# 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 .