Abstract:

Even the most casual mathematical observer should be deeply disturbed upon first seeing the definition of the cross product in R^3, presumably in their first multivariable calculus class (here it is 32A). It is introduced in an innocent manner and yet begs so many questions: What about R^n for n<>3? Are there other sorts of wierd products lurking about out there? For example, the cross product is skew-commutative. Why not look at a commutative product? Then one is led to ask: If the cross-product is relatively unique (it is), and one can come up with some other relatively unique and wierd things, how can we know when we have a complete list? Do these things have reasonable descriptions?