A}+\mathrm{\nabla }f,}$$where${\displaystyle f}$ 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 requireschoosing a gauge.

• Fundamental theorem of vector calculus
• Magnetic vector potential
• Solenoidal vector field
• Closed and Exact Differential Forms

• Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.

• Concepts in physics
• Potentials
• Vector calculus
• Vector physical quantities

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