[BUG] geometric derivative argument

[Tevian Dray]

I'd like to share the following conversation, slightly edited, which has
been held off-list over the last few days.

Tevian

>>>>> Larry Tankersley writes:

    LT> I find that mastering the geometry of the coordinate systems is
    LT> critical for physics students attempting to apply vector
    LT> calculus to E&M. Therefore, the derivatives of the directions
    LT> are motivated using geometric arguments.

Larry's handout is available at:
	http://www.math.oregonstate.edu/bridge/ideas/CoordDeriv.pdf

>>>>> Tevian Dray writes:

    TD> Thanks for the very nice presentation of the geometric
    TD> derivation of d(r-hat), etc.  I believe this is the standard
    TD> argument used in advanced engineering courses: Simply compare
    TD> the basis vectors at nearby points.  It's quite similar to the
    TD> approach used in differential geometry (which unfortunately is
    TD> often presented algebraically rather than geometrically) for
    TD> finding connection coefficients.

>>>>> Tevian Dray also writes:

    TD> Corinne points out that there is a rather subtle argument
    TD> missing from Larry's geometric derivation.  Unit vectors have no
    TD> units!  So you must either argue that your vectors do not
    TD> represent actual lengths, or insert some missing factors of
    TD> "1 cm" (or similar) to carry the units.

    TD> I would probably present this as:
    TD> 	1*d(rhat) = d(1*rhat) = 1*d(phi) phihat
    TD> although it might be better to insert explicit factors of r.

>>>>> Stuart Boersma writes:

    SB> It was suggested that there was no need to make any reference to
    SB> the basis vectors i,j,k in the geometric derivations of the
    SB> various partial derivatives of r_hat, theta_hat, and phi_hat.
    SB> The only place I see that such a reference to a Cartesian
    SB> coordinate system is needed would be in showing that
    SB> r_hat,theta_hat, and phi_hat form an orthogonal basis.  I
    SB> suppose that one can easily argue that theta_hat and phi_hat are
    SB> tangent to spheres centered at the origin and, hence, orthogonal
    SB> to r_hat.  Can one prove that phi_hat and theta_hat are
    SB> orthogonal without either 1) first converting to a Cartesian
    SB> basis or 2) deriving the standard inner product for the
    SB> spherical basis?

>>>>> Tevian Dray writes:

    TD> Depends on what you mean by "prove" and "derive".  Let me first
    TD> ask how you would answer the same question for rectangular
    TD> coordinates!

    TD> If your definition of the dot product is in terms of a
    TD> rectangular basis, then I think you're stuck having to refer
    TD> everything to that definition, and hence first having to convert
    TD> to rectangular coordinates.

    TD> If instead your definition of the dot product is geometric, then
    TD> orthogonality follows from "obvious" geometric properties.
    TD> Standard latitude/longitude lines on the sphere are every bit as
    TD> "obviously" orthogonal as the standard rectangular gridlines.

    TD> If you want a fancier argument, then perhaps the following might
    TD> work.  Recall that the horizontal projection of a sphere to a
    TD> cylinder of the same radius is area preserving.  This is *not* a
    TD> conformal map, but *does* preserve orthogonality of the grid
    TD> lines.  But the grid lines on a cylinder are clearly equivalent
    TD> to those in rectangular coordinates -- just unroll the cylinder.
    TD> This chain would seem to establish that any of these grid lines
    TD> are orthogonal iff all of them are.  Still missing is a clean
    TD> argument that this projection preserves the orthogonality of the
    TD> gridlines, without assuming the result.