Stokes’s Theorem Applied to Path Independence

Stokes’s Theorem Applied to Path Independence

We emphasized in Sec. 10.2 that the value of a line integral generally depends not only on the function to be integrated and on the two endpoints A and B of the path of integration

C , but also on the particular choice of a path from A to B. In Theorem 3 of Sec. 10.2 we proved that if a line integral

F (F 1 dx ⫹ F 2 dy ⫹ F 3 dz )

(involving continuous F 1 ,F 2 ,F 3 that have continuous first partial derivatives) is path independent in a domain D, then

in D. And we claimed in Sec. 10.2 that, conversely, everywhere in D implies path independence of (9) in D provided D is simply connected. A proof of this needs Stokes’s theorem and can now be given as follows. Let C be any closed path in D. Since D is simply connected, we can find a surface S in D bounded by C. Stokes’s theorem applies and gives

冯 r 冮冮 C (curl F) • n dA C

(F 1 dx ⫹ F 2 dy ⫹ F 3 dz

F•r

for proper direction on C and normal vector n on S. Since in D, the surface integral and hence the line integral are zero. This and Theorem 2 of Sec. 10.2 imply that the integral (9) is path independent in D. This completes the proof.

PROBLEM SET 10.9

1–10 DIRECT INTEGRATION OF SURFACE

11. Stokes’s theorem not applicable. Evaluate F•r r ds INTEGRALS ,

(curl F) • n dA 冮冮 directly for

Evaluate the surface integral

F ⫽ (x 2 ⫹y 2 ) ⴚ1 [⫺y, x], C: x 2 ⫹y 2 ori- the given F and S.

ented clockwise. Why can Stokes’s theorem not be 1. F⫽ [z 2 , ⫺x 2 , 0], S the rectangle with vertices (0, 0, 0),

applied? What (false) result would it give? (1, 0, 0), (0, 4, 4), (1, 4, 4)

12. WRITING PROJECT. Grad, Div, Curl in

2. F⫽ [⫺13 sin y, 3 sinh z, x], S the rectangle with vertices Connection with Integrals. Make a list of ideas and (0, 0, 2), (4, 0, 2), (4, p >2, 2), (0, p >2, 2)

results on this topic in this chapter. See whether you

can rearrange or combine parts of your material. Then ⫺ 1 ⬉ x ⬉ 1, 0⬉y⬉1

3. F⫽ [e ⴚz ,e ⴚz cos y, e ⴚz sin y], S : z⫽y 2 >2,

subdivide the material into 3–5 portions and work out 4. F as in Prob. 1, z ⫽ xy (0 ⬉ x ⬉ 1,

the details of each portion. Include no proofs but simple Compare with Prob. 1.

0 ⬉ y ⬉ 4).

typical examples of your own that lead to a better 5. F⫽ [z 2 , 3 2 x , 0], S: 0 ⬉ x ⬉ a, 0 ⬉ y ⬉ a,

understanding of the material. z⫽ 1 6. F⫽ [y 3 , ⫺x 3 , 0],

S: x 2 ⫹y 2 ⬉ 1, z⫽0

13–20 EVALUATION OF F•r r ds

7. F⫽ [e y ,e z ,e x

], S: z ⫽ x 2 (0 ⬉ x ⬉ 2,

0 ⬉ y ⬉ 1) 2 2 2 2 2 8. Calculate this line integral by Stokes’s theorem for the F⫽ [z ,x ,y ], S: z ⫽ 2x ⫹y ,

given F and C. Assume the Cartesian coordinates to be 0, 0⬉z⬉h

right-handed and the z-component of the surface normal to 9. Verify Stokes’s theorem for F and S in Prob. 5.

be nonnegative.

10. Verify Stokes’s theorem for F and S in Prob. 6. 13. [⫺5y, 4x, z] , C the circle x 2 ⫹y 2 16,

Chapter 10 Review Questions and Problems

14. [z 3 ,x 3 ,y 3 y 2 ⫹z 2 9 18. [⫺y, 2z, 0] , C the boundary curve of y 2 ⫹z 2 4, 15. F⫽ [ y 2 ,x 2 , z ⫹ x] around the triangle with vertices

0, 0⬉x⬉h

(0, 0, 0), (1, 0, 0), (1, 1, 0) 19. F⫽ [z, e z , 0] , C the boundary curve of the portion of 2 16. 2 F⫽ [e y , 0, e x ], C as in Prob. 15 the cone ⫹y , x ⭌ 0, y ⭌ 0, 0 ⬉ z ⬉ 1

17. F⫽ [0, z 3 , 0] , C the boundary curve of the cylinder 20. [0, cos x, 0] , C the boundary curve of y 2 ⫹z 2 4, x 2 ⫹y 2

CHAPTER 10 REVIEW QUESTIONS AND PROBLEMS

1. State from memory how to evaluate a line integral. 17. [9z, 5x, 3y] , C the ellipse x 2 ⫹y 2 9, A surface integral.

z⫽x⫹ 2

2. What is path independence of a line integral? What is

18. F⫽ [sin p y , cos p x , sin p x ] , C the boundary curve of its physical meaning and importance?

0 ⬉ y ⬉ 2, z ⫽ x 3. What line integrals can be converted to surface

0 ⬉ x ⬉ 1,

19. F⫽ [z, 2y, x] , C the helix r⫽ [cos t, sin t, t] from integrals and conversely? How?

(1, 0, 0) to (1, 0, 2 p )

4. What surface integrals can be converted to volume 20. F⫽ [ze xz , 2 sinh 2y, xe xz ] , C the parabola y⫽x , integrals? How?

2 , ⫺1 ⬉ x ⬉ 1

5. What role did the gradient play in this chapter? The curl? State the definitions from memory.

6. What are typical applications of line integrals? Of

21–25 DOUBLE INTEGRALS,

surface integrals?

CENTER OF GRAVITY

7. Where did orientation of a surface play a role? Explain. Find the coordinates x ,y of the center of gravity of a 8. State the divergence theorem from memory. Give

mass of density f (x, y) in the region R. Sketch R, show applications.

details.

9. In some line and surface integrals we started from 21. , R the triangle with vertices (0, 0), (2, 0), (2, 2)

vector functions, but integrands were scalar functions. 2 22. ⫹y 2 ,

2 ⫹y 2 ⬉a R: x 2 ,

How and why did we proceed in this way? 2 R: ⫺1 ⬉ x ⬉ 2, x ⬉y⬉x⫹ 2. Why is 10. State Laplace’s equation. Explain its physical impor-

x⬎ 0 ? 24. 1,

tance. Summarize our discussion on harmonic functions. 4 R: 0 ⬉ y ⬉ 1 ⫺ x

k⬎0 , arbitrary, 0⬉y⬉1⫺x , 0⬉x⬉1

25. f ⫽ ky ,

11–20 LINE INTEGRALS (WORK INTEGRALS)

26. Why are and in Prob. 25 independent of k? x 冮 y

(r) • dr

Evaluate for F given F and C by the method that

C seems most suitable. Remember that if F is a force, the

冮冮 F • n dA.

integral gives the work done in the displacement along C.

27–35 SURFACE INTEGRALS

Show details.

11. F⫽ [2x 2 , ⫺4y 2 ], C the straight-line segment from

DIVERGENCE THEOREM

(4, 2) to (⫺6, 10) Evaluate the integral diectly or, if possible, by the divergence 12. [ y cos xy, x cos xy, e z ] , C the straight-line segment

theorem. Show details.

2 2 from to 2 ( p , 1, 0) ( 1

2 , p , 1) 27. [ax, by, cz], S the sphere x ⫹y ⫹z 36 2 13. 2 F⫽ [ y

28. [x ⫹ y ,y⫹z ,z⫹x 2 ], S the ellipsoid with 0⬉x⬉ p >2,

2 , 2xy ⫹ 5 sin x, 0], C the boundary of

0 ⬉ y ⬉ 2, z ⫽ 0 semi-axes of lengths a, b, c 14. 3 [⫺y 3 ,x 3 ⫹e ⴚy , 0] , C the circle 2 ⫹y x 2 25, 29. F⫽ [y ⫹ z, 20y, 2z ], S the surface of 0 ⬉ x ⬉ 2,

0 ⬉ y ⬉ 1, 0⬉z⬉y

2 2 15. 2 [x ,e ,e ] , C: x 9y 9, S: x ⫹y ⫹ 4z 4, 16. F⫽ [x 2 ,y 2 ,y 2 , C the helix

3 2 y ⴚ4z

2 ⫹ 2 2 30. F⫽ [1, 1, 1],

31. [e x ,e x y ] r⫽ [2 cos t, 2 sin t, 3t] F⫽ ,e z ], S the surface of the box ƒ x ƒ ⬉ 1, from to (2, 0, 0) (⫺2, 0, 3 p )

ƒ y ƒ ⬉ 1, ƒzƒ⬉1

CHAP. 10 Vector Integral Calculus. Integral Theorems 32. F⫽ [y 2 ,x 2 ,z 2 ], S the portion of the paraboloid

34. F⫽ [x, xy, z], S the boundary of x 2 ⫹y 2 ⬉ 1, z⫽x 2 ⫹y 2 , z⬉9

0⬉z⬉5

35. F⫽ [x ⫹ z, y ⫹ z, x ⫹ y], S the sphere of radius 3 ⫺ 2⬉v⬉2

33. F⫽ [y 2 ,x 2 ,z 2 ], S: r ⫽ [u, u 2 , v],

0 ⬉ u ⬉ 2,

with center 0

SUMMARY OF CHAPTER 10

Vector Integral Calculus. Integral Theorems

Chapter 9 extended differential calculus to vectors, that is, to vector functions v (x, y, z) or v (t) . Similarly, Chapter 10 extends integral calculus to vector functions. This involves line integrals (Sec. 10.1), double integrals (Sec. 10.3), surface integrals (Sec. 10.6), and triple integrals (Sec. 10.7) and the three “big” theorems for transforming these integrals into one another, the theorems of Green (Sec. 10.4), Gauss (Sec. 10.7), and Stokes (Sec. 10.9).

The analog of the definite integral of calculus is the line integral (Sec. 10.1)

F (r) • dr ⫽ (F dr 1 dx ⫹ F 2 dy ⫹ F 3 dz F (r 冮 dt

C 冮 a dt

(t)) •

where C : r (t), y (t), z (t)i ⫹ y (t)j ⫹ z (t)k (a ⬉ t ⬉ b) is a curve in space (or in the plane). Physically, (1) may represent the work done by a (variable) force in a displacement. Other kinds of line integrals and their applications are also discussed in Sec. 10.1.

Independence of path of a line integral in a domain D means that the integral of a given function over any path C with endpoints P and Q has the same value for all paths from P to Q that lie in D; here P and Q are fixed. An integral (1) is

independent of path in D if and only if the differential form F 1 dx ⫹ F 2 dy ⫹ F 3 dz with continuous F 1 , F 2 , F 3 is exact in D (Sec. 10.2). Also, if

, where [F 1 , F 2 , F 3 ] , has continuous first partial derivatives in a simply connected domain D, then the integral (1) is independent of path in D (Sec. 10.2). Integral Theorems . The formula of Green’s theorem in the plane (Sec. 10.4)

冮冮 0x 0y b

transforms double integrals over a region R in the xy-plane into line integrals over the boundary curve C of R and conversely. For other forms of (2) see Sec. 10.4.

Similarly, the formula of the divergence theorem of Gauss (Sec. 10.7) (3)

div F dV ⫽

F • n dA

Summary of Chapter 10

471

transforms triple integrals over a region T in space into surface integrals over the boundary surface S of T, and conversely. Formula (3) implies Green’s formulas

冮冮冮

2 0g

冮冮 0n

冮冮冮

冮冮 af 0n 0n b dA.

Finally, the formula of Stokes’s theorem (Sec. 10.9) (6)

冮冮

F•r (s) ds

transforms surface integrals over a surface S into line integrals over the boundary curve C of S and conversely.