axes not required

[John Denker <jsd@MONMOUTH.COM>]

At 12:26 PM 9/16/99 -0400, Michael Edmiston wrote in part:
> Nothing plotted in any coordinate system makes any sense until we
> define the axes.

which, as written, is 100% true.

================

Meanwhile, the purpose of this note is to ask people not to misinterpret
that statement.  In particular, consider the superficially-similar statement

% Nothing () makes any sense until we define the axes.

which is 100% false in an interesting and important way.

At any point in space, you can use one set of axes, multiple sets, or (most
interestingly) none at all.

The laws of physics don't care about axes. The laws are best written in
terms of vectors.  Vectors exist independently of their components, just as
numbers have a meaning independent of any particular system of numerals.
Dot product and cross product have geometric meanings, again independent of
the choice of axes.

In general, if you see an alleged law of physics that seems to depend on
the choice of axes, you should look for the corresponding basis-free law.
If none exists, you can assume the alleged law is wrong -- as surely as if
it failed a dimensional-analysis check.

This important fact is sometimes lost because we teach physics to beginners
in terms of the components. Crudely speaking, grinding things out in terms
of the components is like training wheels on a bike:
 -- They are inelegant.
 -- Sometimes they are safer for non-experts.
 -- Sometimes they get in the way.

There are common examples where setting up a coordinate basis is not worth
the trouble.  For instance, the system based on lattitude and longitude has
singularities at the poles.  (Indeed, there *cannot* be a nonsingular
coordinate basis on a sphere.)  But, as mentioned above, the laws of
physics don't care about the axes!  The laws are 100% nonsingular at the
poles.  "F = ma" means "F = ma" -- it's a relationship between vectors, no
components needed.