Mathematical concept in vector calculus
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
, a vector potential is a
vector field
such that
![{\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c51444e65310d3b4c5bbd519573fffea89a38c6b)
Consequence
If a vector field
admits a vector potential
, then from the equality
![{\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e5fbe0f83e82a058195cef389350e8cadb8fb35)
(divergence of the curl is zero) one obtains
![{\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3eea4054b3accf06cda2d3f40654d1fa247c1a)
which implies that
![{\displaystyle \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6)
must be a solenoidal vector field.
Theorem
Let
![{\displaystyle \mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d258f3531c12827b0b0adf3182cf4bcc777c1fd1)
be a solenoidal vector field which is twice continuously differentiable. Assume that
![{\displaystyle \mathbf {v} (\mathbf {x} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25df89e95bf055d5044ee918d871c151c4cbbb37)
decreases at least as fast as
![{\displaystyle 1/\|\mathbf {x} \|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c485b4c2c071ff4391909246060d8141da3b8e26)
for
![{\displaystyle \|\mathbf {x} \|\to \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4cb7f21599f515693bd3e6de124d6a6a5e899d5)
. Define
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae598b73c9e6e4373c760aa41ff51a12a86f1614)
where
![{\displaystyle \nabla _{y}\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbedf5c0a9742000b5a990dbd3e7e0c6f00843ae)
denotes curl with respect to variable
![{\displaystyle \mathbf {y} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25a040b592282dc2a254c3117e792c3c81161f)
. Then
![{\displaystyle \mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1)
is a vector potential for
![{\displaystyle \mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6)
. That is,
![{\displaystyle \nabla \times \mathbf {A} =\mathbf {v} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8feb9b3f8e8bb846be33b72ef6d564272d414a0c)
The integral domain can be restricted to any simply connected region
. That is,
also is a vector potential of
, where
![{\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} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e032b348acf7f036bb238cb2741a53b9d6aaae24)
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,
also qualifies as a vector potential for
, where
.
Substituting
(current density) for
and
(H-field) for
, yields the Biot-Savart law.
Let
be a star domain centered at the point
, where
. Applying Poincaré's lemma for differential forms to vector fields, then
also is a vector potential for
, where
Nonuniqueness
The vector potential admitted by a solenoidal field is not unique. If
is a vector potential for
, then so is
![{\displaystyle \mathbf {A} +\nabla f,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c02c8da495cd470e814da296872ab86d4e9b1f6c)
where
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
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
References
- Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.
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