[BUG] more on spherical basis vectors

[Stuart Boersma]

Quick (?) question:  Can one use the master formula to calculate quantities such as

partial deriv of theta-hat w.r.t theta

without having to resort to any reference to cartesian coordinates (basis vectors or otherwise)?
(theta is the mathematician's theta, the angle in the xy-plane)


Thanks.



>>> 
From: 	Tevian Dray <tevian at math.oregonstate.edu>
To:	Bridge Users Group <bug at science.oregonstate.edu>
Date: 	1/16/2008 1:35 PM
Subject: 	[BUG] spherical basis vectors

I discovered a new use for small white boards this week.  I was
explaining to my (small, honors) vector calculus class about adapted
bases, and we had reached spherical coordinates.  As I now usually do,
I had everyone stand up, close their eyes, and put their arms out along
r-hat, theta-hat and phi-hat, with theta-hat being the hardest.  But for
the first time, I put the origin on the ceiling, rather than the floor,
so that the students were in the Southern Hemisphere.  This was
challenging even for me, so I looked for a way to make it clear.

We had been working with small white boards, so I asked the class to use
their boards to show me where the tangent plane to the sphere would be
at their location.  My class is already accustomed to my asking
questions using these boards, so I guess I shouldn't have been
surprised: They all started drawing pictures of spheres on their
boards, while still standing up.

No, I said, use the board to represent the tangent plane.

Oh, they said.

I was then able to ask whether theta-hat (and phi-hat) should be in this
plane (yes), and where r-hat should be (perpendicular to this plane).
Despite the initial confusion, I believe the boards helped to solidify
these concepts for the students, and will happily repeat the experience.

Tevian


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