Editing Vector potential (section)

==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>