The forms I want to have fun with are calleddifferential forms. As well as being pretty neat in their own right, my pals tell me they are finding increasing application in Physics, especially field theory. I will try to show how they can be used to write Maxwell's field equations in a way that is even more succinct than the Heavyside vector versions we learn in school.

First a disclaimer; I intend to cover no more ground than we will strictly need to to track down our quarry, so there will be a lot left unsaid. But of course, if anyone wants to chip in with extra detail, that would be really cool. Plus, as this subject is relatively new to me, there may be some absolute howlers here, so please correct me if I get it wrong.

So, let us start with an arbitrary n-dimensional vector space over the field which, for economies sake I will write as . We now define another vector space which we will write as , and call this the space ofp-forms. For now, we will just take this notation to be a representation of the following instructions:

from the set of basis vectors for select at a time and call this another vector space(with the usual axioms). A coupla things should be immediately obvious:

, since we are just selecting each of the basis vectors of , one at a time. Also is

either meaningless (select no basis vectors!), or it is something like a set of scalars. In fact it is so defined.

A word of caution; here, as often elsewhere, no real distinction is made between scalars proper and scalar-valued functions. Thus we may have . Also obvious should be that we must insist that .

An obvious question then is what is the dimension of the space . Consider first . Elementary combinatorics tells us that, when there are choices, and we will write the dimension of this space as (say "from 3 choose 2").

Now quite obviously, this doesn't generalize, so let's write the general form as - a quick calculation shows the equivalence when

So the elements in are called p-forms, hence the elements in are 1-forms. I haven't said why these are calleddifferentialforms, but this is already highly suggestive; I will leave like that for now, and see if anyone takes the bait!.