15.3E: Conservative Vector Fields (Exercises)

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

Answer:

True

2. True or False? Function \(\vecs r(t)=\vecs a+t(\vecs b−\vecs a)\), where \(0≤t≤1\), parameterizes the straight-line segment from \(\vecs a\) to \(\vecs b\).

Answer:

True

3. True or False? Vector field \(\vecs F(x,y,z)=(y\sin z)\,\mathbf{\hat i}+(x\sin z)\,\mathbf{\hat j}+(xy\cos z)\,\mathbf{\hat k}\) is conservative.

Answer:

True

4. True or False? Vector field \(\vecs F(x,y,z)=y\,\mathbf{\hat i}+(x+z)\,\mathbf{\hat j}−y\,\mathbf{\hat k}\) is conservative.

5. Justify the Fundamental Theorem of Line Integrals for \(\displaystyle \int _C\vecs F·d\vecs r\) in the case when \(\vecs{F}(x,y)=(2x+2y)\,\mathbf{\hat i}+(2x+2y)\,\mathbf{\hat j}\) and \(C\) is a portion of the positively oriented circle \(x^2+y^2=25\) from \((5, 0)\) to \((3, 4).\)

Answer:

\(\displaystyle \int _C \vecs F·d\vecs r=24\) units of work

6. [T] Find \(\displaystyle \int _C\vecs F·d\vecs r,\) where \(\vecs{F}(x,y)=(ye^{xy}+\cos x)\,\mathbf{\hat i}+\left(xe^{xy}+\frac{1}{y^2+1}\right)\,\mathbf{\hat j}\) and \(C\) is a portion of curve \(y=\sin x\) from \(x=0\) to \(x=\frac{π}{2}\).

7. [T] Evaluate line integral \(\displaystyle \int _C\vecs F·d\vecs r\), where \(\vecs{F}(x,y)=(e^x\sin y−y)\,\mathbf{\hat i}+(e^x\cos y−x−2)\,\mathbf{\hat j}\), and \(C\) is the path given by \(\vecs r(t)=(t^3\sin\frac{πt}{2})\,\mathbf{\hat i}−(\frac{π}{2}\cos(\frac{πt}{2}+\frac{π}{2}))\,\mathbf{\hat j}\) for \(0≤t≤1\).

Answer:

\(\displaystyle \int _C\vecs F·d\vecs r=\left(e−\frac{3π}{2}\right)\) units of work

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

8. \(\vecs{F}(x,y)=2xy^3\,\mathbf{\hat i}+3y^2x^2\,\mathbf{\hat j}\)

9. \(\vecs{F}(x,y)=(−y+e^x\sin y)\,\mathbf{\hat i}+((x+2)e^x\cos y)\,\mathbf{\hat j}\)

Answer:

Not conservative

10. \(\vecs{F}(x,y)=(e^{2x}\sin y)\,\mathbf{\hat i}+(e^{2x}\cos y)\,\mathbf{\hat j}\)

11. \(\vecs{F}(x,y)=(6x+5y)\,\mathbf{\hat i}+(5x+4y)\,\mathbf{\hat j}\)

Answer:

Conservative, \(f(x,y)=3x^2+5xy+2y^2+k\)

12. \(\vecs{F}(x,y)=(2x\cos(y)−y\cos(x))\,\mathbf{\hat i}+(−x^2\sin(y)−\sin(x))\,\mathbf{\hat j}\)

13. \(\vecs{F}(x,y)=(ye^x+\sin(y))\,\mathbf{\hat i}+(e^x+x\cos(y))\,\mathbf{\hat j}\)

Answer:

Conservative, \(f(x,y)=ye^x+x\sin(y)+k\)

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

14. \(\displaystyle ∮_C(y\,\mathbf{\hat i}+x\,\mathbf{\hat j})·d\vecs r,\) where \(C\) is any path from \((0, 0)\) to \((2, 4)\)

15. \(\displaystyle ∮_C(2y\,dx+2x\,dy),\) where \(C\) is the line segment from \((0, 0)\) to \((4, 4)\)

Answer:

\(\displaystyle ∮_C(2y\,dx+2x\,dy)=32\) units of work

16. [T] \(\displaystyle ∮_C\left[\arctan\dfrac{y}{x}−\dfrac{xy}{x^2+y^2}\right]\,dx+\left[\dfrac{x^2}{x^2+y^2}+e^{−y}(1−y)\right]\,dy\), where \(C\) is any smooth curve from \((1, 1)\) to \((−1,2).\)

17. Find the conservative vector field for the potential function \(f(x,y)=5x^2+3xy+10y^2.\)

Answer:

\(\vecs{F}(x,y)=(10x+3y)\,\mathbf{\hat i}+(3x+20y)\,\mathbf{\hat j}\)

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

18. \(\vecs{F}(x,y)=(12xy)\,\mathbf{\hat i}+6(x^2+y^2)\,\mathbf{\hat j}\)

19. \(\vecs{F}(x,y)=(e^x\cos y)\,\mathbf{\hat i}+6(e^x\sin y)\,\mathbf{\hat j}\)

Answer:

\(\vecs F\) is not conservative.

20. \(\vecs{F}(x,y)=(2xye^{x^2y})\,\mathbf{\hat i}+6(x^2e^{x^2y})\,\mathbf{\hat j}\)

21. \(\vecs F(x,y,z)=(ye^z)\,\mathbf{\hat i}+(xe^z)\,\mathbf{\hat j}+(xye^z)\,\mathbf{\hat k}\)

Answer:

\(\vecs F\) is conservative and a potential function is \(f(x,y,z)=xye^z+k\).

22. \(\vecs F(x,y,z)=(\sin y)\,\mathbf{\hat i}−(x\cos y)\,\mathbf{\hat j}+\,\mathbf{\hat k}\)

23. \(\vecs F(x,y,z)=\dfrac{1}{y}\,\mathbf{\hat i}-\dfrac{x}{y^2}\,\mathbf{\hat j}+(2z−1)\,\mathbf{\hat k}\)

Answer:

\(\vecs F\) is conservative and a potential function is \(f(x,y,z)=\dfrac{x}{y}+z^2-z+k.\)

24. \(\vecs F(x,y,z)=3z^2\,\mathbf{\hat i}−\cos y\,\mathbf{\hat j}+2xz\,\mathbf{\hat k}\)

25. \(\vecs F(x,y,z)=(2xy)\,\mathbf{\hat i}+(x^2+2yz)\,\mathbf{\hat j}+y^2\,\mathbf{\hat k}\)

Answer:

\(\vecs F\) is conservative and a potential function is \(f(x,y,z)=x^2y+y^2z+k.\)

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

26. \(\vecs{F}(x,y)=(e^x\cos y)\,\mathbf{\hat i}+6(e^x\sin y)\,\mathbf{\hat j}\)

27. \(\vecs{F}(x,y)=(2xye^{x^2y})\,\mathbf{\hat i}+6(x^2e^{x^2y})\,\mathbf{\hat j}\)

Answer:

\(\vecs F\) is conservative and a potential function is \(f(x,y)=e^{x^2y}+k\)

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

28. Evaluate \(\displaystyle \int _C\vecs ∇f·d\vecs r\), 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 \(\displaystyle \int _C\vecs ∇f·d\vecs r\), where \(f(x,y)=xy+e^x\) and \(C\) is a straight line from \((0,0)\) to \((2,1)\).

Answer:

\(\displaystyle \int _C\vecs F·d\vecs r=\left(e^2+1\right)\) units of work

30. [T] Evaluate \(\displaystyle \int _C\vecs ∇f·d\vecs r,\) where \(f(x,y)=x^2y−x\) and \(C\) is any path in a plane from (1, 2) to (3, 2).

31. Evaluate \(\displaystyle \int _C\vecs ∇f·d\vecs r,\) where \(f(x,y,z)=xyz^2−yz\) and \(C\) has initial point \((1, 2, 3)\) and terminal point \((3, 5, 2).\)

Answer:

\(\displaystyle \int _C\vecs F·d\vecs r=38\) units of work

For the following exercises, let \(\vecs{F}(x,y)=2xy^2\,\mathbf{\hat i}+(2yx^2+2y)\,\mathbf{\hat j}\) and \(G(x,y)=(y+x)\,\mathbf{\hat i}+(y−x)\,\mathbf{\hat j}\), and let \(C_1\) be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and\(C_2\) be the curve consisting of a line segment from \((0, 0)\) to \((1, 1)\) followed by a line segment from \((1, 1)\) to \((3, 1).\)

32. Calculate the line integral of \(\vecs F\) over \(C_1\).

33. Calculate the line integral of \(\vecs G\) over \(C_1\).

Answer:

\(\displaystyle ∮_{C_1}\vecs G·d\vecs r=−8π\) units of work

34. Calculate the line integral of \(\vecs F\) over \(C_2\).

35. Calculate the line integral of \(\vecs G\) over \(C_2\).

Answer:

\(\displaystyle ∮_{C_2}\vecs F·d\vecs r=7\) units of work

36. [T] Let \(\vecs F(x,y,z)=x^2\,\mathbf{\hat i}+z\sin(yz)\,\mathbf{\hat j}+y\sin(yz)\,\mathbf{\hat k}\). Calculate \(\displaystyle ∮_C\vecs F·d\vecs{r}\), where \(C\) is a path from \(A=(0,0,1)\) to \(B=(3,1,2)\).

37. [T] Find line integral \(\displaystyle ∮_C\vecs F·dr\) of vector field \(\vecs F(x,y,z)=3x^2z\,\mathbf{\hat i}+z^2\,\mathbf{\hat j}+(x^3+2yz)\,\mathbf{\hat k}\) along curve \(C\) parameterized by \(\vecs r(t)=(\frac{\ln t}{\ln 2})\,\mathbf{\hat i}+t^{3/2}\,\mathbf{\hat j}+t\cos(πt),1≤t≤4.\)

Answer:

\(\displaystyle \int _C\vecs F·d\vecs r=150\) units of work

For exercises 38 - 40, show that the following vector fields are conservative.  Then calculate \(\displaystyle \int _C\vecs F·d\vecs r\) for the given curve.

38. \(\vecs{F}(x,y)=(xy^2+3x^2y)\,\mathbf{\hat i}+(x+y)x^2\,\mathbf{\hat j}\); \(C\) is the curve consisting of line segments from \((1,1)\) to \((0,2)\) to \((3,0).\)

39. \(\vecs{F}(x,y)=\dfrac{2x}{y^2+1}\,\mathbf{\hat i}−\dfrac{2y(x^2+1)}{(y^2+1)^2}\,\mathbf{\hat j}\); \(C\) is parameterized by \(x=t^3−1,\;y=t^6−t\), for \(0≤t≤1.\)

Answer:

\(\displaystyle \int _C\vecs F·d\vecs r=−1\) units of work

40. [T] \(\vecs{F}(x,y)=[\cos(xy^2)−xy^2\sin(xy^2)]\,\mathbf{\hat i}−2x^2y\sin(xy^2)\,\mathbf{\hat j}\); \(C\) is the curve \(\langle e^t,e^{t+1}\rangle,\) for \(−1≤t≤0\).

 

41. The mass of Earth is approximately \(6×10^{27}g\) and that of the Sun is 330,000 times as much. The gravitational constant is \(6.7×10^{−8}cm^3/s^2·g\). The distance of Earth from the Sun is about \(1.5×10^{12}cm\). Compute, approximately, the work necessary to increase the distance of Earth from the Sun by \(1\;cm\).

Answer:

\(4×10^{31}\) erg

42. [T] Let \(\vecs{F}(x,y,z)=(e^x\sin y)\,\mathbf{\hat i}+(e^x\cos y)\,\mathbf{\hat j}+z^2\,\mathbf{\hat k}\). Evaluate the integral \(\displaystyle \int _C\vecs F·d\vecs r\), where \(\vecs r(t)=\langle\sqrt{t},t^3,e^{\sqrt{t}}\rangle,\) for \(0≤t≤1.\)

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 \(\vecs r_2(t)=t\,\mathbf{\hat i},\quad −a≤t≤a.\)

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.