Given a function f: M → R, define a special element of Tp*M. Call it
ωf. Define
ωf by the following:
ωf = (∂f/∂xβ) dxβ. In other words, the coefficients ωβ = ∂f/∂xβ.
Now, let's see what happens if we act with
ωf on a vector
vp in TpM:
ωf[ vα∂/∂xα ] = (∂f/∂xβ) vβ =
vp(f).
ωf[
v ] is the directional derivative of f in the direction of
v. It exactly gives us the same result we would get if we acted on f with
v as a directional derivative.
We give
ωf a new name:
ωf =
df = (∂f/∂xβ) dxβ.
Now,
df[
v ] =
v(f) is the directional derivative of f in the v-direction.
Now, we can see that our notation for "
df" connects with our notation for "dxβ ". We have used boldface to distinguish them thus far, but soon that will not be necessary. As a special case, let f be the coordinate function f = xβ. Then,
df =
d(xβ) = (∂xβ/∂xα) dxα, where "dxα " is still our dual vector notation.
Now, ∂xβ/∂xα = δαβ is just our friendly Kronecker Delta again, which is fairly easy to see, since our coordinates are independent of each other. Therefore, we have the relation
d(xβ) = dxβ. Our notation is consistent.
So, in a fairly unorthodox manner, the notation has lead us to a map
d from functions into forms.
d is known as the exterior derivative.
