[Phys-l] splice: independent/dependent variables +- cyclic triple chain rule

[John Denker <jsd@av8n.com>]

We can splice together three recent threads:

Yesterday we had an opportunity to use the cyclic triple derivative rule.
You can find two derivations of this (one graphical, one algebraic) at
  http://www.av8n.com/physics/spontaneous.htm#sec-cyclic-triple-derivative

Any situation involving this rule is a good illustration of why you don't
want to worry too much about "independent" variables and "dependent" variables,
and you also don't want to worry too much about what is on the "horizontal axis"
and the "vertical axis".

In particular, consider this figure:
  http://www.av8n.com/physics/spontaneous.htm#fig-cyclic-triple-derivative

The figure has some interesting properties:
 -- There are no axes at all, strictly speaking.
 -- The contours of constant x are not vertical.
 -- The contours of constant y are not horizontal.
 -- The x and y contours are not perpendicular.
 -- There are also z contours that come into play.

On one line of the derivation, we treat z as a function of x and y, while on
the very next line we treat x as a function of y and z.  All three variables
participate in equivalent, symmetrical ways in the result, i.e. the cyclic
triple chain rule.

  If you insist on using the terms "independent" and "dependent", all of the
  variables are equally "dependent" and all of them are equally "independent".
  Maybe you can say they take turns being "dependent" or "independent", but 
  I'm not 100% sure what that means.

In summary:
 -- In thermodynamics and lots of other fields, and in life in general, the 
  number of variables is often huge compared to the number of degrees of 
  freedom.  Choosing which ones to call "independent" is usually not worth 
  the trouble.  Neither the math nor the physics requires you to choose.

 -- The whole concept of "axis" is suboptimal and constricting.  The smart 
  approach is to think in terms of the _contours_ of constant P, constant V, 
  constant T, constant S, et cetera.

  In simple cases, you can of course get away with a plain "horizontal axis"
  and "vertical axis" ... but you don't want to invest too much in that 
  approach.  Even in cases where plain axes might suffice, it is a good habit 
  to include the grid also (unless there is some peculiar compelling reason 
  not to).  Modern plotting software makes it super-easy to include the grid.
  Even excel can do it.

 -- Never write a partial derivative without specifying what's being held constant.
  This business is confusing enough without any additional _completely avoidable_
  confusion.

 -- There are nice graphical ways of showing the meaning of partial derivatives,
  including the cyclic triple chain rule.