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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

• Theorem

• Nonuniqueness

• See also

• References

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.

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 π ∫ ∫ R 3 ∇ ∇ y × × v ( y ) ‖ x − − y ‖ d 3 y {\displaystyle \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} } where ∇ ∇ y × × {\displaystyle \nabla _{y}\times } denotes curl with respect to variable y {\displaystyle \mathbf {y} } . 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 Ω {\displaystyle \mathbf {\Omega } } . That is, A ′ {\displaystyle \mathbf {A'} } also is a vector potential of v {\displaystyle \mathbf {v} } , where A ′ ( x ) = 1 4 π π ∫ ∫ Ω Ω ∇ ∇ y × × v ( y ) ‖ x − − y ‖ d 3 y . {\displaystyle \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 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 Ω v ( y ) × × ( x − − y ) 4 π π | x − − y | 3 d 3 y {\displaystyle \mathbf {A''} (\mathbf {x} )=\int _{\Omega }{\frac {\mathbf {v} (\mathbf {y} )\times (\mathbf {x} -\mathbf {y} )}{4\pi |\mathbf {x} -\mathbf {y} |^{3}}}d^{3}\mathbf {y} } .

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 Ω {\displaystyle \mathbf {\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((\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 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.

• 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.