Apply the Divergence Theorem to the part of Ω(S) between S(a) and S to show that
|Ω(S)| = dS
where r is the radius vector from P to any point on S, r = | r |, and the unit normal vector n is directed away from P.
This shows that the definition of the measure of a solid angle is independent of the radius a 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 4π steradians.
To show: The value of .
Write the expression for Divergence Theorem.
Consider the portion of and is represented by and is the boundary.
Also consider the lateral surface of is represented by .
Modify equation (1) as follows.
Consider in terms of , and .
Consider in terms of , and represented as follows.
Find the value of .
Substitute for in equation (2),
The surface of other than and is results in .
Find the value of .
Expand in terms of , and as follows
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