Page 269 - Schaum's Outline of Theory and Problems of Advanced Calculus

260 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS [CHAP. 10

ð
2
2
10.39. Evaluate ð2x þ yÞ ds, where C is the curve in the xy plane given by x þ y ¼ 25 and s is the arc length
C
parameter, from the point ð3; 4Þ to ð4; 3Þ along the shortest path. Ans: 15
ð
2
10.40. If F ¼ð3x 2yÞi þð y þ 2zÞj x k,evaluate F dr from ð0; 0; 0Þ to ð1; 1; 1Þ, where C is a path consisting
C
3
2
of (a)the curve x ¼ t; y ¼ t ; z ¼ t ,(b)a straight line joining these points, (c)the straight lines from
2
2
ð0; 0; 0Þ to ð0; 1; 0Þ,thento ð0; 1; 1Þ and then to ð1; 1; 1Þ,(d)the curve x ¼ z ; z ¼ y .
Ans: ðaÞ 23=15; ðbÞ 5=3; ðcÞ 0; ðdÞ 13=15
10.41. If T is the unit tangent vector to a curve C (plane or space curve) and F is a given force field, prove that under
ð ð
appropriate conditions F dr ¼ F T ds where s is the arc length parameter. Interpret the result
C C
physically and geometrically.
GREEN’S THEOREM IN THE PLANE, INDEPENDENCE OF THE PATH
þ
3
2
2
10.42. Verify Green’s theorem in the plane for ðx xy Þ dx þð y 2xyÞ dy where C is a square with vertices at
C
ð0; 0Þ; ð2; 0Þ; ð2; 2Þ; ð0; 2Þ and counterclockwise orientation. Ans. common value ¼ 8
10.43. Evaluate the line integrals of (a)Problem 10.36 and (b)Problem 10.37 by Green’s theorem.
10.44. (a)Let C be any simple closed curve bounding a region having area A.Prove that if a 1 ; a 2 ; a 3 ; b 1 ; b 2 ; b 3 are
constants,
þ
ða 1 x þ a 2 y þ a 3 Þ dx þðb 1 x þ b 2 y þ b 3 Þ dy ¼ðb 1 a 2 ÞA
C
(b) Under what conditions will the line integral around any path C be zero? Ans. (b) a 2 ¼ b 1

10.45. Find the area bounded by the hypocycloid x 2=3 þ y 2=3 ¼ a 2=3 .
2
3
3
[Hint: Parametric equations are x ¼ a cos t; y ¼ a sin t; 0 @ t @ 2 .] Ans: 3 a =8
þ ð
2
10.46. If x ¼ cos ; y ¼ sin ,prove that 1 1 d and interpret.
2 xdy ydx ¼ 2
þ
3
2
2
10.47. Verify Green’s theorem in the plane for ðx x yÞ dx þ xy dy, where C is the boundary of the region
C
2
2
2
2
enclosed by the circles x þ y ¼ 4 and x þ y ¼ 16. Ans: common value ¼ 120
ð
ð2;1Þ
3
4
2
10.48. (a)Prove that ð2xy y þ 3Þ dx þðx 4xy Þ dy is independent of the path joining ð1; 0Þ and ð2; 1Þ.
ð1;0Þ
(b) Evaluate the integral in (a). Ans: ðbÞ 5
ð
2
2 2
3
2
10.49. Evaluate ð2xy y cos xÞ dx þð1 2y sin x þ 3x y Þ dy along the parabola 2x ¼ y from ð0; 0Þ to
C
2
ð =2; 1Þ. Ans. =4
10.50. Evaluate the line integral in the preceding problem around a parallelogram with vertices at ð0; 0Þ; ð3; 0Þ,
ð5; 2Þ; ð2; 2Þ. Ans: 0
2
2
2
10.51. (a)Prove that G ¼ð2x þ xy 2y Þ dx þð3x þ 2xyÞ dy is not an exact differential. (b)Prove that e y=x G=x
2 2
is an exact differential of and find .(c)Find a solution of the differential equation ð2x þ xy 2y Þ dxþ
2
ð3x þ 2xyÞ dy ¼ 0.
2
2
Ans: ðbÞ ¼ e y=x ðx þ 2xyÞþ c; ðcÞ x þ 2xy þ ce y=x ¼ 0