[BUG] units

[Tevian Dray]

An interesting issue regarding units came up recently in the vector calculus
class here at OSU (which I am not teaching).

The following question arose during a discussion between the TA and some
students, all of whom (including the TA) were struggling with what to do about
units.  What are the units of d/dV M/V, where M is a constant mass in kg, and
V is volume measured in cubic meters.  Writing this as d/dV [ M(kg)/V(m^3) ]
led some students to speculate that the answer should be M(kg)/V^2(m^4),
feeling that the exponent should follow the rules of exponents when taking
derivatives!  (Also looks as though two minus signs have canceled...)

It is easy for us to laugh and say that the units should of course be kg/m^6,
and even to point out that this is obvious from the initial statement of the
problem (since "dV" has units of m^3).  But this is not obvious to students,
all the more so since many are unfamiliar with Leibniz notation.  There is a
full section on this topic in the Hughes Hallett single variable calculus book
(Section 2.4), from which I am currently teaching differential calculus, which
attempts to get students to interpret f'(x).  I conjecture that these
important ideas, regarding both units and interpretation, would be much easier
for students to master if the Leibniz notation were used throughout, that is,
dy/dx instead of f'(x).  The inverse problem, also in the text, of
interpreting finv'(x) is equally easy if written as dx/dy -- and quite
difficult otherwise.

Moving on to vector calculus, the question arose whether the paths given in
the valley lab have reasonable units.  Students were not bothered by the
equation y=x, where it doesn't matter what units of length are used, only that
the same are used for both x and y.  Nor were they bothered by x=1, where it
does matter, but where it's obvious that x and 1 must have the same units.
Even x^2+y^2=2 caused little concern; again, the size of the circle is
determined by the choice of units, but it's clear that "2" must have
dimensions of length squared.

However, the real question was how to interpret the parabolic path given by
y=x^2.  By this time, students (and the TA) were quite willing to assume that
there is a dimensionful constant "1" on the RHS.  But the shape of this path,
and not just its location, depend on the choice of units!  OK, this is just a
question of scale, and yes, there's a difference between a circular path of
radius 10 feet and one with radius 10 miles.  But the parabolic path seems to
provide a better example for driving home the point that you don't in fact
know the shape of the path until you know what units are being used.

As an aside, I have avoided putting dimensionful constants into the equations
for the paths in the valley lab for fear of cluttering up the lab too much.
On the one hand, it's surely better to always use dimensionful constants.
On the other, leaving them out in this case provides a golden opportunity for
further discussion.  I welcome comments on this choice.

In any case, at least some of this discussion will make it into the
Instructor's Guide at the next opportunity.

Tevian