A quick review of spherical coordinates:
Derive the spherical to rectangular conversion formulas shown to the right:
Note the axes orientation.
substituting [eq 1] into [eq 2] and [eq 3] yields:
NEXT, Derive the surface area of a sphere equation:
Remembering that arc length (S) is just the radius times the subtended angle (where q is in radians):
the differential area dA element side lengths dl and dw are thus given by:
- dλ = r*df since r is measured from center of sphere.
- But the radius used to calculate dw is NOT the radius from the sphere center but rather (a) measured from the Z axis where: a = r*sin(f).
using double integration where the limits are as shown and r a constant:
