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 {\displaystyle \mathbf {v} } , a vector potential is a C 2 {\displaystyle C^{2}} vector field A {\displaystyle \mathbf {A} } such that
v = ∇ × A . {\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}
Consequence If a vector field v {\displaystyle \mathbf {v} } admits a vector potential A {\displaystyle \mathbf {A} } , then from the equality
∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}
(divergence of the curl is zero) one obtains
∇ ⋅ v = ∇ ⋅ ( ∇ × A ) = 0 , {\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0,}
which implies that v {\displaystyle \mathbf {v} } must be a solenoidal vector field.
Theorem Let
v : R 3 → R 3 {\displaystyle \mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}}
be a solenoidal vector field which is twice continuously differentiable. Assume that v ( x ) {\displaystyle \mathbf {v} (\mathbf {x} )} decreases at least as fast as 1 / ‖ x ‖ {\displaystyle 1/\|\mathbf {x} \|} for ‖ x ‖ → ∞ {\displaystyle \|\mathbf {x} \|\to \infty } . Define
A ( x ) = 1 4 π ∫ R 3 ∇ s × v ( s ) ‖ x − s ‖ d 3 s {\displaystyle \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 ∇ s × {\displaystyle \nabla _{\mathbf {s} }\times } denotes curl with respect to variable s {\displaystyle \mathbf {s} } . Then A {\displaystyle \mathbf {A} } is a vector potential for v {\displaystyle \mathbf {v} } . That is,
∇ × A = v . {\displaystyle \nabla \times \mathbf {A} =\mathbf {v} .}
The integral domain can be restricted to any simply connected region Ω {\displaystyle \Omega } . That is, A ′ {\displaystyle \mathbf {A'} } also is a vector potential of v {\displaystyle \mathbf {v} } , where
A ′ ( x ) = 1 4 π ∫ Ω ∇ s × v ( s ) ‖ x − s ‖ d 3 s . {\displaystyle \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, A ″ ( x ) {\displaystyle \mathbf {A''} (\mathbf {x} )} also qualifies as a vector potential for v {\displaystyle \mathbf {v} } , where
A ″ ( x ) = ∫ Ω v ( s ) × ( x − s ) 4 π | x − s | 3 d 3 s {\displaystyle \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 j {\displaystyle \mathbf {j} } (current density) for v {\displaystyle \mathbf {v} } and H {\displaystyle \mathbf {H} } (H-field) for A {\displaystyle \mathbf {A} } , yields the Biot–Savart law. Let Ω {\displaystyle \Omega } be a star domain centered at the point p {\displaystyle \mathbf {p} } , where p ∈ R 3 {\displaystyle \mathbf {p} \in \mathbb {R} ^{3}} . Applying Poincaré's lemma for differential forms to vector fields, then A ‴ ( x ) {\displaystyle \mathbf {A'''} (\mathbf {x} )} also is a vector potential for v {\displaystyle \mathbf {v} } , where
A ‴ ( x ) = ∫ 0 1 s [ ( x − p ) × v ( s x + ( 1 − s ) p ) ] d s {\displaystyle \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}
Nonuniqueness The vector potential admitted by a solenoidal field is not unique. If A {\displaystyle \mathbf {A} } is a vector potential for v {\displaystyle \mathbf {v} } , then so is
A + ∇ f , {\displaystyle \mathbf {A} +\nabla f,}
where f {\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 requires choosing a gauge.
See also Fundamental theorem of vector calculus Magnetic vector potential Solenoidal vector field Closed and Exact Differential Forms
References Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.