Question concerning differential forms?

Question

In regards to the response by JCS here:
http://answers.yahoo.com/question/index;_ylt=AkPiG6NOPG3U92S_lYqNE688.Bd.;_ylv=3?qid=20080402130723AAQdXZt

In particular, where he says:
"dxdy = dx∧dy (it's a 2-form)

Where the "∧" is an algebraic operation called the exterior product; again, it's a generalization of the more familiar exterior product on R^3."

Why is it not instead the following, at least for a 2-form?
dxdy = |dx∧dy|

And in more general terms, are n-forms going to be something like
dx1dx2...dxn = |dx∧dx2∧...∧dxn|

And how is this generating Stokes theorem from de Rham's Theorem?

I have read the proof of de Rham's Theorem, after working out a satisfactory justification of it, but was unable to figure out the connection between the differential form in question and the Stokes Theorem in R^3...

Also, I would appreciate hints on arriving at Gauss' Theorem and Green's Theorem from de Rham's, but in particular, Stokes Theorem is the focus.

Thank you very much...

Mαtt · Accepted Answer

dxdy ≠ |dx∧dy|

because it is not a magnitude.

differential forms are a way to extend the theorems from vector calculus to R^n.  The wedge produce or exterior product extends the cross product to R^n.  dxdy = dx∧dy is just a notational difference.  Some books use dxdy and others use dx∧dy.  I think that the wedge notation is better because it is clearer, for example
dx∧dx = 0
where as dxdx implies (dx)²

Also
dx∧dy = -dx∧dy
Recall that the wedge product is anti-commutative

Response:

From analytic geometry you know that if you have two vectors in R^2 |v×u| is the area of the parallelogram, and in R^3 |v×u| is the volume of the paralleliped.  In terms of differential forms dx ∧dy is not an ‘area’ or ‘volume’, it is more like a unit vector if you will like 5i + 6j.  It carries no value but is used to determine what quantities contribute when you compute the exterior derivative, used in the generalization of Stokes’s Theorem.

I am not familiar with all of the details on de Rham's Theorem, but you should realize that Green’s Theorem and Stokes theorems are just special cases of the Gauss Divergence theorem, (multi-dimensional integration by parts).  In fact, once I finished vector calculus I never heard it referred to anything other than the Divergence theorem.

Question concerning differential forms?

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