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 , 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
edit
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
∇
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
{\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
∫
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
v
(
y
)
×
(
x
−
y
)
4
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
{\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}
Nonuniqueness
edit
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
edit
References
edit
Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.