What is the difference between an x axis, x plane, and an x contour?
I assume "x contour" is shorthand for "contour of constant x".
In three _Cartesian_ dimensions, contours of constant x are
planar, so I assume that is what "x plane" refers to. However,
if there any number of dimensions other than 2, the contours
of constant x are not planar, so "x plane" is not a viable
concept AFAICT. Also, there are lots of situations where the
geometry is not Cartesian -- or where there is no geometry
at all -- in which case the concept of "plane" goes out the
window, but the concept of contour survives.
In particular, in thermodynamic state-space, there is no notion
of distance or angle. There is topology, but no geometry.
There is no natural way to define a "flat" plane. Even if you
arbitrarily impose the idea that the contours of constant P
and constant T and constant V are somehow flat, then the
contour of constant S will not be flat. To make progress,
you need a formalism that can handle "some" non-flat contours,
and at that point you might as well use the same formalism
for all contours, not even pretending that any of them are
flat.
In three Cartesian dimensions, the X axis is perpendicular
to the contours of constant X, so these are easily seen to
be different concepts. In two dimensions, there is a slight
pedagogical problem, because the contours of constant X look
like lines and the X axis looks like a line, so students may
wonder why one kind of line is better than another kind of
line. That answer, of course, is that as soon as there are
more than two variables in two dimensions, *or* there are
more than two dimensions, then axes get you into trouble
and contours get you out of trouble.
For the last N years, whenever I make a plot using a computer,
I habitually turn on the "grid" feature so as to plot the
contours. I haven't fully imposed this policy on all my
older plots, but I am gradually upgrading any plots I get
my hands on. I do this even when it's not important,
because I want people to get in the habit of seeing the
contours.
This is example #754 of the fact that somebody who knows enough
to /pass/ the course doesn't know nearly enough to teach the
course. That's because there are always N different ways of
doing things, and it's hard to know which way will get you
into trouble later, and which way lays the foundation for
further progress. Drawing the grid -- even when it is not
important -- lays a good foundation.