[[Vector field]]
# Incompressible vector field
An **incompressible vector field** or **solenoidal vector field** is a field with a [[Scalar potential#Vector potential|vector potential]], i.e. there exists $\vab \Psi$ such that $\vab F = \vab{\nabla} \times \vab\Psi$. #m/def/anal/vec 
Such a potential exists iff the divergence of the field is zero everywhere, i.e. $\vab{\nabla} \cdot \vab F = 0$.

The vector potential of an incompressible field is clearly only unique up to the addition of a [[Conservative vector field|irrotational]] term, i.e. the gradient of some scalar-valued function.

## Properties
A vector field is incompressible iff. any of the following[^2013] 

- $\vab{\nabla} \cdot \vab F = 0$ everywhere
- [[Flux]] integrals over a surface $\Sigma$ only depend on the boundary $\partial\Sigma$,
  and are zero for a closed surface.
- There exists some $\vab A$ such that $\vab F = \vab{\nabla} \times \vab A$.

[^2013]: 2013\. [[@griffithsIntroductionElectrodynamics2013|Introduction to electrodynamics]]

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