
Let S be a smooth parametric surface and P be a point such that each line that starts at P intersects S at most once. The Solid angle
Apply the Divergence Theorem to the part of
Where r is the radius
This shows that the definition of the measure of a solid angle is independent of the radius a of the sphere. (Note the analogy with the definition of radian measure.) The total solid angle subtended by a sphere at its center is thus

To show:
Answer to Problem 1P
Solution:
Explanation of Solution
1) Concept:
i) The Divergence Theorem:
Let
ii)
2) Given:
The measure of solid angle is
3) Calculation:
We are given that,
The measure of solid angle is
Let
Let
Let
Let
By using divergence theorem,
Note that on
Therefore,
Consider,
By definition of dot product,
By using quotient rule of differentiation,
Simplifying,
Factor out common terms and then cancel out common factor from the numerator and denominator,
Therefore,
Therefore,
From
On
Therefore,
From
Therefore, from
Therefore,
Conclusion:
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Chapter 16 Solutions
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