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| {{about|the general concept in the mathematical theory of vector fields|the vector potential in electromagnetism|Magnetic vector potential|the vector potential in fluid mechanics|Stream function}}
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| In [[vector calculus]], a '''vector potential''' is a [[vector field]] whose [[Curl (mathematics)|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.
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| Formally, given a vector field '''v''', a ''vector potential'' is a vector field '''A''' such that
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| :<math> \mathbf{v} = \nabla \times \mathbf{A}. </math>
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| If a vector field '''v''' admits a vector potential '''A''', then from the equality
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| :<math>\nabla \cdot (\nabla \times \mathbf{A}) = 0</math> | |
| ([[divergence]] of the [[Curl (mathematics)|curl]] is zero) one obtains
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| :<math>\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0,</math>
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| which implies that '''v''' must be a [[solenoidal vector field]].
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| ==Theorem== | |
| Let
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| :<math>\mathbf{v} : \mathbb R^3 \to \mathbb R^3</math>
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| be a [[solenoidal vector field]] which is twice [[smooth function|continuously differentiable]]. Assume that '''v'''('''x''') decreases sufficiently fast as ||'''x'''||→∞. Define
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| :<math> \mathbf{A} (\mathbf{x}) = \frac{1}{4 \pi} \nabla \times \int_{\mathbb R^3} \frac{ \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d\mathbf{y}. </math>
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| Then, '''A''' is a vector potential for '''v''', that is,
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| :<math>\nabla \times \mathbf{A} =\mathbf{v}. </math>
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| 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]].
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| ==Nonuniqueness== | |
| The vector potential admitted by a solenoidal field is not unique. If '''A''' is a vector potential for '''v''', then so is
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| :<math> \mathbf{A} + \nabla m </math> | |
| where ''m'' is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.
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| This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires [[Gauge fixing|choosing a gauge]].
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| == See also == | |
| * [[Fundamental theorem of vector analysis]]
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| * [[Magnetic potential]]
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| * [[Solenoid]]
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| == References ==
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| * ''Fundamentals of Engineering Electromagnetics'' by David K. Cheng, Addison-Wesley, 1993.
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| [[Category:Concepts in physics]]
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| [[Category:Potentials]] | |
| [[Category:Vector calculus]]
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