I've been reading bits of the book by Mary L. Boas,
_Mathematical Methods in the Physical Sciences_
It's a pretty decent book. It covers more-or-less the
same topics as Kreyszig, but is IMHO better in every
way. Maybe I'll write a proper review at some point.
Today I just want to talk about partial derivatives.
In Example 6 on page 165, an otherwise-reasonable
discussion falls off the rails. Boas makes a big fuss
about (partial r / partial x) not being the reciprocal
of (partial x / partial r). There is a detailed
discussion of why not. The essential idea is that you
are "supposed" to know that
(partial r / partial x) "naturally" means
(partial r / partial x) at constant y
while
(partial x / partial r) "naturally" means
(partial x / partial r) at constant theta
I call this the "doctrine of natural abscissas".
This makes my flesh crawl. It's like fingernals screeking on
the chalkboard. It's the wrong approach. The argument in
favor of "natural abscissas" is nowhere near strong enough
to outweigh the counterarguments.
1) One counterargument goes something like this: Suppose we
generalize the polar coordinates from D=2 to D=3. Should we
use cylindrical polar coordinates (r, theta, z) or spherical
polar coordinates (r, theta, phi)? IMHO this question is
unanswerable. Therefore the expression (partial x / partial r)
is hopelessly ambiguous in D=3.
2) If that example isn't scary enough, consider thermodynamics.
You can easily have a situation where there are umpteen variables
(energy, entropy, enthalpy, free energy, free enthalpy, volume,
pressure, temperature, number, mass, molar volume, molar energy,
etc. etc.) each of which might be known as a function of a few
of the others.
The problem is that students who have been exposed to the
doctrine of "natural abscissas" drive themselves crazy
trying to figure out what variables the energy is "naturally"
a function of, what variables the enthalpy is "naturally" a
function of, et cetera. These questions are unanswerable.
============
The correct approach is simple:
-- Treat every partial derivative as a directional derivative.
-- Always specify the direction explicitly ... except possibly
when the direction is really, really obvious, and maybe even
then.
Returning to Boas's Example 6: (partial x / partial theta) at
constant y *is* the reciprocal of (partial theta / partial x) at
constant y (assuming both are nonzero). There is no surprise here,
no confusion, and nothing to explain.
In serious work, you need to specify the directions. In
elementary work, maybe you can slightly simplify the
notation by suppressing the directions ... but why bother?
Cumbersome and clear is better than streamlined and
confusing.
=====================
Tangential remark: This illustrates why teachers need to have
expertise *beyond* the level of the courses they are teaching.
It seems to me, all too often, people think that taking a course
in topic "X" makes them qualified to turn around and teach
topic "X". Alas, that is really pernicious. You can see how
it leads to trouble in the case of partial derivatives: They
could study (partial x / partial theta) and then teach it for
years, without ever realizing how much trouble it causes in
more advanced work.
In this way, misconceptions can be passed down from generation
to generation. They never get corrected, and indeed new errors
creep in. There's no feedback.
Typically there are ten ways of introducing concept "X". All
ten ways are equivalent at the introductory level ... but only
one or two generalize well to more advanced levels. It costs
nothing to things the right way ... you just have to know
which of the ten ways is right. This requires at least a
nodding familiarity with the advanced levels.
Bystanders will, alas, be unable to see why the additional
expertise is needed, because on a day-to-day basis the
advanced topics are not mentioned in the introductory class.
But nevertheless the expertise *is* needed. It provides
unspoken constraints on what is and isn't said.