Re: integral nomenclature

["John S. Denker" <jsd@MONMOUTH.COM>]

Carl E. Mungan wrote:

> Consider the integral Int{f(x) dx}. As physicists, probably many of
> us read and think of this as a summation of quantity f multiplied by
> small intervals dx (where f is evaluated say in the middle of each
> interval for specificity).

Yes, that's more-or-less the definition of Riemann integral.

[Alternative definitions include the Lebesgue integral.
See below for more on this.]

> We note that dx is dimensional and carries units (typically).

Yes, true and important.

> But I'm sure many of us have experience with students leaving off the
> dx,

Ick.

> ... text is clearly saying:
>
> "The integral is denoted Int{f(x) dx} or Int{f}. The symbol Int is an
>  integral sign. It is a stylized S to suggest the connection with
> sums. f is called the integrand. The symbol dx is a device for
> keeping track of the variable."

> AARGH #1: The text appears to condone the dropping of dx.
> AARGH #2: dx is not even given a name. It's just a "device."

> 1. What would you name "dx" in an integral specifically?

It's called the _measure_.


I don't care if they classify it as a device.
The whole integral could be considered a device.
The measure is an important part of the device,
no matter what you call it.
       Integral F(x,y) dx is not the same as
       Integral F(x,y) dy, obviously.


> 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 like to take the bull by the horns and
introduce totally non-trivial measures.

Since high school I've been writing the
moment-of-inertia formula as
       Integral r^2 dm

where the measure involves mass, not length.  This
forces people to think, because the first 500
integrals they saw used length as the measure.

Plain old mass = Integral dm
First moment   = Integral r dm
  which is useful for finding the center of mass
Second moment  = Integral r^2 dm
  i.e. moment of inertia

If you know how the mass is distributed as a function
of r you can convert the foregoing into an integral
w.r.t r, but that is not necessary, not always
convenient, and not the way I think of the concept.
Conceptually, it is a sum over elements of mass,
weighted by r^2.

The math faculty will thank you for this, because
it lays the foundation for Lebesgue measure theory
and Lebesgue integrals.

Also note that modern probability theory is
completely dependent on measure theory.  I didn't
have a clue about this until I was out of school.
It's a funny story:  I had a post-doc who was a
card-carrying mathematician.  We were writing a
paper that had a ton of probabilistic stuff in it.
I had to ask him what a probability measure was.  He
just about fainted.  He said that mathematicians always
marveled at the way physicists invented things like
delta functions and used things like probabilities
without appreciating the mathematical niceties.

We were going on a trip that involved about ten
hours of driving.  He said if I would teach him
quantum mechanics he'd teach me measure theory.
So that's what we did.  Pretty much filled up
the drive-time.  Five hours of QM plus five hours
of measure theory.

And it turned out that measure theory was exactly
what we needed to completely blow open the problem
we were working on.  We couldn't have done it
without a physicist's intuitive feel for what was
going on, and we couldn't have done it without a
mathematician's more-sophisticated, more-general
and more-flexible approach.