Thomas Calculus 13th [Solutions]

Section 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 1167
dr
dr
1 dt
1 dt
1 1 2 2 2 2 2
33. F ( a sin t ) i ( a cos t ) j, ( a sin t ) i ( a cos t ) j F a sin t a cos t a
2 2
1
0
1 1
Circ a dt a ; M a sin t, N a cos t, dx a sin t dt, dy a cos t dt
0
2 2
Flux1 1 1
sin cos sin cos 0;
C M dy N dx a t t a t t dt 2 2
F2 tj, i F2
0
dt
dt
Circ 0; M 0, N t , dx dt , dy 0 Flux M dy N dx t dt 0; therefore,
2 2 2 2
C
2 2
2
1 2 a
1 2
Circ Circ Circ and Flux Flux Flux 0
2 2 2 2 dr
dr
1 1
3 3 3 3
dt
dt
1 1
34. F a sin t i a cos t j, ( a sin t ) i ( a cos t ) j F a sin t a cos t
35. (a)
(b)
3 3 3 3 4 3 2 2 2 2
1
0
3 1 1
Circ a sin t a cos t dt a ; M a sin t , N a cos t , dy a cos t dt , dx a sin t dt
3 2 3 2 2 3
1 1 1
0
3
Flux cos sin sin cos ;
C M dy N dx a t t a t t dt a 2 2
F2 t j, i F2
0
dt
dt
2 a 2 2 3
2 2 2 2
C
2 2
a
3
4 3
1 2 a
3
1 2
Circ 0; M 0, N t , dy 0, dx dt Flux M dy N dx t dt a ; therefore,
Circ Circ Circ and Flux Flux Flux 0
2 2
r (cos t) i (sin t) j, 0 t , and F ( x y) i x y j ( sin t) i (cos t) j and
d r
dt
2 2 dr
2
dt
F (cos t sin t ) i cos t sin t j F sin t cos t sin t cos t F T ds
2 1 2 sin 2t
sin t cos t sin t cos t dt sin t
t
sin t
0 2 2 4
0
2
a
a
dr
C
2
dr
2 2 d r
2
dt
r (1 2 t) i, 0 t 1, and F ( x y) i x y j 2i and F (1 2 t) i (1 2 t)
j
F
dr
dt
4 t 2 F T ds (4 t 2) dt 2 t 2 t 0
C
1 2
1
0 0
2 2 dr
2
dt
1
(c) r1 (1 t) i tj, 0 t 1, and F ( x y) i x y j i j and F (1 2 t) i 1 2t 2t
j
dr
2 2 dr
1 2 2
1 1
F
dt
t t t t 1 F
C dt
3 2
1 0
t dt r t i t j
(2 1) 1 2 2 2 Flow 2 ; ( 1) ,
2 2 d r
2 2
dt
2
0 t 1, and ( x y) x y and t t 2t
1
F i j i j F i j
2 dr
2 2 dr
1 2
i t t j F t t t t F t t dt
t
2 2
2 2 1 1 2 2 1 2 2 Flow2
2 2
dt
C dt
2 0
2 3
1
2
t
1 2 1
3 3 1 2
0
3 3
Flow Flow Flow 1
2 2
1
36. From (1,0) to (0,1): r1 (1 t) i tj, 0 t 1, and F ( x y) i x y j i j,
2 2 1 2 2 3
1
2 1
1 1 1 1 1
0 3 0 3
2 2 dr
r2 ti (1 t) j, 0 t 1, and F ( x y) i x y j i j,
dt
F i 1 2t 2 t j, and n | v | i j F n | v | 2t 2t Flux 2t 2 t dt t t ;
From (0, 1) to ( 1, 0):
2
2 2 2
2 2 2 2
F (1 2 t) i 1 2t 2 t j, and n | v | i j F n | v | (2t 1) 1 2t 2t 2 4t 2t
1 2 2 3
1
2 2
2 t t dt t t t
0 3 0 3
Flux 2 4 2 2 2 ;
dr
dt
dr
dr
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