Exterior Algebra


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In the introduction, we considered a set of three basis vectors \(\ee_1\), \(\ee_2\), and \(\ee_3\). In addition, we pontificated a bit on why restricting ourselves to vectors can cause issues, and argued for the need for a richer structure to match the richness of the geometry. But how should we go about doing this?

In 3-dimensions, it seems a bit unfair that only "arrows" can be represented. After all, our world is filled with objects that have area and volume too. Suppose we wanted to represent a unit area in the x-y plane. Let's give it a name, say \(\ee_{12}\). It seems reasonable that the areas \(\ee_{13}\) and \(\ee_{23}\). But wait! Our choice of of index order seems a bit arbitrary. What about \(\ee_{21}\), \(\ee_{31}\) and \(\ee_{32}\). If we follow our nose a bit, it seems reasonable that the area represented by \(\ee_{12}\) should equal \(-\ee_{21}\). After all, they possess opposite orientations from one another.

What about elements with two repeated indices? Like \(\ee_{11}\) or \(\ee_{33}\)? Well, such elements can't reasonably span any area, so let's get back to how we should handle those in a moment. Like the unit vectors, it seems sensible to allow us to scale these area-elements with a weight (like \(3\ee_{12}\)) and add them together to create areas of arbitrary weight and orientation. So, for example, let's try to add \(\ee_{12} + 2\ee_{23}\). What should its orientation be?


For volumes, we have one choice that spans a non-zero volume which is \(\ee_{123}\). We could have also chosen a different basis ordering, which again we'll get back to later. Let's call our single-index elements vectors as before, the two-index elements bivectors, and the three-index elements trivectors.