[Phys-L] Other Things Being Equal ... or not

[John Denker <jsd@av8n.com>]

1) I shudder whenever somebody says "other things being equal".  The
question arises:  *which* other things?

Here is a pedagogical example that is easy to visualize.  Consider a 
lever with four labeled points:

       ================
       A    B    C    D

and we want to know whether point C rises when I raise point A.  Well, that
rather depends on whether the fulcrum is at point B or point D!  To repeat:
it does not suffice to say that we are raising A.  Raising A at constant B 
is very different from raising A at constant D.


2) Ambiguities are notoriously common in thermodynamics.  Suppose we
change the volume of gas by a factor of two ...
 -- along a contour of constant temperature, or
 -- along a contour of constant pressure, or
 -- along a contour of constant entropy, or
 -- whatever

... and even that is assuming no particles are crossing the boundary, assuming 
no chemical reactions are taking place, et cetera.

The rule is, whenever you specify a change, you have to specify the direction
of change.  In an N-dimensional space, that usually requires specifying which
N-1 things are being held constant.


3) The way that partial derivatives are taught in introductory courses is
terrible.  It teaches bad habits.

Specifically, partial derivatives are introduced in the context of a "preferred" 
XYZ coordinate system, such that ∂E/∂X "implies" constant Y and Z, while ∂E/∂Y 
"implies" constant X and Z, et cetera.  This leads to disaster in more general 
situations (such as thermodynamics) where there is no "preferred" or "natural" 
set of coordinates.

Believe it or not, some thermo books try to introduce a set of "preferred"
variables, which just makes everything worse.  In practice people are interested
in various different directions of change, so building a formalism that assumes
otherwise is (at best) a waste of time, and reinforces bad habits.


On 05/17/2012 08:53 AM, LaMontagne, Bob wrote:
> I think we agree on making our questions as unambiguous as possible.
> Doubling mass or doubling speed - pretty clear because we are only
> changing a single parameter. 

5) I'm not at all convinced that those are "clear" in general.  In the case of
a pendulum in motion, adding mass at the bottom of the arc is very different
from adding mass at the top of the arc.

Also, for an electrical LC oscillator, taking L to be analogous to mass, we
can change L suddenly, change it adiabatically, change it at this-or-that
phase in the cycle, et cetera.  So the claim that "we are only changing a 
single parameter" is nowhere near sufficient to keep us out of trouble.

Ditto for changing the velocity.

Again:  In an N-dimensional space, you usually need to specify which N-1
things are being held constant.


> I think we agree on making our questions as unambiguous as possible.

6) Well, yes and no.  

In the real world, unambiguous questions are exceedingly rare.  Even the
most innocuous questions can be considered ill-posed:
  me:    Could I have a hamburger, please?
  clerk: Would you like fries with that?

Evidently when I asked for 
       ∂(hamburger)/∂t 
I should have specified
       ∂(hamburger)/∂t in direction of constant fries.

My point is, school should teach people how to function in the real world.
Therefore we need a two-step process:
  a) Teach students how to handle ill-posed questions, then
  b) Given them plenty of practice with such questions.

Simply avoiding ill-posed questions in school is a tremendous mistake.


7) None of my remarks should be construed as a criticism of the questions Richard 
Tarara posted on 05/15/2012 07:23 PM ... because AFAICT it was clear which variables 
were being held constant.  I just checked again.  Maybe I'm missing something, but 
I'm usually pretty sensitive to ill-posedness, and I'm just not seeing it here.

There is an unstated assumption that the mass of the bowling ball remains substantially
unchanged from one throw to another, but that seems entirely reasonable to me.