## 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 vector gives the value if and otherwise.

Of course, the define a basis of at every point , just like the define a basis of at every point . This was exactly what we needed to compare vectors — at least to some extent — at points within a local coordinate patch, and let us define the tangent bundle as a -dimensional manifold.

In exactly the same way, we can define the cotangent bundle . Given the coordinate patch we define a coordinate patch covering all the cotangent spaces with . The coordinate map is defined on a cotangent vector by

Everything else in the construction of the cotangent bundle proceeds exactly as it did for the tangent bundle, but we’re missing one thing: how to translate from one basis of coordinate differentials to another.

So, let’s say and are two coordinate maps at , defining coordinate differentials and . How are these two bases related? We can calculate this by applying to :

where are the components of the Jacobian matrix of the transition function . What does this mean? Well, consider the linear functional

This has the same values on each of the as does, and we conclude that they are, in fact, the same cotangent vector:

On the other hand, recall that

That is, we use the Jacobian of one transition function to transform from the basis to the basis of , but the transpose of the same Jacobian to transform from the basis to the basis of . And this is actually just as we expect, since the transpose is actually the adjoint transformation, which automatically connects the dual spaces.