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Re: non-rectangular div/grad/curl (was: another math question)



Justin Parke wrote:

I am wondering if, when I introduce vector operators like
the gradient, divergence, and curl, I should go beyond
just rectangular coordinates and include cylindrical and
spherical as well.

Interesting question. Impossible to answer without
knowing a lot more about the students' background.


... if none of the calculus tectbooks do this then perhaps
I am overlooking a good reason why it ought not be done.

Thoughts?

Thought #1: Don't underestimate how hard the task is.
Consider the three topics:
a) scalar-valued functions of non-rectangular coordinates
b) div/grad/curl in rectangular coordinates
c) div/grad/curl in non-rectangular coordinates.

Obviously (a) and (b) are necessary before proceeding
to (c), but they are far from sufficient. It is not
an easy step from (a+b) to (c).


Thought #2: There are various levels of understanding:
High: Students can derive the formulas for div/grad/curl
in funny coordinates.
Medium: Students can more-or-less follow your derivation,
look up the formulas when needed, apply them in
simple quantitative cases, and be able to recognize
real-world situations where the ideas come into
play (e.g. laying out a city with all major streets
north/south and all 1 mile apart).
Low: Students have at least seen the formulas, so they
are on notice that (a+b) does not make (c), i.e.
so they don't blithely assume the length^2 of the
gradient is (d/dr)^2 + (d/d theta)^2
Zilch: Students have never been exposed to the topic of
div/grad/curl in non-rectangular coordinates.


Obviously "low" is better than "zilch" and doesn't cost much.
"Medium" is better than "low" but takes some time and effort.


Thought #3: Beware, there are multiple schemes for setting
up a basis in non-rectangular coordinates, two of which are
perfectly plausible and commonly used:
-- theta basis vector corresponds to unit change in theta
-- theta basis vector has unit length.
Some books (e.g. Jackson _Electrodynamics_) tacitly assume
one scheme; other books tacitly assume another scheme; some
cover both. Covering both causes confusion immediately;
covering only one will cause confusion later.


Thought #4: The usual notation for partial derivatives is
a stinking mess. In rectangular coordinates you can usually
assume that (partial/partial z) implies constant x and constant
y, but when you start using more intricate coordinate systems
(e.g. classical mechanics, thermodynamics, and/or general
relativity) the notation can be interpreted in N different
ways, N-1 of which are wrong, and the student has no way of
figuring out which is the correct interpretation. For more
on this point, see Gerry Sussman and Jack Wisdom, _Structure
and Interpretation of Classical Mechanics_. The entire
book is available online; start with the preface:

http://www-swiss.ai.mit.edu/~gjs/6946/sicm-html/book-Z-H-4.html#%_chap_Temp_2