Thomas Calculus 13th [Solutions]

Section 16.5 Surfaces and Area 1191
2x 2y 2 3z 16, or x y 6z 8 2. The parametrization
x 4sin cos , y 4sin sin , z 4cos the surface is
29. The parametrization
2
r( , z) (3sin 2 ) i 6sin j zk at
3 3
P 9
0 , , 0
2 2
3
and z 0. Then
r (6cos 2 ) i (12sin cos ) j 3i 3 3j and
i j k
rz
k at P0 r rz
3 3 3 0 3 3i 3j
0 0 1
the tangent plane is
x 3sin 2 and
2 2 2
x y z 16, z 0.
3 3
3 3i 3j x i y 9 j z 0 k 0 3x y 9. The parametrization
2 2
2 2 2 2 2
y 6sin x y 9sin 2 6sin
2 2 4 2 2 2 2 2
9 4sin cos 36sin 6 6sin 6y x y 6y 9 9 x ( y 3) 9
2
2
30. The parametrization r( x, y)
xi yj x k at
P0 (1, 2, 1) rx
i 2xk i 2k and ry
i j k
at P0 rx
ry
1 0 2 2i k the tangent
0 1 0
plane is 2 i k ( x 1) i ( y 2) j ( z 1) k 0
and
2x z 1. The parametrization x x,
y y
2
z x the surface is
z
2
x
j
31. (a) An arbitrary point on the circle C is ( x, z) ( R r cos u, r sin u) ( x, y, z ) is on the torus with
x ( R r cos u) cos v, y ( R r cos u)sin v , and z r sin u, 0 u 2 , 0 v 2
(b) ru ( r sin u cos v) i ( r sin u sin v) j ( r cos u)
k and rv ( R r cos u)sin v i ( R r cos u) cos v j
i j k
ru rv r sin u cos v r sin u sin v r cosu
( R r cos u)sin v ( R r cos u)cos v 0
( R r cos u)( r cos v cos u) i ( R r cos u)( r sin v cos u) j ( r sin u)( R r cos u)
k
2 2 2 2 2 2 2 2 2 2
| ru rv | ( R r cos u) r cos v cos u r sin v cos u r sin u | ru rv
| r( R r cos u)
2 2 2 2
2
A rR r cos u du dv 2 rR dv 4 rR
0 0 0
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