Exterior derivative

From Conservapedia

Let $f:M\rightarrow \mathbb{R}$ be a smooth function on a manifold. The differential (or exterior derivative), $d f$ , is a covector field on M defined as follows: for v a tangent vector at a point $p$

$d f (v) = D v (f)$

i.e., $d f (v)$ is the directional derivative of f in the direction v.

Note that if $x 1,..., x n$ are a local coordinate system for M at p, then $d x 1,..., d x n$ define a local co-frame near p. Thus, near p, we may write the differential of f as a linear combination:

$d f = g 1 d x 1 + ... + g n d x n$

In fact, since $dx_i(\frac{\partial}{\partial x_j}) = \delta^i_j$ , we get that:

$df = \frac{\partial f}{\partial x_1} dx_1 + ... + \frac{\partial f}{\partial x_n} dx_n$

Exterior derivative of differential forms

If $ω$ is a differential k-form (i.e., a smooth section of $Λ k T * M$ ), the exterior derivative $d ω$ is a differential (k+1)-form defined as follows:

If we can write $ω$ in local coordinates as

$f_{i_1\cdots i_k} dx_{i_1}\wedge\cdots\wedge dx_{i_k}$

then in this coordinate system, $d ω$ equals

$df_{i_1\cdots i_k}\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k}$

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

Cosmological properties of the differential

The operator d has the important property that $d\circ d = 0$ . This essentially follows from the equality of mixed partial derivatives. The following simplest example illustrates the general proof: Let $f (x, y)$ be a smooth function in two variables. Then