Page 254 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 245

(b)Let A ¼ðx 1 ; y 1 Þ; B ¼ðx 2 ; y 2 Þ. From part (a),
ð
ðx;yÞ
Pdx þ Qdy
ðx; yÞ¼
ða;bÞ
Then omitting the integrand Pdx þ Qdy,we have
B ðx 2 ;y 2 Þ ðx 2 ;y 2 Þ ðx 1 ;y 1 Þ
ð ð ð ð
¼ ¼ ¼ ðx 2 ; y 2 Þ ðx 1 ; y 1 Þ¼ ðBÞ ðAÞ
A ðx 1 ;y 1 Þ ða;bÞ ða;bÞ
ð
ð3;4Þ 2 3 2 2
10.14. (a) Prove that ð6xy y Þ dx þð6x y 3xy Þ dy is independent of the path joining ð1; 2Þ and
ð1;2Þ
ð3; 4Þ.(b) Evaluate the integral in (a).
2
2
3
2
2
(a) P ¼ 6xy y ; Q ¼ 6x y 3xy . Then @P=@y ¼ 12xy 3y ¼ @Q=@x and by Problem 10.12 the line
integral is independent of the path.
(b) Method 1: Since the line integral is independent of the path, choose any path joining ð1; 2Þ and ð3; 4Þ,
for example that consisting of lines from ð1; 2Þ to ð3; 2Þ [along which y ¼ 2; dy ¼ 0] and then ð3; 2Þ to
ð3; 4Þ [along which x ¼ 3; dx ¼ 0]. Then the required integral equals
ð 3 ð 4
2
ð54y 9y Þ dy ¼ 80 þ 156 ¼ 236
ð24x 8Þ dx þ
x¼1 y¼2
@P @Q @ @
2
3
2
2
Method 2: Since ; we must have ¼ 6xy y ; ¼ 6x y 3xy :
@y ¼ @x ð1Þ @x ð2Þ @y
3
2 2
2 2
3
From (1), ¼ 3x y xy þ f ð yÞ. From (2), ¼ 3x y xy þ gðxÞ. The only way in which
3
2 2
these two expressions for are equal is if f ð yÞ¼ gðxÞ¼ c,aconstant. Hence ¼ 3x y xy þ c.
Then by Problem 10.13,
ð ð
ð3;4Þ ð3;4Þ
2 3 2 2 2 2 3
ð6xy y Þ dx þð6x y 3xy Þ dy ¼ dð3x y xy þ cÞ
ð1;2Þ ð1;2Þ
2 2 3 ð3;4Þ
¼ 3x y xy þ cj ¼ 236
ð1;2Þ
Note that in this evaluation the arbitrary constant c can be omitted. See also Problem 6.16, Page 131.
We could also have noted by inspection that
2 3 2 2 2 2 3 2
ð6xy y Þ dx þð6x y 3xy Þ dy ¼ð6xy dx þ 6x ydyÞ ð y dx þ 3xy dyÞ
2 2 3 2 2 3
¼ dð3x y Þ dðxy Þ¼ dð3x y xy Þ
2 2
3
from which it is clear that ¼ 3x y xy þ c.
þ
x
2
2
2 x
10.15. Evaluate ðx y cos x þ 2xy sin x y e Þ dx þðx sin x 2ye Þ dy around the hypocycloid
x 2=3 þ y 2=3 ¼ a 2=3 :
2
2
2 x
P ¼ x y cos x þ 2xy sin x y e ; Q ¼ x sin x 2ye x
x
2
Then @P=@y ¼ x cos x þ 2x sin x 2ye ¼ @Q=@x,sothat by Problem 10.11 the line integral around any
closed path, in particular x 2=3 þ y 2=3 ¼ a 2=3 is zero.
SURFACE INTEGRALS
10.16. If
is the angle between the normal line to any point ðx; y; zÞ of a surface S and the
positive z-axis, prove that