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Conventions (was vector directions)



Conventions might be used as shortcuts. Is it good or
bad? Here is one example. Consider sign conventions
in geometrical optics, for example, the Gaussian formula
for finding an image produced by a single spherical
boundary between two media of different n. The formula
is derived (for paraxial rays) on the basis one specific
example. Then we say something like this "provided
our sign convention is respected the formula is valid
for any other possible situation." The actual situations
are different, depending on: (a) where the source is with
respect to C and F, (b) which index of refraction is larger
and (c) is the surface seen as concave or convex? Each
situation leads to a different set of triangles.

The natural approach would be to begin by saying
that s, s', f and r are positive (lengths of a segments)
and to derive a formula from each picture. Then one
can see how several similar, but not identical, formulas
can be reduced to one by choosing a sing convention.
This approach, however, is time consuming. Most often
the optical sign convention is used as a shortcut.
Ludwik Kowalski

On Friday, Oct 3, 2003, John S. Denker wrote:

There are two issues here:
-- For the ultra-beginning students, it is important
to pick some sort of convention and stick to it
FOR A WHILE.
-- It is important to let 'em know from the very start
that the local convention is not the only way of doing
things, nor even the only standard way of doing things.

There are many different standards, and infinitely many
legitimate non-standard ways of doing things. You don't
need to discuss all the alternatives, but you should make
it clear that your chosen convention is local and temporary.

I really think Joel's approach is the right one: tell 'em
1a) You can express anything relative to anything,
1b) provided you explain what you've done.
2) Here's a suggestion; here's the local default,
the local convention.

I suppose there are limits to item (1). There is the
usual continuum:
conventional -- eccentric -- smartaleck -- perverse
and it is a matter of judgement how much unconventionality
you want to put up with, but usually any students who are
smart enough and engaged enough to work out seriously
unconventional approaches (while complying with proviso
1b!) aren't the ones you need to worry about.

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

As another datapoint on the "many conventions"
bandwagon, consider the naming of wind velocities.
A northerly wind is coming *from* the north. This
stands in contrast to almost everything else; for
instance a boat on a northerly heading is going
*to* the north. In physics velocity vectors
point in the direction things are moving *to*.

I mention this not to be perverse, but to point
out a source of genuine confusion for students.
You cannot teach your favorite convention assuming
the kids are starting from scratch. They may have
other conventions established in their minds, in
which case some unlearning is required. That's
always hard.

As a tangent: putting too much faith in
local conventions can get you into trouble,
as e.g. the Mars Observer folks found out.

On 10/03/2003 03:46 PM, Arlyn DeBruyckere wrote:
The last time I was at an airport the runways were still set as
clockwise from North, that would make the typical x axis by +90
degrees.

Wow, if you're looking for counterexamples to the
"radians CW from X" convention, runway numbering is
perhaps the ultimate counterexample. Runway numbering,
as illustrated at
http://www.av8n.com/physics/img48/rwy.png
conforms to the following conventions:
-- numbered CCW from north.
-- relative to magnetic north, not true north.
-- numbered using units of tens of degrees.
-- backwards w.r.t location in the sense that
the "35" number is found on the south end of
runway 17/35.

These conventions are very well suited to the
application. This sure makes the point that
the story in the high-school math book isn't
the whole story.