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.

Cohomological 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

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Since , the equality of mixed partials shows that .