[ajkjk]

I think there's a big difference between "presenting a formula" and "presenting rules for doing calculations". People are very good at extrapolating complicated results from a few simple rules: that's why taking derivatives and doing (elementary) integrals is fairly easy and also fairly easy to remember a long time after you've taken a calculus course. On the other hand, a bunch of literal miscellaneous formulas is very hard to hold on to --- for instance that's how introductory physics is taught, a bunch of disjoint relationships that you have to make sense of in your mind to make any use of.

In fact all anyone really wants for vector calculus is a bunch of "tools" they can use that will generally give the right answer if applied mindlessly. I think that's why GA is relatively popular, because it says how to do basic geometric operations (rotations, reflections, etc) without any thought.