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:



Exterior derivative of differential forms

If  is a differential k-form (i.e., a smooth section of ), the exterior derivative  is a differential (k+1)-form defined as follows:

If we can write  in local coordinates as



then in this coordinate system,  equals



More generally, we define the differential  by extending the above definition by linearity.

Cosmological properties of the differential

The operator d has the important property that . This essentially follows from the equality of mixed partial derivatives. The following simplest example illustrates the general proof: Let  be a smooth function in two variables. Then



Thus



Since , the equality of mixed partials shows that .