Thomas Calculus 13th [Solutions]

Section 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 1161
ds: ( D(g)^2 D(h)^2 D(k)^2)^(1/2): #(a)
'ds'
ds(t)*'dt';
F: f(g,h,k): #(b)
'F(t)'
F(t);
Int( f, s C..NULL ) Int( simplify(F(t)*ds(t)), t a..b ); #(c)
`` value(rhs(%));
Mathematica: (functions and domains may vary)
Clear[x, y, z, r, t, f]
2
f[x_,y_,z_]: Sqrt(1 30x 10y]
{a, b} {0, 2};
x[t_]: t
y[t_]:
2
t
2
z[t_]: 3t
r[t_]: {x[t], y[t], z[t]}
v[t_]: D[r[t],t]
mag[vector_]: Sqrt[vector.vector]
Integrate[f[x(t),y(t),z[t]] mag[v[t]], {t, a, b}]
N[%]
16.2 VECTOR FIELDS AND LINE INTEGRALS: WORK, CIRCULATION, AND FLUX
1.
( , , ) 2 2 2
1/2 f 2 2 2
3/2
2 2 2
3/2
f x y z x y z 1 x y z
x 2
(2 x ) x x y z ; similarly,
f 2 2 2
3/2 f 2 2 2
3/2
xi yj zk
y x y z and z x y z f
y
z
x y z
2 2 2
3/2
2 2 2 1 2 2 2 f
f ( x, y, z) ln x y z ln x y z 1 1 (2 x ) x ; similarly,
2 x 2 x y z x y z
2.
2 2 2 2 2 2
f y f
and z
xi yj zk
f
y 2 2 2 z 2 2 2 2 2 2
x y z x y z x y z
g x y z e z
x 2 y 2 g 2x
g 2 y
x x y y
and g z
2 2 y z
e g x i j e k
x y z x 2 y 2 x 2 y
2
3. ( , , ) ln ,
2 2 2 2
4. g( x, y, z) xy yz xz g y z, g x z, and g y x g ( y z) i ( x z) j ( x y)
k
x y z
5. | F | inversely proportional to the square of the distance from ( x, y ) to the origin
2 2
M ( x, y) N( x, y)
k , 0;
y
k F points toward the origin F is in the direction of n x i j F an , for
2 2
x y
2 2 2 2
x y x y
some constant a 0. Then
M ( x, y ) ax and
2 2
x y
ky
a k F kx i j , for any constant k 0
2 2
2 2
3/2
2 2
3/2
x y x y x y
ay
2 2
N( x, y) M ( x, y) N( x, y)
a
2 2
x y
Copyright
2014 Pearson Education, Inc.