Thomas Calculus 13th [Solutions]

Section 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 1165
22.
t
6
2 2
r (sin t) i (cos t) j k, 0 t 2 , and F 6zi y j 12xk F ti cos t j (12 sin t) k and
dr
dt
1
6
dr
dt
2
2 2 3
2
t t t t t dt t t t
1
t t
0 3
0
(cos t) i (sin t) j k F t cos t sin t cos t 2 sin t
work cos sin cos 2 sin cos sin cos 2 cos 0
23.
2 2 2 3 2
x t and y x t r ti t j, 1 t 2, and F xyi ( x y) j F t i t t j and
dr 3 2 3 3 2 2 3 2
i 2 j F
dr 2 2 3 2 ( ) F
dr
3 2
dt dt C
C dt 1
4 3
2
3
t
2
t 12
16 3 2 45 18 69
4 3 1 3 4 3 4 3 4
t t t t t t xy dx x y dy dt t t dt
24. Along (0,0) to (1,0): r t i, 0 t 1, and F ( x y ) i ( x y ) j F t i t j and i F t ;
Along (1,0) to (0,1): r (1 t) i tj, 0 t 1, and F ( x y) i ( x y) j F (1 2 t) i j and
dr
dt
i j F
dr
dt
2 t;
Along (0,1) to (0,0): r (1 t) j, 0 t 1, and F ( x y) i ( x y) j F ( t 1) i (1 t) j and
dr
j F
dr
dr dt C
2
1
2t
t 2 1 1
0
dr
dt
1 1 1 1
t 1 ( x y) dx ( x y) dy t dt 2 t dt ( t 1) dt (4t 1) dt
0 0 0 0
dr
dt
25.
2 2 4 5
, 2 1, and
dr
2 and
dr
2
dy
dy
r x i y j y i y j y F x i y j y i y j y i j F y y
C
1 1 5 6 2
1
F
dr
2
1 1 1 1 64 4 3 63 39
2 dy 2 3 2 2 3 2 3 2 2 3 2
F T ds dy y y dy y y
26.
r (cos t) i (sin t) j, 0 t , and F yi xj F (sin t) i (cos t) j and ( sin t) i (cos t)
j
dr
dt
2 2
2
F sin t cos t 1 F dr
( 1) dt
C
0
/2
2
d r
dt
27.
r ( i j) t( i 2 j) (1 t) i (1 2 t) j, 0 t 1, and F xyi ( y x) j F 1 3t 2t i tj
and
2
2 1
2 2 3
1
dr i 2j F
dr 1 5t 2t work F
dr
dt 1 5t 2t dt t
5
t
2
t
25
dt dt C dt 0 2 3 0 6
28. r (2 cos t) i (2 sin t) j, 0 t 2 , and F f 2( x y) i 2( x y)
j
d r
dt
2 2 2 2
F 4(cos t sin t) i 4(cos t sin t) j and ( 2 sin t) i (2 cos t)
j
F
dr
dt
8 sin t cos t sin t 8 cos t cos t sin t 8 cos t sin t 8 cos 2t
dr
dt
2 2
0
0
work f d r F dt 8 cos 2t dt 4 sin 2t
0
C
C
d r
dt
dr
1 2 1 dt
2
29. (a) r (cos t) i (sin t) j, 0 t 2 , F1 xi yj, and F2
yi xj ( sin t) i (cos t) j,
2 2
F (cos t ) i (sin t ) j, and F ( sin t ) i (cos t ) j F 0 and F sin t cos t 1
dr
dt
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