Let be a smooth parametric surface and let be a point such

Chapter 16, Problem 16.612

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Let be a smooth parametric surface and let be a point such that each line that starts at intersects at most once. The solid angle subtended by at is the set of lines starting at and passing through . Let be the intersection of with the surface of the sphere with center and radius . Then the measure of the solid angle (in steradians) is defined to be Apply the Divergence Theorem to the part of between and to show that where is the radius vector from to any point on , , and the unit normal vector is directed away from . This shows that the definition of the measure of a solid angle is independent of the radius of the sphere. Thus the measure of the solid angle is equal to the area subtended on a unit sphere. (Note the analogy with the definition of radian measure.) The total solid angle subtended by a sphere at its center is thus steradians.