Cotangent Vectors, Differentials, and the Cotangent Bundle
There’s another construct in differential topology and geometry
that isn’t quite so obvious as a tangent vector, but which is
every bit as useful: a cotangent vector. A cotangent vector at a
point is just an element of the dual space to , which we write
as . We actually have a really nice example of cotangent vectors
already: a gadget that takes a tangent vector at and gives back
a number. It’s the differential, which when given a vector
returns the directional derivative in that direction. And we can
generalize that right away. Indeed, if is a smooth germ at ,
then we have a linear functional defined for all tangent vectors
. We will call this functional the differential of at , and
write . If we have local coordinates at , then each coordinate
function is a smooth function, which has differential . These
actually furnish the dual basis to the coordinate vectors .
Indeed, we calculate That is, evaluating the coordinate
differential on the coordinate