# Exterior Algebra

Danger

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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?

TODO!

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.

TODO!