Time remaining:
Compare each of the following lengths to the length of a great circle of a s

Mathematics
Tutor: None Selected Time limit: 0 Hours

3. compare each of the following lengths to the length of a great circle of a sphere

A. distance between any pair of pole points

B. Distance between any pole point and its equator 

Jan 8th, 2015


Consider the class of all regular paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by

\theta = \theta(t),\quad \phi = \phi(t),\quad a\le t\le b

provided we allow φ to take on arbitrary real values. The infinitesimal arc length in these coordinates is

ds=r\sqrt{\theta'^2+\phi'^{2}\sin^{2}\theta}\, dt.

So the length of a curve γ from p to q is a functional of the curve given by

S[\gamma]=r\int_a^b\sqrt{\theta'^2+\phi'^{2}\sin^{2}\theta}\, dt.

Note that S[γ] is at least the length of the meridian from p to q:

S[\gamma] \ge r\int_a^b|\theta'(t)|\,dt \ge r|\theta(b)-\theta(a)|.

Since the starting point and ending point are fixed, S is minimized if and only if φ' = 0, so the curve must lie on a meridian of the sphere φ = φ0 = constant. In Cartesian coordinates, this is

x\sin\phi_0 - y\cos\phi_0 = 0

which is a plane through the origin, i.e., the center of the sphere.

Jan 8th, 2015

Studypool's Notebank makes it easy to buy and sell old notes, study guides, reviews, etc.
Click to visit
The Notebank
...
Jan 8th, 2015
...
Jan 8th, 2015
Mar 1st, 2017
check_circle
Mark as Final Answer
check_circle
Unmark as Final Answer
check_circle
Final Answer

Secure Information

Content will be erased after question is completed.

check_circle
Final Answer