. After a long journey, you come across the curve
C
on a sphere as in the picture. Assume that
C
is an equilateral spherical triangle of side length
s=50 mm
on the sphere
x 2
+y 2
+z 2
=R 2
, where
R=110 mm
. This means that
C
is made up of three arcs, each of which is a part of a great circle 9 and has arc length
50 mm
. Let
S
be the spherical triangle bounded by
C
, oriented outwards. Compute the flux of the vector field
F=2xi+2yj+2zk
across
S
. Hint: you may use the following facts without justification: if
T
is a equilateral spherical triangle of side length
s
on the unit sphere, then (1) the angle
α
at each corner of the triangle satisfies
cosα= tans
tan(s/2)
, and (2) the area of
T
is equal to
3α−π
. Challenge: (not graded) prove these facts.
