Thomas Calculus 13th [Solutions]

1232 Chapter 16 Integrals and Vector Fields
i j k
r( x, y) xi yj ( ax by)
k be a parametrization of the surface. Then rx ry 1 0 a ai bj k
0 1 b
2 2
d rx
r y dx dy a b 1 dx dy.
Also,
i j k
F
x y z
i j k and n
z x y
2 2
a b 1
xy xy xy
2 2
ai
bj k
a b
2 2 1
1 2 2
F n d a b a b 1 dx dy ( a b 1) dx dy ( a b 1) dx dy.
Now
S R R R
2 2 2 1 2 1 2
x y ( ax by) 4 a x b y ab xy 1 the region R
4 4 2
xy is the interior of the ellipse
2 2
2 2
Ax Bxy Cy 1 in the xy -plane, where A a 1, B ab ,and C b 1.
The area of the ellipse is
4 2 4
2
2 4
4 ( a b 1)
( a b 1)
F dr h( a, b) . Thus we optimize H ( a, b) :
2 2 2 2 2
4AC B a b 1 C
2 2
a b 1
a b 1
H
a
2( a b 1) b 1 a ab
a
2
2 2
b
1
2
0
and
H
b
2( a b 1) a 1 b ab
a
2
2 2
b
1
2
2
0 a b 1 0, or b 1 a ab 0 and
2 2 2
a 1 b ab 0 a b 1 0, or a b ( b a) 0 a b 1 0, or ( a b)( a b 1) 0
a b 1 0 or a b . The critical values a b 1 0 give a saddle.
2
If a b, then 0 b 1 a ab
2
a 1 a
2
a 0 a 1 b 1. Thus, the point ( a, b ) ( 1, 1) gives a local extremum for
F
C
dr
z x y x y z 0 is the desired plane, if c 0.
Note: Since h ( 1, 1) is negative, the circulation about n is clockwise, so n is the correct pointing normal
for the counterclockwise circulation. Thus F ( n ) d actually gives the maximum circulation.
S
If c 0, one can see that the corresponding problem is equivalent to the calculation above when b 0, which
does not lead to a local extreme.
11. (a) Partition the string into small pieces. Let i s be the length of the i th piece. Let( xi
, y i ) be a point in the
th
i piece. The work done by gravity in moving the i th piece to the x-
axis is approximately
Wi ( gxi yi is) yi where xi yi is is approximately the mass of the i th piece. The total work done by
2 2
gravity in moving the string to the x-axis is Wi gxi yi is Work gxy ds
i i
C
(b) Work
2 /2 2 2 2 /2 2
gxy ds g(2cos t)(4sin t) 4 sin t 4cos tdt 16g cos t sin t dt
C 0 0
16g
/2
3
sin t 16 g
3 3
0
(c) x
x( xy) ds
C
and y
y( xy)
ds
C
; the mass of the string is
xy ds
xy ds
xy ds and the weight of the string is
C
C
C
g xy ds . Therefore, the work done in moving the point mass at ( x, y) to the x-axis is
C
W g 2 16 .
C
xy ds y g C
xy ds 3
g
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