covectorveld

Covectorveld, or covector field in English, is a concept in differential geometry. It assigns to each point p of a differentiable manifold M a covector α_p in the cotangent space T_p^M, and the assignment varies smoothly with p. Equivalently, a covectorveld is a smooth section of the cotangent bundle T^M. For any vectorveld X, the scalar function p ↦ α_p(X_p) is smooth on M.

In local coordinates x^1, ..., x^n on a chart, a covectorveld can be written as α = ∑_{i=1}^n α_i dx^i, where the α_i are smooth functions. Under a change of coordinates, the components transform by α'_j = ∑_i α_i ∂x^i/∂x'^j, which is the dual transformation law to that of tangent vectors.

Special cases and relationships: the differential df of a scalar function f is a covectorveld. If a Riemannian metric g is present, there is a natural correspondence between gradient fields ∇f and covectorvelden via df = g(∇f, ·) in appropriate notation. Covectorvelden, together with vectorvelden, yield scalar fields through their pointwise pairing, and they can be pulled back along smooth maps or related to higher-order differential forms via the exterior calculus.

The cotangent bundle perspective clarifies their role: covectorvelden are the smooth sections of T^M, serving as the dual objects to vectorvelden. They are central in integration, in the definition of differential forms, and in connections and curvature constructions where covariant indices are lowered.