irrotational

Irrotational is a term used to describe a vector field whose curl is zero throughout a region. In three-dimensional space, this is written as ∇×F = 0. In fluid dynamics, an irrotational flow has zero vorticity, meaning there is no local rotation of fluid elements.

A fundamental consequence is that an irrotational field is locally conservative. In a simply connected region, an irrotational vector field F can be expressed as the gradient of a scalar potential: F = ∇φ. As a result, the line integral of F between two points depends only on the endpoints, not on the chosen path.

The converse holds in simply connected domains: if F = ∇φ for some scalar φ, then ∇×F = 0. However, on regions with holes or nontrivial topology, the curl can vanish everywhere but the field may fail to be the gradient of a single-valued potential. A classic example is the vector field F = (−y/(x²+y²), x/(x²+y²)) defined on R² without the origin; ∇×F = 0 where defined, yet F is not globally conservative because tracing around the origin changes the value of the potential.

In physics, irrotational fields arise in electrostatics and gravity where fields are gradients of scalar potentials (E = −∇Φ, g = −∇Φg) under static conditions. In electromagnetism, time variation can induce a nonzero curl of the electric field (Faraday’s law), so the field need not be irrotational in general.

In fluid dynamics, an irrotational, incompressible flow implies a velocity potential φ with ∇²φ = 0, i.e., φ is a harmonic function. This leads to the simplification of many problems via potential flow theory.