This article is about the general concept in the mathematical theory of vector fields. For the vector potential in electromagnetism, see
Magnetic vector potential . For the vector potential in fluid mechanics, see
Stream function .
In vector calculus , a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential gradient is a given vector field.
Formally, given a vector field v , a vector potential is a
C
2
{\displaystyle C^{2}}
A such that
v
=
∇
×
A
.
{\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}
Consequence Edit
If a vector field v admits a vector potential 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,}
v must be a solenoidal vector field .
Let
v
:
R
3
→
R
3
{\displaystyle \mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}}
solenoidal vector field which is twice continuously differentiable . Assume that v (x )||x . Define
A
(
x
)
=
1
4
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} .}
A is a vector potential for v
∇
×
A
=
v
.
{\displaystyle \nabla \times \mathbf {A} =\mathbf {v} .}
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 .
Nonuniqueness Edit
The vector potential admitted by a solenoidal field is not unique. If A v
A
+
∇
f
,
{\displaystyle \mathbf {A} +\nabla f,}
f This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge .
References Edit
Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993. 
