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Well... what is it that we are trying to teach? Are we teaching that
"anything goes" or are we teaching that one of the goals of learning
science is learning common practice so as to make it more
likely that we
can communicate clearly.
If we are converting from i-j notation to r,theta notation, or
converting from i-j-k notation to r, theta,phi notation (or
vice-versa)
there are some conventions we assume we are using. For example we
assume orthogonal axes and we even assume a right-handed coordinate
system.
This is the convention in math books, physics books, high-school or
college, and also what practicing physicists use in publications, etc.
Why wouldn't we want students to understand this and use this
convention?
Yes... we also want to teach that physics is independent of the
coordinate system used. Yes, we don't really care which way
the student
orients the axes as long as they maintain a right-handed system. But
once a standard coordinate system has been chosen and the orientation
has been chosen, we ought to follow conventions.
I do not totally disagree with Joel's assertion that we can accept any
convention as long as the student makes it clear what convention is
being used. Clarity is a very significant goal in our
teaching. But if
clarity is what we are after, and assuming this implies accuracy and
ease of communication, then we also ought to teach (and
expect students
to be able to use) the conventions that are fairly well established.