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15.5E: Divergence and Curl (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

For the following exercises, determine whether the statement is True or False.

1. If the coordinate functions of F:R3R3 have continuous second partial derivatives, then curl(divF) equals zero.

2. (xˆi+yˆj+zˆk)=1.

Answer
False

3. All vector fields of the form F(x,y,z)=f(x)ˆi+g(y)ˆj+h(z)ˆk are conservative.

4. If curlF=0, then F is conservative.

Answer
True

5. If F is a constant vector field then divF=0.

6. If F is a constant vector field then curlF=0.

Answer
True

For the following exercises, find the curl of F.

7. F(x,y,z)=xy2z4ˆi+(2x2y+z)ˆj+y3z2ˆk

8. F(x,y,z)=x2zˆi+y2xˆj+(y+2z)ˆk

Answer
curlF=ˆi+x2ˆj+y2ˆk

9. F(x,y,z)=3xyz2ˆi+y2sinzˆj+xe2zˆk

10. F(x,y,z)=x2yzˆi+xy2zˆj+xyz2ˆk

Answer
curlF=(xz2xy2)ˆi+(x2yyz2)ˆj+(y2zx2z)ˆk

11. F(x,y,z)=(xcosy)ˆi+xy2ˆj

12. F(x,y,z)=(xy)ˆi+(yz)ˆj+(zx)ˆk

Answer
curl F=ˆi+ˆj+ˆk

13. F(x,y,z)=xyzˆi+x2y2z2ˆj+y2z3ˆk

14. F(x,y,z)=xyˆi+yzˆj+xzˆk

Answer
curl F=yˆizˆjxˆk

15. F(x,y,z)=x2ˆi+y2ˆj+z2ˆk

16. F(x,y,z)=axˆi+byˆj+cˆk for constants a,b,c.

Answer
curl F=0

For the following exercises, find the divergence of F.

17. F(x,y,z)=x2zˆi+y2xˆj+(y+2z)ˆk

18. F(x,y,z)=3xyz2ˆi+y2sinzˆj+xe2zˆk

Answer
divF=3yz2+2ysinz+2xe2z

19. F(x,y)=(sinx)ˆi+(cosy)ˆj

20. F(x,y,z)=x2ˆi+y2ˆj+z2ˆk

Answer
divF=2(x+y+z)

21. F(x,y,z)=(xy)ˆi+(yz)ˆj+(zx)ˆk

22. F(x,y)=xx2+y2ˆi+yx2+y2ˆj

Answer
divF=1x2+y2

23. F(x,y)=xˆiyˆj

24. F(x,y,z)=axˆi+byˆj+cˆk for constants a,b,c.

Answer
divF=a+b

25. F(x,y,z)=xyzˆi+x2y2z2ˆj+y2z3ˆk

26. F(x,y,z)=xyˆi+yzˆj+xzˆk

Answer
divF=x+y+z

For exercises 27 & 28, determine whether each of the given scalar functions is harmonic.

27. u(x,y,z)=ex(cosysiny)

28. w(x,y,z)=(x2+y2+z2)1/2

Answer
Harmonic

29. If F(x,y,z)=2ˆi+2xj+3yk and G(x,y,z)=xˆiyˆj+zˆk, find curl(F×G).

30. If F(x,y,z)=2ˆi+2xj+3yk and G(x,y,z)=xˆiyˆj+zˆk, find div(F×G).

Answer
div(F×G)=2z+3x

31. Find divF, given that F=f, where f(x,y,z)=xy3z2.

32. Find the divergence of F for vector field F(x,y,z)=(y2+z2)(x+y)ˆi+(z2+x2)(y+z)ˆj+(x2+y2)(z+x)ˆk.

Answer
divF=2r2

33. Find the divergence of F for vector field F(x,y,z)=f1(y,z)ˆi+f2(x,z)ˆj+f3(x,y)ˆk.

For exercises 34 - 36, use r=|r| and r(x,y,z)=x,y,z.

34. Find the curlr

Answer
curlr=0

35. Find the curlrr.

36. Find the curlrr3.

Answer
curlrr3=0

37. Let F(x,y)=yˆi+xˆjx2+y2, where F is defined on {(x,y)R|(x,y)(0,0)}. Find curlF.

For the following exercises, use a computer algebra system to find the curl of the given vector fields.

38. [T] F(x,y,z)=arctan(xy)ˆi+lnx2+y2ˆj+ˆk

Answer
curl F=2xx2+y2ˆk

39. [T] F(x,y,z)=sin(xy)ˆi+sin(yz)ˆj+sin(zx)ˆk

For the following exercises, find the divergence of F at the given point.

40. F(x,y,z)=ˆi+ˆj+ˆk at (2,1,3)

Answer
divF=0

41. F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,3)

42. F(x,y,z)=exyˆi+exzˆj+eyzˆk at (3,2,0)

Answer
divF=22e6

43. F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,1)

44. F(x,y,z)=exsinyˆiexcosyˆj at (0,0,3)

Answer
divF=0

For exercises 45- 49, find the curl of F at the given point.

45. F(x,y,z)=ˆi+ˆj+ˆk at (2,1,3)

46. F(x,y,z)=xyzˆi+yˆj+xˆk at (1,2,3)

Answer
curl F=ˆj3ˆk

47. F(x,y,z)=exyˆi+exzˆj+eyzˆk at (3,2,0)

48. F(x,y,z)=xyzˆi+yˆj+zˆk at (1,2,1)

Answer
curl F=2ˆjˆk

49. F(x,y,z)=exsinyˆiexcosyˆj at (0,0,3)

50. Let F(x,y,z)=(3x2y+az)ˆi+x3ˆj+(3x+3z2)ˆk. For what value of a is F conservative?

Answer
a=3

51. Given vector field F(x,y)=1x2+y2y,x on domain D=R2{(0,0)}={(x,y)R2|(x,y)(0,0)}, is F conservative?

52. Given vector field F(x,y)=1x2+y2x,y on domain D=R2{(0,0)}, is F conservative?

Answer
F is conservative.

53. Find the work done by force field F(x,y)=eyˆixeyˆj in moving an object fromP(0,1) to Q(2,0). Is the force field conservative?

54. Compute divergence F(x,y,z)=(sinhx)ˆi+(coshy)ˆjxyzˆk.

Answer
divF=coshx+sinhyxy

55. Compute curl F=(sinhx)ˆi+(coshy)ˆjxyzˆk.

For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity ω=a,b,c. If P is a point in the body located at r=xˆi+yˆj+zˆk, the velocity at P is given by vector field F=ω×r.

A three dimensional diagram of an object rotating about the x axis in a counterclockwise manner with constant angular velocity w = <a,b,c>. The object is roughly a sphere with pointed ends on the x axis, which cuts it in half. An arrow r is drawn from (0,0,0) to P(x,y,z) and down from P(x,y,z) to the x axis.

56. Express F in terms of ˆi,ˆj, and ˆk vectors.

Answer
F=(bzcy)ˆi+(cxaz)ˆj+(aybx)ˆk

57. Find divF.

58. Find curlF

Answer
curl F=2ω

In the following exercises, suppose that F=0 and G=0.

59. Does F+G necessarily have zero divergence?

60. Does F×G necessarily have zero divergence?

Answer
F×G does not have zero divergence.

In the following exercises, suppose a solid object in R3 has a temperature distribution given by T(x,y,z). The heat flow vector field in the object is F=kT, where k>0 is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is F=kT=k2T.

61. Compute the heat flow vector field.

62. Compute the divergence.

Answer
F=200k[1+2(x2+y2+z2)]ex2+y2+z2

63. [T] Consider rotational velocity field v=0,10z,10y. If a paddlewheel is placed in plane x+y+z=1 with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.

A three dimensional diagram of a rotational velocity field. The arrows are showing a rotation in a clockwise manner. A paddlewheel is shown in plan x + y + z = 1 with n extended out perpendicular to the plane.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

 


This page titled 15.5E: Divergence and Curl (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by OpenStax.

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