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I would argue that this "extra credit" part of the assignment
is far and away the most worthwhile part ... because if you do
this, you get the other parts practically for free.
For example, suppose we want to answer the very physically-
significant question of how much a pointer precesses if
flatlanders carry it along during a trip around the spherical
triangle. This involves "parallel transport" i.e. at each
step the new pointer orientation is aligned with (parallel to)
the old pointer orientation.
Lemma: To convince yourself that this is an interesting
question, consider the example of the spherical triangle
that has the following properties:
-- sides: length s = (pi/2) R
-- interior angles: three right angles
-- area: (pi/2) R^2
-- solid angle: one octant i.e. (pi/2)
Now we argue that the amount of precession is proportional to
the area of the triangle. ...
...
This tells us that the amount of precession is proportional
to area. The remaining work is to determine the constant of
proportionality. We can obtain this from the aforementioned
90-90-90 triangle. It has an area of (pi/2) R^2 and a
precession of (pi/2). So we conclude that the bottom-line
answer is incredibly simple:
precession = area / R^2