FW: [BUG] multiple integration

[Corinne A. Manogue]

This thread is fascinating! The attempt to find a beautiful mathematical
prescription to the dilemna of orientation.  However, there is a much
simpler pedagogical answer:

Only in mathematics do integrals appear without a physical context.  And
the physical context will ALWAYS tell you which sign you mean.  There is a
fantastic pedagogical opportunity here to get students to concentrate on
what the integral MEANS.  This will help their science applications in the
short run.  In the long run, I believe, it will also help them with
mathematical subtleties like the difference between Lebesgue and Riemann
integrals.

In the context of area, dS must be positive, and one must chose the limits
of integration to respect that.  In the context of mass, dS must similarly
be positive and the limits of integration again must be positive to
respect that.  Since the mass density is always positive and the answer
better be positive one is actually safe calculating any old which way and
taking the absolute value of the answer, but that's not sufficient.  What
about the context of charge?  You must chop up your space and the charge
density may be positive in some regions and negative in others.  Now, the
signs really matter.  But dS must still be positive, exactly as before.
It is the integrand that changes sign.  It really works to teach students
that dS in these contexts "really has" an absolute value in it and that
they can choose their limits to enforce this.

Flux, with an oriented dS(vector) is a completely different physical
context.  Orientation definitely matters and we must help students have
strategies for sorting through this.

Please don't pass up the opportunity to have students think/experiment
with the MEANING of mathematical expressions by attempting to give them a
prescription that allows them to get the right answer with their brain
turned off.

Corinne

---------- Forwarded message ----------
Date: Tue, 8 Feb 2005 09:57:37 -0600
From: "Smith, Alexander J." <SMITHAJ at uwec.edu>
To: bug at science.oregonstate.edu
Subject: RE: FW: [BUG] multiple integration

It sounds like the best way to get at your point Tevian is to introduce
the Lebesgue integral instead of the Riemann integral. This is often done
in European curriculums. This way students run around with d_mu in their
heads instead of dx.  The integral over X of d_mu is 1 and orientation is
irrelevant. But we are stuck with the Riemann integral where the integral
of dx over an interval depends on the orientation.

Alex
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