[BUG] more on spherical basis vectors

[Tevian Dray]

>>>>> Stuart Boersma writes:

    SB> Can one use the master formula to calculate
    SB> quantities such as
    SB>		partial deriv of theta-hat w.r.t theta
    SB> without having to resort to any reference to cartesian
    SB> coordinates (basis vectors or otherwise)?  (theta is the
    SB> mathematician's theta, the angle in the xy-plane)

I don't immediately see how to use the master formula to discuss
derivatives of vector fields, but there is a simple argument in polar
coordinates which does yield the derivatives of (r-hat and) theta-hat:

Let's start with r-hat.  From the statements
	r-vec = r r-hat
	d(r-vec) = dr-vec = dr r-hat + r d(theta) theta-hat
and the product rule, it follows immediately that
	d(r-hat) = d(theta) theta-hat

Now take differentials of the equations
	theta-hat dot theta-hat = 1
	theta-hat dot r-hat = 0
to determine the components of d(theta-hat).

Unfortunately, this argument is less straightforward in spherical
coordinates.  Determining the theta-phi cross terms (e.g. the partial
derivative of theta-hat w.r.t. phi, in either convention) requires
either using the differential form argument that d^2=0, or converting
the polar result above for d(theta-hat) to spherical coordinates
(or, of course, resorting to rectangular coordinates).  I'm not
aware of any elementary argument wholly within spherical coordinates. 
(Well, the differential form argument can presumably be expressed in
terms of mixed partial derivatives.)

Tevian