Thomas Calculus 13th [Solutions]

1166 Chapter 16 Integrals and Vector Fields
F
2 2 2 2
1 2 n i j F
0 0
1 n
Circ 0 dt 0 and Circ dt 2 ; (cos t) (sin t) cos t sin t 1 and
2 n
1
2 2
dt
0
2
0
d r
dt
0 Flux 2 and Flux 0 dt 0
(b) r (cos t) i (4 sin t) j, 0 t 2 ( sin t) i (4 cos t) j, F1
(cos t) i (4 sin t) j , and
dr
dr
2 1 dt
2 dt
1
F ( 4 sin t ) i (cos t ) j F 15 sin t cos t and F 4 Circ 15 sin t cos t dt
2
15 2
2
sin t 0 and Circ
4 1
2 2 4 dt 8 ; n cos t i sin t j F
0 17 17
1 n
0
4 2 4 2 15
2 2
F
4
2 n 1 F
17 17 17 0
1 n v
0 17
cos t sin t and sin t cos t Flux | | dt 17 dt
2 2 2
2
15 15
2 F
0
2 n v dt t t dt t
0 17 2 0
8 and Flux | | sin cos 17 sin 0
2
0
30. r ( a cos t) i ( a sin t) j, 0 t 2 , F1 2xi 3 yj, and F2
2 xi ( x y) j ( a sin t) i ( a cos t) j,
F (2a cos t) i (3a sin t) j, and F (2a cos t) i ( a cos t a sin t) j n | v| ( a cos t) i ( a sin t) j,
1 2
1 | |
2 2
2a cos t
2 2
3a sin t, and 2 | |
2 2
2a cos t
2
a sin t cos t
2 2
a sin t
1
2 sin 2
2
sin 2
2
2 2 2 2 2 t 2 t
a t a t dt a
t
a
t
0 2 4
0
2 4
0
2
a
F n v F n v
Flux 2 cos 3 sin 2 3 , and
2
2 2 2 2 2 2
0
2 t sin 2t
2 2
a 2
2
2 t sin 2t
2
2
2 4
0
2 0 2 4
0
Flux 2a cos t a sin t cos t a sin t dt
2a sin t a a
d r
dt
dr
dr
1 dt
1 dt
1 1
1 1
31. F ( a cos t ) i ( a sin t ) j, ( a sin t ) i ( a cos t ) j F 0 Circ 0; M a cos t ,
1 1
C
1 1
0
2 2 2 2
N a sin t, dx a sin t dt, dy a cos t dt Flux M dy N dx a cos t a sin t dt
0
2 2
a dt
a
;
dr
dr
a
2 dt 2 dt 2
a
2 2
F ti i F t t dt M t N dx dt dy
2 2
, Circ 0; , 0, , 0
Flux M dy N dx 0 dt 0;
2
C
2 2
therefore, Circ Circ1 Circ2
0 and
a
a
Flux Flux Flux a
1 2
2 2 2 2 dr
dr
1 1
3 2 3 2
dt
dt
1 1
32. F a cos t i a sin t j, ( a sin t ) i ( a cos t ) j F a sin t cos t a cos t sin t
3 2 3 2 2a
2 2 2 2
1
0
3 1 1
Circ a sin t cos t a cos t sin t dt ; M a cos t , N a sin t , dy a cos t dt ,
3 3 3 3 4 3
1
C
1 1
0
3
dx a sin t dt Flux M dy N dx a cos t a sin t dt a ;
2 dr
dr
2 a 2 a 2
2 dt 2 dt 2
a 3 2 2
3
2 2
, Circ
2
; , 0, 0,
F t i i F t t dt M t N dy dx dt
4
2
C
2 2 1 2 1 2 3
Flux M dy N dx 0; therefore, Circ Circ Circ 0 and Flux Flux Flux a
3
2
3
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