Rotationnel

Rotationnel, in French mathematical literature often called the curl, is a differential operator in vector calculus that measures the local rotation of a vector field. It maps a differentiable vector field F to another vector field curl F. The operator is denoted by the symbol ∇×, where ∇ is the del operator.

In three-dimensional Cartesian coordinates, if F = (F1, F2, F3), then curl F = (∂F3/∂y − ∂F2/∂z, ∂F1/∂z − ∂F3/∂x, ∂F2/∂x − ∂F1/∂y). The curl is undefined for fields that are not differentiable at a point. In non-Cartesian coordinates the components are more complex, but the same geometric meaning applies.

Geometrically, the curl at a point is proportional to the circulation density of the field around a small loop enclosing that point; its direction is given by the right-hand rule and is perpendicular to the plane of the loop.

Key properties include linearity and its role in fundamental vector identities. Notably, the divergence of a curl vanishes: ∇·(∇×F) = 0, and the curl of a gradient vanishes: ∇×(∇φ) = 0 for any smooth scalar field φ.

The curl is central to several theorems and applications. Stokes’ theorem relates a line integral around a closed curve to a surface integral of the curl over any surface spanning the curve: ∮C F·dr = ∬S (∇×F)·n dS. In physics and engineering, curl appears in electromagnetism (Maxwell’s equations) and in fluid dynamics as the vorticity of a flow. More broadly, curl is linked to differential geometry via exterior calculus.