James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Problems Plus 1. Let S 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 VsSd subtended by S at P is the set of lines
starting at P and passing through S. Let Ssad be the intersection of VsSd 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
| VsSd |
−
area of Ssad
a 2
Apply the Divergence Theorem to the part of VsSd between Ssad and S to show that
| VsSd | − yy r n dS
r 3
S
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 4 steradians.
S
S(a)
P
a
2. Find the positively oriented simple closed curve C for which the value of the line integral
is a maximum.
y C
sy 3 2 yd dx 2 2x 3 dy
3. Let C be a simple closed piecewise-smooth space curve that lies in a plane with unit normal
vector n − ka, b, cl and has positive orientation with respect to n. Show that the plane area
enclosed by C is
1
2 y C
sbz 2 cyd dx 1 scx 2 azd dy 1 say 2 bxd dz
;
4. Investigate the shape of the surface with parametric equations x − sin u, y − sin v,
z − sinsu 1 vd. Start by graphing the surface from several points of view. Explain the
appearance of the graphs by determining the traces in the horizontal planes z − 0, z − 61,
and z − 6 1 2 .
5. Prove the following identity:
=sF Gd − sF =dG 1 sG =dF 1 F 3 curl G 1 G 3 curl F
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