# 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 .