Exercises:

For each vector field F,  find the flux of curl( F)  through the paraboloid

r( u,v) = á vcos( u) ,vsin( u) ,1-v2 ñ ,    u  in  [ 0,2p],  v  in  [ 0,1]

 first by direct evaluation and then by Stoke's theorem using the boundary curve T of the surface.

1. F( x,y,z) = á y,-x,z ñ 2. F( x,y,z) = á-y,-x,z ñ 3. F( x,y,z) = á x+y,x-y,z ñ 4. F( x,y,z) = á xy,yz,zx ñ

For each vector field F,  find the flux of curl( F)  through the upper unit hemisphere first by direct evaluation and then by Stoke's theorem using the boundary curve T of the surface.

5. F( x,y,z) = á y,-x,z ñ 6. F( x,y,z) = á-y,-x,z ñ 7. F( x,y,z) = á x+y,x-y,z ñ 8. F( x,y,z) = á xy,yz,zx ñ 9. F( x,y,z) =  x (x2+y2+z2) 3/2 ,  y ( x2+y2+z2)3/2 ,  z ( x2+y2+z2) 3/2 10. F( x,y,z) = á x,y,z ñ

Use Green's theorem to find the flux of F through the given region in the xy-plane.

11. F( x,y,z) = á y,-x,z ñ 12. F( x,y,z) = áy,-x,z ñ Region: y = 0,  y = x-x2,     Region: y = x, y = x2             x  in  [ 0,1]             x in [ 0,1] 13. F( x,y,z) = á y,-x,z ñ 14. F( x,y,z) = á x,y,z ñ Region: on and in unit circle Region: on and in unit circle

Use Stoke's theorem to evaluate the following line integrals using either the paraboloid in example 1 or the unit upper hemisphere in example 4.

15. T   ydx - xdy 16. T  y2dx - x2dy 17. T   xydx + xdy 18. C  ydx + xdy

 

Show that each of the following curves is contained in a plane by showing that r×v has constant direction. Then use (1) to find the area enclosed by the curve.

19. r( t) = á 2sin( t), 0, 2cos( t) ñ ,  t  in  [ 0,2p] 20. r( t) = á 0, 3sin( t), 2cos( t) ñ ,  t  in  [ 0,2p] 21. r( t) = á 6sin( t), 8sin( t), 10cos( t) ñ ,  t  in  [0,2p] 22. r( t) = á sin( t), sin( t), cos( t) ñ,  t  in  [ 0,2p] 23. r( t) = á cos( t), sin( t), sin( t) ñ ,  t  in  [ 0,2p] 24. r( t) = á cos(t), sin( t), cos( t) ñ,  t  in  [ 0,2p]

25. Use Stoke's theorem and an appropriate surface to evaluate ò_C ydx+xdz, where C is the curve parameterized by

r( t) = á cos( t) ,1,sin(t) ñ ,  t  in  [ 0,2p]

26. Use Stoke's theorem and an appropriate surface to evaluate ò_C xdx-zdy, where C is the curve parameterized by

r( t) = á cos( t) ,1,sin(t) ñ ,  t  in  [ 0,2p]

27. Use Stoke's theorem and an appropriate surface to evaluate ò_C ydx-zdy+xdz, where C is the curve parameterized by

r( t) = á cos( t) ,sin(t) ,1+sin( t) ñ ,  t  in  [ 0,2p]

28. Use Stoke's theorem and an appropriate surface to evaluate ò_C xyz  dx, where C is the curve parameterized by

r( t) = á cos( t) ,sin(t) ,1+sin( t) ñ ,  t  in  [ 0,2p]

29. Show that if F is sufficiently smooth on and inside a sphere centered at the origin, then

ó õ ó õ   Sphere curl( F) ·dS = 0

30. Show that the curl of

F( x,y,z) =  -y x2+y2 ,    x x2+y2 , z

is zero if ( x,y) ¹ ( 0,0) . Then show that

T   F ·dr

is not zero. Why does this not violate Stoke's theorem?   

31. Apply Stoke's theorem to F( x,y,z) = á bz-cy,cx-az,ay-bx ñ to show that if S is a region in a plane with normal N = á a,b,c ñ , then

Area  of  S =  ±1 2 a2+b2+c2   ¶S  ( bz-cy) dx+( cx-az) dy+( ay-bx) dz

32. Show that Green's theorem can also be written in the form

ó õ ó õ S ( Mx+Ny) dA =  ¶S  Mdy - Ndx

Then explain why if F( x,y) = á M(x,y) ,N( x,y) ñ , then

ó õ ó õ S div( F) dA =  ¶S  F · N  ds

where N is the unit normal vector to the curve ¶S.

Suppose that r( x,y,z) is the charge density of an electron cloud and that j( x,y,z) is the current density of the cloud (i.e., the number of ''moving charges'' per unit volume). Then the resulting Electric Field E( x,y,z,t)  and the Magnetic Field H( x,y,z,t) are related to these densities (and each other) by Maxwell's equations, which in empty space are

(1) div( E) = 4pr (2) div( H) = 0 (3) curl( E) =  -1 c  ¶H ¶t (4) curl( H) =  4p c j-  1 c  ¶E ¶t

where c is the speed of light in a vacuum. Maxwell's equations are at the heart of the study of electricity and magnetism, and Stoke's theorem is often used to study Maxwell's equations.

33. Use Stoke's theorem to show that if S is a surface, then

ó õ ó õ S    -1 c  ¶H ¶t  · dS = ó (ç) õ ¶S  E·dr

(Note: This yields the important idea that

-1 c  ¶ ¶t ( Flux  of  H  through  S) =   Circulation  of  E  around  ¶S

34. What type of result can be obtained by applying Stoke's theorem to Maxwell's equation (4)?

35. It can be shown that div( H) = 0 implies that there exists a vector field A such that

H = curl( A)

Use Stoke's theorem to show that

ó õ ó õ S  H·dS =  ¶S   A · dr

36. Show that if r( x,y) is the charge density of a 2-dimensional electric field E, then

ó õ ó õ S r( x,y) dA =  1 4p ¶S  E·N  ds