From: Dan Piponi <dan@manex-group.com>
Newsgroups: sci.math.research
Subject: Re: Differential forms and integration
Date: Mon, 11 May 1998 11:51:53 -0700

Mark W. Meckes wrote:

> Okay, I can follow all the algebraic definitions and derivations, I've read
> basic discussions of geometric motivation, but I'm looking for some real
> intuition here.  Just why is a totally antisymmetric type (0,q) tensor
> field the right kind of thing to integrate?

I'll see if I can give some kind of idea of what's going on with n-forms
with some really easy (but *approximate*) pictures. (Some people find
these pictures very helpful but others don't feel they gain any
intuition - it depends on how your mind works!)

Do you know the onion-skin picture of 1-forms? The idea is simply - take
a pin representing a vector (I assume you already know the pointy object
representation of vectors :-) and stick it in an onion. Rotate the
onion-pin complex around, scale it, in fact apply any invertible linear
operation to it and you find that the number of layers of onion skin the
pin goes through is constant. The number of layers is a scalar. In other
words an onion gives a mapping from vectors to scalars otherwise known
as 1-forms.

So now we have a way to picture a 1-form field: draw n-1 dimensional
surfaces representing layers. For example on a sheet of paper with the
usual coordinates you can draw dx as vertical lines one unit apart, draw
dy with horizontal lines one unit apart or dr=dsqrt(x*x+y*y) as
concentric circles one unit apart. Stick some vectors on these diagrams
and the number of lines they pass through (counting doubling back over a
line negatively - strictly speaking the lines are oriented) and you'll
get the evaluation of the one-form field on the vector at that point
(within a certain tolerance as these diagrams are approximate).

Now we can do our first integral. To integrate a 1-form along a curve
simply count how many layers it goes through (again counting doubling
back negatively). Notice how there isn't a really obvious way to combine
a 2-dimensional region, say, with a 1-form field. 1-forms seem to
interact most naturally with 1-dimensional curves.

Now consider the exterior product. To find the 2-form a/\b draw the
pictures for a and b and take the intersections of the lines (note how
/\ even *looks* like the intersection symbol). So, for example, dx/\dy
is represented (approximately) by points on the grid with integer
coordinates. Note how there is now a natural way to combine a
2-dimensional region with a 2-form - simply count the number of points
inside the region. Notice how there isn't really an obvious way to
combine a curve, say, with a 2-form. 2-forms seem to interact most
naturally with 2-dimensional regions.

With a bit of imagination it isn't too hard to extend the above
discussion to higher dimensions as well as draw pictures for exterior
derivatives and Stokes' Theorem.

Hope this helps...

--
Dan Piponi PhD
http://www.grin.net/~tanelorn