OK below is way more than I intended to write. TL;DR: exterior products are ânaturalâ if you think in terms of abstract algebra, so maybe it would be best to do a bit of abstract algebra first and then come back to it.
The way that makes most sense to me (warning: Iâm an algebra person) is the universal property way. In my opinion this is way has the most natural motivation, and no formulas are âdropped from the sky.â Because of the abstraction, this is not the easiest/quickest approach if you havenât seen abstract algebra, but if youâre just satisfying a curiosity it may be worthwhile to learn some algebra anyway. I wonât go over the construction here, just say how the parts of it can be motivated.
The exterior product is an example of a multilinear, alternating map (definitions below). It is in fact the âmost genericâ multilinear, alternating map, in a precise sense described below. So to understand exterior products, we should first understand these maps.
Lets use torque as an example. Before we give a definition for what torque is, we list some properties we think it ought to have.
We have some vector space V = R^3 representing space.
We have some vector space W = ??? whose elements are all possible torques.
Whatever a torque is, we should be able to get one from a pair of vectors r and F in V, representing force and displacement. That is, we should have a function from V x V to W.
This function should be bilinear: if we scale either the force or displacement, the resulting torque should scale by the same amount. If we hold displacement constant and add two forces (as vectors in V), the resulting torque should be the sum (as vectors in W) of the individual torques. Same for if we hold force constant and add displacements.
This function should be alternating: if the force and displacement are in the same direction, the resulting torque should be zero.
So we know torque should be an alternating bilinear map out of V x V.
The exterior product comes in here: Any alternating bilinear map f : V x V -> W factors uniquely as a linear map g : â§Â˛V -> W composed with the exterior product V x V -> â§Â˛ V. That is, the wedge product turns alternating, bilinear maps into just plain old linear maps, and vice versa. By abstract nonsense, this property completely determines the space â§Â˛V up to unique isomorphism.Â
So if we donât have any more assumptions to make, we just decide that a torque should be an element of â§Â˛V. One can show that in this case â§Â˛V is three-dimensional (this is where the work is), and so this agrees with the classical definition.
This is done in many âstandardâ books on abstract algebra (I know at least Dummit & Foote and Lang do it this way; Aluffi has a more modern book I havenât read which has a lot of exposition on universal properties as well.)Â Unfortunately because of the scope of these books this is always covered after ~500 pages of other material, but the only hard prereqs needed from earlier chapters are definitions and the general âalgebra vibe.â