AWL/Thomas_ch16p1143-1228 4100 9/17/04 1:50 PM Page 1209
16.7 Stokes Theorem
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EXERCISES 16.7
Using Stokes Theorem to Calculate Circulation
In Exercises 16, use the surface integral in Stokes Theorem to calculate the circulation of the field F around the curve C in the indicated direction. 1. F = x 2i + 2xj + z 2k C: The ellipse 4x 2 + y 2 = 4 in the xy-plane, counterclockwise when viewed from above 2. F = 2yi + 3xj - z 2k C: The circle x 2 + y 2 = 9 in the xy-plane, counterclockwise when viewed from above 3. F = yi + xzj + x 2k C: The boundary of the triangle cut from the plane x + y + z = 1 by the first octant, counterclockwise when viewed from above 4. F = s y 2 + z 2 di + s x 2 + z 2 dj + s x 2 + y 2 dk C: The boundary of the triangle cut from the plane x + y + z = 1 by the first octant, counterclockwise when viewed from above 5. F = s y 2 + z 2 di + s x 2 + y 2 dj + s x 2 + y 2 dk C: The square bounded by the lines x = ; 1 and y = ; 1 in the xy-plane, counterclockwise when viewed from above and let 6. F = x 2y 3i + j + zk C: The intersection of the cylinder x 2 + y 2 = 4 and the hemisphere x 2 + y 2 + z 2 = 16, z 0, counterclockwise when viewed from above.
Flux of the Curl
7. Let n be the outer unit normal of the elliptical shell S: F = yi + x 2j + s x 2 + y 4 d3>2 sin e 2xyz k. 4x 2 + 9y 2 + 36z 2 = 36, z 0,
Find the value of 6
S
* F # n ds.
(Hint: One parametrization of the ellipse at the base of the shell is x = 3 cos t, y = 2 sin t, 0 t 2p.) 8. Let n be the outer unit normal (normal away from the origin) of the parabolic shell S: 4x 2 + y + z 2 = 4, y 0,
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and let
Chapter 16: Integration in Vector Fields
17. F = 3yi + s 5 - 2x dj + s z 2 - 2 dk 1 1 b i + s tan-1 y dj + ax + b k. 2+x 4+z
F = a-z +
Find the value of 6
S
18. F = y2i + z2j + xk * F # n ds.
S: rs f, u d = A 23 sin f cos u B i + A 23 sin f sin u B j + A 23 cos f B k, 0 f p> 2, 0 u 2p S: rs f, u d = s 2 sin f cos u di + s 2 sin f sin u dj + s 2 cos f dk, 0 f p> 2, 0 u 2p
9. Let S be the cylinder x 2 + y 2 = a 2, 0 z h, together with its top, x 2 + y 2 a 2, z = h. Let F = - yi + xj + x 2k. Use Stokes Theorem to find the flux of * F outward through S. 10. Evaluate 6
S
Theory and Examples
19. Zero circulation Use the identity * = 0 (Equation (8) in the text) and Stokes Theorem to show that the circulations of the following fields around the boundary of any smooth orientable surface in space are zero. a. F = 2xi + 2yj + 2zk b. F = s xy 2z 3 d c. F = * s xi + yj + zk d 20. Zero circulation Let s x, y, z d = s x 2 + y 2 + z 2 d-1>2. Show that the clockwise circulation of the field F = around the circle x 2 + y 2 = a 2 in the xy-plane is zero by a. taking r = s a cos t di + s a sin t dj, 0 t 2p, and integrating F # dr over the circle. b. by applying Stokes Theorem. 21. Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2 , oriented as shown here. Show that F 2y dx + 3z dy - x dz d. F =
* s yi d # n ds,
where S is the hemisphere x 2 + y 2 + z 2 = 1, z 0. 11. Flux of curl F Show that 6
S
* F # n ds
has the same value for all oriented surfaces S that span C and that induce the same positive direction on C. 12. Let F be a differentiable vector field defined on a region containing a smooth closed oriented surface S and its interior. Let n be the unit normal vector field on S. Suppose that S is the union of two surfaces S1 and S2 joined along a smooth simple closed curve C. Can anything be said about 6
S
* F # n ds?
C
z 2 2x C 2y z 2
Give reasons for your answer.
Stokes Theorem for Parametrized Surfaces
In Exercises 1318, use the surface integral in Stokes Theorem to calculate the flux of the curl of the field F across the surface S in the direction of the outward unit normal n. 13. F = 2zi + 3xj + 5yk S: rs r, u d = s r cos u di + s r sin u dj + s 4 - r 2 dk, 0 r 2, 0 u 2p 14. F = s y - z di + s z - x dj + s x + z dk S: rs r, u d = s r cos u di + s r sin u dj + s 9 - r 2 dk, 0 r 3, 0 u 2p 15. F = x2yi + 2y3zj + 3zk S: rs r, u d = s r cos u di + s r sin u dj + rk, 0 r 1, 0 u 2p 16. F = s x - y di + s y - z dj + s z - x dk S: rs r, u d = s r cos u di + s r sin u dj + s 5 - r dk, 0 r 5, 0 u 2p
x O 1 a1
y
depends only on the area of the region enclosed by C and not on the position or shape of C. 22. Show that if F = xi + yj + zk, then * F = 0. 23. Find a vector field with twice-differentiable components whose curl is xi + yj + zk or prove that no such field exists. 24. Does Stokes Theorem say anything special about circulation in a field whose curl is zero? Give reasons for your answer. 25. Let R be a region in the xy-plane that is bounded by a piecewisesmooth simple closed curve C and suppose that the moments of
Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4100 AWL/Thomas_ch16p1143-1228
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7:27 AM
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16.7 Stokes Theorem inertia of R about the x- and y-axes are known to be Ix and Iy. Evaluate the integral where r = 2x 2 + y 2, in terms of Ix and Iy.
C
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is zero but that F F # dr
F
s r 4 d # n ds,
C
26. Zero curl, yet field not conservative Show that the curl of F= -y x2 + y2 i+ x j + zk x2 + y2
is not zero if C is the circle x 2 + y 2 = 1 in the xy-plane. (Theorem 6 does not apply here because the domain of F is not simply connected. The field F is not defined along the z-axis so there is no way to contract C to a point without leaving the domain of F.)
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