Loading [MathJax]/jax/element/mml/optable/LetterlikeSymbols.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

15.3E: Conservative Vector Fields (Exercises)

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

1. True or False? If vector field F is conservative on the open and connected region D, then line integrals of F are path independent on D, regardless of the shape of D.

Answer
True

2. True or False? Function r(t)=a+t(ba), where 0t1, parameterizes the straight-line segment from a to b.

Answer
True

3. True or False? Vector field F(x,y,z)=(ysinz)ˆi+(xsinz)ˆj+(xycosz)ˆk is conservative.

Answer
True

4. True or False? Vector field F(x,y,z)=yˆi+(x+z)ˆjyˆk is conservative.

5. Justify the Fundamental Theorem of Line Integrals for CF·dr in the case when F(x,y)=(2x+2y)ˆi+(2x+2y)ˆj and C is a portion of the positively oriented circle x2+y2=25 from (5,0) to (3,4).

Answer
CF·dr=24 units of work

6. [T] Find CF·dr, where F(x,y)=(yexy+cosx)ˆi+(xexy+1y2+1)ˆj and C is a portion of curve y=sinx from x=0 to x=π2.

7. [T] Evaluate line integral CF·dr, where F(x,y)=(exsinyy)ˆi+(excosyx2)ˆj, and C is the path given by r(t)=(t3sinπt2)ˆi(π2cos(πt2+π2))ˆj for 0t1.

CNX_Calc_Figure_16_03_201.jpg

Answer
CF·dr=(e3π2) units of work

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

8. F(x,y)=2xy3ˆi+3y2x2ˆj

9. F(x,y)=(y+exsiny)ˆi+((x+2)excosy)ˆj

Answer
Not conservative

10. F(x,y)=(e2xsiny)ˆi+(e2xcosy)ˆj

11. F(x,y)=(6x+5y)ˆi+(5x+4y)ˆj

Answer
Conservative, f(x,y)=3x2+5xy+2y2+k

12. F(x,y)=(2xcos(y)ycos(x))ˆi+(x2sin(y)sin(x))ˆj

13. F(x,y)=(yex+sin(y))ˆi+(ex+xcos(y))ˆj

Answer
Conservative, f(x,y)=yex+xsin(y)+k

For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.

14. C(yˆi+xˆj)·dr, where C is any path from (0,0) to (2,4)

15. C(2ydx+2xdy), where C is the line segment from (0,0) to (4,4)

Answer
C(2ydx+2xdy)=32 units of work

16. [T] C[arctanyxxyx2+y2]dx+[x2x2+y2+ey(1y)]dy, where C is any smooth curve from (1,1) to (1,2).

17. Find the conservative vector field for the potential function f(x,y)=5x2+3xy+10y2.

Answer
F(x,y)=(10x+3y)ˆi+(3x+20y)ˆj

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

18. F(x,y)=(12xy)ˆi+6(x2+y2)ˆj

19. F(x,y)=(excosy)ˆi+6(exsiny)ˆj

Answer
F is not conservative.

20. F(x,y)=(2xyex2y)ˆi+6(x2ex2y)ˆj

21. F(x,y,z)=(yez)ˆi+(xez)ˆj+(xyez)ˆk

Answer
F is conservative and a potential function is f(x,y,z)=xyez+k.

22. F(x,y,z)=(siny)ˆi(xcosy)ˆj+ˆk

23. F(x,y,z)=1yˆixy2ˆj+(2z1)ˆk

Answer
F is conservative and a potential function is f(x,y,z)=xy+z2z+k.

24. F(x,y,z)=3z2ˆicosyˆj+2xzˆk

25. F(x,y,z)=(2xy)ˆi+(x2+2yz)ˆj+y2ˆk

Answer
F is conservative and a potential function is f(x,y,z)=x2y+y2z+k.

For the following exercises, determine whether the given vector field is conservative and find a potential function.

26. F(x,y)=(excosy)ˆi+6(exsiny)ˆj

27. F(x,y)=(2xyex2y)ˆi+(x2ex2y)ˆj

Answer
F is conservative and a potential function is f(x,y)=ex2y+k

For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals.

28. Evaluate Cf·dr, where f(x,y,z)=cos(πx)+sin(πy)xyz and C is any path that starts at (1,12,2) and ends at (2,1,1).

29. [T] Evaluate Cf·dr, where f(x,y)=xy+ex and C is a straight line from (0,0) to (2,1).

Answer
CF·dr=(e2+1) units of work

30. [T] Evaluate Cf·dr, where f(x,y)=x2yx and C is any path in a plane from (1, 2) to (3, 2).

31. Evaluate Cf·dr, where f(x,y,z)=xyz2yz and C has initial point (1,2,3) and terminal point (3,5,2).

Answer
CF·dr=38 units of work

For the following exercises, let F(x,y)=2xy2ˆi+(2yx2+2y)ˆj and G(x,y)=(y+x)ˆi+(yx)ˆj, and let C1 be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C2 be the curve consisting of a line segment from (0,0) to (1,1) followed by a line segment from (1,1) to (3,1).

Calc_15_3_E_32F.jpeg              CNX_Calc_Figure_16_03_203.jpg

32. Calculate the line integral of F over C1.

33. Calculate the line integral of G over C1.

Answer
C1G·dr=8π units of work

34. Calculate the line integral of F over C2.

35. Calculate the line integral of G over C2.

Answer
C2F·dr=7 units of work

36. [T] Let F(x,y,z)=x2ˆi+zsin(yz)ˆj+ysin(yz)ˆk. Calculate CF·dr, where C is a path from A=(0,0,1) to B=(3,1,2).

37. [T] Find line integral CF·dr of vector field F(x,y,z)=3x2zˆi+z2ˆj+(x3+2yz)ˆk along curve C parameterized by r(t)=(lntln2)ˆi+t3/2ˆj+tcos(πt),1t4.

Answer
CF·dr=159 units of work

For exercises 38 - 40, show that the following vector fields are conservative. Then calculate CF·dr for the given curve.

38. F(x,y)=(xy2+3x2y)ˆi+(x+y)x2ˆj; C is the curve consisting of line segments from (1,1) to (0,2) to (3,0).

39. F(x,y)=2xy2+1ˆi2y(x2+1)(y2+1)2ˆj; C is parameterized by x=t31,y=t6t, for 0t1.

Answer
CF·dr=1 units of work

40. [T] F(x,y)=[cos(xy2)xy2sin(xy2)]ˆi2x2ysin(xy2)ˆj; C is the curve et,et+1, for 1t0.

41. The mass of Earth is approximately 6×1027g and that of the Sun is 330,000 times as much. The gravitational constant is 6.7×108cm3/s2·g. The distance of Earth from the Sun is about 1.5×1012cm. Compute, approximately, the work necessary to increase the distance of Earth from the Sun by 1cm.

Answer
4×1031 erg

42. [T] Let F(x,y,z)=(exsiny)ˆi+(excosy)ˆj+z2ˆk. Evaluate the integral CF·dr, where r(t)=t,t3,et, for 0t1.

43. [T] Let C:[1,2]→ℝ^2 be given by x=e^{t−1},y=\sin\left(\frac{π}{t}\right). Use a computer to compute the integral \displaystyle \int _C\vecs F·d\vecs r=\int _C 2x\cos y\,dx−x^2\sin y\,dy, where \vecs{F}(x,y)=(2x\cos y)\,\mathbf{\hat i}−(x^2\sin y)\,\mathbf{\hat j}.

Answer
\displaystyle \int _C\vecs F·d\vecs s=0.4687 units of work

44. [T] Use a computer algebra system to find the mass of a wire that lies along the curve \vecs r(t)=(t^2−1)\,\mathbf{\hat j}+2t\,\mathbf{\hat k}, where 0≤t≤1, if the density is given by d(t) = \dfrac{3}{2}t.

45. Find the circulation and flux of field \vecs{F}(x,y)=−y\,\mathbf{\hat i}+x\,\mathbf{\hat j} around and across the closed semicircular path that consists of semicircular arch \vecs r_1(t)=(a\cos t)\,\mathbf{\hat i}+(a\sin t)\,\mathbf{\hat j},\quad 0≤t≤π, followed by line segment \vecs r_2(t)=t\,\mathbf{\hat i},\quad −a≤t≤a.

CNX_Calc_Figure_16_03_204.jpg

Answer
\text{circulation}=πa^2 and \text{flux}=0

46. Compute \displaystyle \int _C\cos x\cos y\,dx−\sin x\sin y\,dy, where \vecs r(t)=\langle t,t^2 \rangle, \quad 0≤t≤1.

47. Complete the proof of the theorem titled THE PATH INDEPENDENCE TEST FOR CONSERVATIVE FIELDS by showing that f_y=Q(x,y).

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 15.3E: Conservative Vector Fields (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by OpenStax.

Support Center

How can we help?