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Concept
Complex lamellar vector field
Résumé
In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different ways, many of which involve the curl. A lamellar vector field is a special case given by vector fields with zero curl.
The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The lamellae to which "lamellar vector field" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field. This language is particularly popular with authors in rational mechanics.
In vector calculus, a complex lamellar vector field is a vector field in three dimensions which is orthogonal to its own curl. That is,
The term lamellar vector field is sometimes used as a synonym for the special case of an irrotational vector field, meaning that
Complex lamellar vector fields are precisely those that are normal to a family of surfaces. An irrotational vector field is locally the gradient of a function, and is therefore orthogonal to the family of level surfaces (the equipotential surfaces). Any vector field can be decomposed as the sum of an irrotational vector field and a complex lamellar field.
In greater generality, a vector field F on a pseudo-Riemannian manifold is said to be hypersurface-orthogonal if through an arbitrary point there is a smoothly embedded hypersurface which, at all of its points, is orthogonal to the vector field. By the Frobenius theorem this is equivalent to requiring that the Lie bracket of any smooth vector fields orthogonal to F is still orthogonal to F.
The condition of hypersurface-orthogonality can be rephrased in terms of the differential 1-form ω which is dual to F. The previously given Lie bracket condition can be reworked to require that the exterior derivative dω, when evaluated on any two tangent vectors which are orthogonal to F, is zero.
Source officielle
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