FW: [BUG] multiple integration

[Tevian Dray]

>>>>> Alexander J Smith writes:

    AJS> I guess one answer to when Delta A=dx dy is taught is in the details
    AJS> of the hypotheses of Fubini's Theorem.

    AJS> I have my students make extensive use of Matthias's Maple package
    AJS> intdraw at this point in a course. If you try intdraw(x=0..1,y=1..0);
    AJS> then you get an error, which is good. It draws attention to this
    AJS> issue that goes back the a hypothesis in Fubini.

I acknowledge that the orientation is built into Fubini's theorem, which is
often used as the "definition" of multiple integration in terms of iterated
single integrals.  But I'm not so sure that this is a good thing.  Why
shouldn't one be able to integrate from large parameter values to small ones?
That's all x and y are, parameters; double integrals in the plane are special
cases of surface integrals (and not, I would argue, the other way around).

Yes, one must be careful about the orientation, but this is neatly
accomplished by regarding dA as the magnitude of the vector surface element
(and again I would argue, not the other way around, that is, not by
constructing the latter out of the former together with the unit normal
vector).  I would further argue that it's up to the user to put appropriate
signs on dx etc to match the given limits, rather than forcing the choice of
limits to match the given orientation.  This is nothing new -- the magnitude
of a vector always involves absolute values, yet nothing prevents dx from
being negative except convention.

The point, I think, is that multiple integrals do NOT in fact depend on the
orientation (and hence differ from the integration of differential forms).
Fubini's Theorem says, in effect, IF you choose an orientation AND you choose
your limits in a certain way THEN you can do multiple integrals as iterated
single integrals with those limits.  But I don't think it says you have to
make those choices, which I for one view as overly restrictive.

Tevian