Exterior derivative

Let be a smooth function on a manifold. The differential (or exterior derivative), , is a covector field on M defined as follows: for v a tangent vector at a point

i.e., is the directional derivative of f in the direction v.

Note that if are a local coordinate system for M at p, then define a local co-frame near p. Thus, near p, we may write the differential of f as a linear combination:

In fact, since , we get that: