I was able to actually stick with my plan a little earlier and spend the day parsing through some stuff in Kobayashi and Nomizu. That’s not a bad way to spend a Sunday.
Somewhere in the middle of that, I ended up stumbling upon something I’d always been somewhat privy to and I did so almost by accident. In the text, I ran across the definition in the following context:
Consider two manifolds and and a mapping of the prior into the latter. Then for a point , the differential of at is a linear mapping which is defined as follows: Given a vector , choose a path with . Then is the vector tangent to the curve at .
The notation reminded me of something I saw during my very first foray into Differential Geometry, namely the Hodge star/dual operator. It was a notion that was so novel when I first saw it that I contemplated preparing a seminar talk at BGSU for my peers, none of whom were geometers of any kind; now that I’ve rediscovered it, I’m having similar ideas for my non-geometer peers here at FSU. But I’m getting ahead of myself…
Let be a manifold of dimension and let be a -form on for . Here, denotes a multi-index which – without loss of generality – can be assumed to be increasing. Then the hodge star operator is the operator which takes to the -form where , where is the multi-index consisting of all numbers not in , and where denotes the sign of .
Here’s what all that says:
Say we’re in with the standard orthonormal directional 1-forms , , and . As expected, the 0-form gets mapped to and the volume form satisfies . Also, the Hodge star operator maps each of the aforementioned coordinate 1-forms to the following 3-1 = 2-forms:
, , and .
Similarly, the following summarizes the behavior of the Hodge star on the 2-forms:
, , .
As a less trivial example1, consider the case of (again) with the metric . This space + metric combination is the Lorentzian manifold known as Minkowski Space. Here, the basis has the form but the presence of the negative square of the component changes the behavior of the Hodge star operator as seen below:
As expected, the properties of the Hodge star operator run far more deeply than the ones described above. To see more, see the paper cited in  below. There are also lots of other resources online as well.
1. Example taken from Tevian Dray of Oregon State.