Thomas Calculus 13th [Solutions]

Section 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 1171
C : r t i (1 t ) j, 0 t 1 F 2 t i 2(1 t ) k and i j F 2 t
3
3
1 2
1
t dt t
0 0
Flow 2 1 Circulation ( 1 ) 1 0
d r
dt
dr
dt
52.
dy f f dy f
1 2 2 2
F dr
x
dx
y z
dz dx dz
, where f ( x, y, z) x , y x F dr
d
f r ( t)
by
dt dt dt dt x dt y dt z dt 2
dt dt
dr
dt
b
d
a dt
the chain rule Circulation F dt f r( t ) dt f r( b ) f r ( a ) . Since C is an entire
ellipse, r( b) r ( a),
thus the Circulation 0.
C
53. Let x t be the parameter
dr
dt
2 2
y x t and
2
z x t r ti t j tk , 0 t 1 from (0, 0, 0) to (1,1, 1)
3 2 3 3 3 3 3 1 3
i 2tj k and F xyi yj yzk t i t j t k F dr
t 2t t 2t Flow 2t dt
1
dt
0 2
54. (a)
2 3
dr f dx f dy z dz df
dt x dt y dt z dt dt
F ( xy z ) F , where
2 3
f ( x , y , z ) xy z F dr
dt
C
dt
(b)
b d
dt a
F
C
dr
dt
f r( t) dt f r( b) f r ( a) 0 since C is an entire ellipse.
(2,1, 1) 2 3 2 3
(2,1, 1)
d
2 3 2 3
xy z dt xy z
(1,1,1) dt
(1,1,1)
(2)(1) ( 1) (1)(1) (1) 2 1 3
55-60. Example CAS commands:
Maple:
with( LinearAlgebra );#55
F: r - r[1]*r[2]^6|3*r[1]*(r[1]*r[2]^5+2>;
r: t- 2*cos(t)| sin(t) ;
a,b: 0.2*Pi;
dr: map(diff,r(t),t): # (a)
F(r(t));
# (b)
q1: simplify( F(r(t)) . dr ) assuming t::real; # (c)
q2: Int( q1, t a..b );
value( q2 );
Mathematica: (functions and bounds will vary):
Exercises 55 and 56 use vectors in 2 dimensions
Clear[x, y, t, f, r, v]
6 5
f[x_,y_]: {x y ,3x(x y 2)}
{a, b} {0, 2 };
x[t_]: 2 Cos[t]
y[t_]: Sin[t]
r[t_]: {x[t], y[t]}
v[t_]: r'{t]
integrand f[x[t], y[t]]. v[t]//Simplify
Integrate[integrand, (t, a, b)]
N[%]
If the integration takes too long or cannot be done, use NIntegrate to integrate numerically. This is suggested for
Exercises 57 - 60 that use vectors in 3 dimensions. Be certain to leave spaces between variables to be multiplied.
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