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[Phys-L] Re: integral nomenclature



Way back on Fri, 21 Feb 2003 14:30:04 -0500, Carl Mungan asked
some deep questions:

1. What would you name "dx" in an integral specifically? I sometimes
call it the "differential" but this makes more sense in connection
with derivative, DE's, and such.

2. What advice have you gained in your years of teaching physics that
help students understand that "dx" is not something you can insert or
remove from an integral on an as-needed basis?

I gave a passable answer back then, but my understanding has improved,
and I can give a deeper answer now.

In brief:
1) The "d" can fruitfully be interpreted quite literally as an
exterior derivative.

2) The expression inside the integral needs to be an operator
that maps each element of the domain-of-integration to a
number (or the like) that can be summed by the integral.

3) An expression involving "d" is the usual way, but not the
only way, of meeting requirement (2). The "d" need not
appear explicitly, if requirement (2) is met in some other
way.

For example, the usual discrete sum corresponds exactly to
a Lebesgue integral using a measure that assigns unit weight
to each integer. No "d" is required.

4) This is useful for resolving some nasty contradictions that
come up in the usual (mis)interpretations of thermodynamics.

In particular, it is perfectly OK to integrate q, where
q is T dS. The integral will not contain an explicit "d",
because q already meets requirement (2). It is neither
necessary nor permissible to write T dS as dq or dQ,
whether or not it is within an integration.

I am pleased with this, because it clears up some questions
that have been bugging me for years and years.

For the next level of detail, see
http://www.av8n.com/physics/thermo-laws.htm#sec-temptation