Vector field

Incompressible vector field

An incompressible vector field or solenoidal vector field is a field with a vector potential, i.e. there exists ⃗𝚿 such that ⃗𝐅 =⃗∇ ×⃗𝚿. vec Such a potential exists iff the divergence of the field is zero everywhere, i.e. ⃗∇ ⋅⃗𝐅 =0.

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

Properties

A vector field is incompressible iff. any of the following¹

• ⃗∇ ⋅⃗𝐅 =0 everywhere
• Flux integrals over a surface Σ only depend on the boundary 𝜕Σ, and are zero for a closed surface.
• There exists some ⃗𝐀 such that ⃗𝐅 =⃗∇ ×⃗𝐀.

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Footnotes

1. 2013. Introduction to electrodynamics ↩