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{{Short description|Mathematical concept in vector calculus}}
{{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}}

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.

Formally, given a vector field <math>\mathbf{v}</math>, a ''vector potential'' is a <math>C^2</math> vector field <math>\mathbf{A}</math> such that
<math display="block"> \mathbf{v} = \nabla \times \mathbf{A}. </math>

==Consequence==
If a vector field <math>\mathbf{v}</math> admits a vector potential <math>\mathbf{A}</math>, then from the equality
<math display="block">\nabla \cdot (\nabla \times \mathbf{A}) = 0</math>
([[divergence]] of the [[Curl (mathematics)|curl]] is zero) one obtains
<math display="block">\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0,</math>
which implies that <math>\mathbf{v}</math> must be a [[solenoidal vector field]].

==Theorem==
Let
<math display="block">\mathbf{v} : \R^3 \to \R^3 </math>
be a [[solenoidal vector field]] which is twice [[smooth function|continuously differentiable]]. Assume that <math>\mathbf{v}(\mathbf{x})</math> decreases at least as fast as <math> 1/\|\mathbf{x}\| </math> for <math> \| \mathbf{x}\| \to \infty </math>. Define
<math display="block"> \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} </math>
where <math>\nabla_y \times</math> denotes curl with respect to variable <math>\mathbf{y}</math>. Then <math>\mathbf{A}</math> is a vector potential for <math>\mathbf{v}</math>. That is,
<math display="block">\nabla \times \mathbf{A} =\mathbf{v}. </math>

The integral domain can be restricted to any simply connected region <math>\mathbf{\Omega}</math>. That is, <math>\mathbf{A'}</math> also is a vector potential of <math>\mathbf{v}</math>, where
<math display="block"> \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}. </math>

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]], <math>\mathbf{A''}(\mathbf{x})</math> also qualifies as a vector potential for <math>\mathbf{v}</math>, where

:<math>\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}</math>.

Substituting <math>\mathbf{j}</math> ([[current density]]) for <math>\mathbf{v}</math> and <math>\mathbf{H}</math> ([[H-field]]) for <math>\mathbf{A}</math>, yields the Biot-Savart law.

Let <math>\mathbf{\Omega}</math> be a [[star domain]] centered at the point <math>\mathbf{p}</math>, where <math>\mathbf{p}\in \R^3</math>. Applying [[Poincaré's lemma]] for [[differential forms]] to vector fields, then <math>\mathbf{A'''}(\mathbf{x})</math> also is a vector potential for <math>\mathbf{v}</math>, where

<math>\mathbf{A'''}(\mathbf{x})
=\int_0^1 s ((\mathbf{x}-\mathbf{p})\times ( \mathbf{v}( s \mathbf{x} + (1-s) \mathbf{p} ))\ ds
</math>

==Nonuniqueness==
The vector potential admitted by a solenoidal field is not unique. If <math>\mathbf{A}</math> is a vector potential for <math>\mathbf{v}</math>, then so is
<math display="block"> \mathbf{A} + \nabla f, </math>
where <math>f</math> 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 [[Gauge fixing|choosing a gauge]].

== See also ==
* [[Fundamental theorem of vector calculus]]
* [[Magnetic vector potential]]
* [[Solenoidal vector field]]
* [[Closed and exact differential forms#Application in electrodynamics|Closed and Exact Differential Forms]]

== References ==
* ''Fundamentals of Engineering Electromagnetics'' by David K. Cheng, Addison-Wesley, 1993.

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[[Category:Concepts in physics]]
[[Category:Potentials]]
[[Category:Vector calculus]]
[[Category:Vector physical quantities]]

Vector potential - Revision history

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