Reference no: EM131110760

Let be a smooth parametric surface and let P be a point such that each line that starts at P intersects S at most once. The solid angle Ω(S) subtended by S at P is the set of lines starting at P and passing through . Let S(a) be the intersection of Ω(S) with the surface of the sphere with center P and radius a. Then the measure of the solid angle (in steradians) is defined to be

|Ω(s)| = areaf s(a) / a²

Apply the Divergence Theorem to the part of Ω(S) between S(a) and to show that

|Ω(s)| = ∫_s∫ (r·n) ds / r³

where is the radius vector from P to any point on S, r = |r|, and the unit normal vector is directed away from P.