covectoriel

Covectoriel is a term rooted in linear algebra and differential geometry, referring to covectors and the cotangent structures that arise from them. A covector is a linear functional on a vector space; the collection of covectors attached to a point forms the cotangent space, and the covector bundle (the cotangent bundle) collects these spaces over every point of a differentiable manifold.

In the setting of a differentiable manifold M, for each p ∈ M the tangent space T_p M consists of velocity vectors at p, while the dual space T*_p M consists of linear functionals on T_p M. The cotangent bundle T* M = ⨿_p T*_p M is the bundle whose fibre at p is T*_p M. A covector field, or 1-form, assigns to every p a covector ω_p ∈ T*_p M. In local coordinates x^i with dual basis {dx^i}, a covector is written as ω = ∑_i ω_i dx^i, and its components transform with the inverse Jacobian under a change of coordinates.

Differential forms and operations: the action of a covector on a vector is a scalar, ω(v). The differential df of a function f is a canonical covector field, with df_p(v) = v(f). The exterior derivative d maps 1-forms to 2-forms, and further algebraic operations such as the wedge product build higher-degree forms.

Relation to vectors and metrics: in Euclidean space with the standard metric, covectors can be identified with row vectors; more generally, a metric or an isomorphism between TM and T*M (musical isomorphisms) relates vectors and covectors. Covectoriel objects thus underpin many coordinate-free formulations in geometry and physics, including gradients viewed as differentials and the calculus of forms.