Summary: American Mathematical Monthly, 2004, 111, no. 2, p. 172.
`Problem and Solution' section
Using Stokes's Theorem on the Sphere
10957 [2002, 664]. Proposed by Victor Alexandrov, Sobolev Institute of Mathematics,
Let S 2 be a unit sphere in R 3 . Let D be a domain in S 2 with piecewise smooth
N denote the the function on the sphere that maps each point to the
unit inward normal vector at that point. Let n denote the function on the smooth part
of the boundary #D of D that maps each such point to the inward unit vector normal
to #D, and parallel to the plane tangent to the sphere at that point. Let # be the usual
measure on a sphere, and let s be the arc length measure on the boundary of D. Prove
2 ## D
N d# + # #D
n ds = 0.
Solution by Knut Dale, Telemark University College, Bø, Norway.
We replace the unit sphere with a sphere of radius R centered at the origin. We also