Given a manifold (or generalized smooth space) , the cotangent bundle of is the vector bundle over dual to the tangent bundle of . A cotangent vector or covector on is an element of . The cotangent space of at a point is the fiber of over ; it is a vector space. A covector field on is a section of . (More generally, a differential form on is a section of the exterior algebra of ; a covector field is a differential -form.)
Given a covector at and a tangent vector at , the pairing is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for to be dual to . More globally, given a covector field and a tangent vector field , the paring is a scalar function on .
Given a point in and a differentiable (real-valued) partial function defined near , the differential of at is a covector on at ; given a tangent vector at , the pairing is given by
thinking of as a derivation on differentiable functions defined near . (It is really the germ at of that matters here.) More globally, given a differentiable function , the differential of is a covector field on ; given a vector field , the pairing is given by
thinking of as a derivation on differentiable functions.
One can also define covectors at to be germs of differentiable functions at , modulo the equivalence relation that if is constant on some neighbourhood of . In general, a covector field won't be of the form , but it will be a sum of terms of the form . More specifically, a covector field on a coordinate patch can be written
in local coordinates . This fact can also be used as the basis of a definition of the cotangent bundle.