9.3E: EXERCISES

Exercise \(\PageIndex{1}\): True or False?

1. 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.

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

3. The vector field \(\vecs{F}(x,y,z)=(y sinz) \, \hat{ \mathbf i}+(x sinz) \, \hat{ \mathbf j}+(xy cosz) \, \hat{ \mathbf k}\) is conservative.

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

Answer

1. T, 3. T

Exercise \(\PageIndex{2}\): Line integral over vector field

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

Answer

\(\int _C \vecs{F} · \vecs{dr}=24\)

6. [T] Find \(\int _C \vecs{F} · \vecs{dr},\) where \( \vecs{F}(x,y)=(ye^{xy}+ \ cos(x)) \, \hat{ \mathbf i} + (xe^{xy}+\dfrac{1}{y^2+1}) \, \hat{ \mathbf j} \) and \(C\) is a portion of curve

\(y=sinx\) from \(x=0\) to \(x=\dfrac{π}{2}\).

7. [T] Evaluate line integral \(\int _C \vecs{F} · \vecs{dr} \), where \(\vecs{F}(x,y)=(e^xsiny−y) \, \hat{ \mathbf i}+(e^xcosy−x−2)\, \hat{ \mathbf j} \), and \(C\) is the path given by

\(r(t)=(t^3sin\dfrac{πt}{2}) \, \hat{ \mathbf i}−(\dfrac{π}{2}cos(\dfrac{πt}{2}+\dfrac{π}{2})) \, \hat{ \mathbf j}\) for \(0≤t≤1\).

Answer

\(\int _C \vecs{F} · \vecs{dr}=e−\dfrac{3π}{2}\)

Exercise \(\PageIndex{3}\)

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

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

9. \(\vecs{F}(x,y)=(−y+e^xsiny) \, \hat{ \mathbf i} +[(x+2)e^xcosy] \, \hat{ \mathbf j} \)

Answer

Not conservative

10. \(\vecs{F}(x,y)=(e^{2x}siny) \, \hat{ \mathbf i} +[e^{2x}cosy] \, \hat{ \mathbf j} \)

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

Answer

Conservative, \(\vecs{F}(x,y)=3x^2+5xy+2y^2\)

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

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

Answer

Conservative, \(\vecs{F}(x,y)=ye^x+xsin(y)\)

Exercise \(\PageIndex{4}\)

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

14. \(∮_C(y \, \hat{ \mathbf i} +x \, \hat{ \mathbf j} )·dr,\) 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

\(∮_C(2ydx+2xdy)=32\)

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

17. Find the conservative vector field for the potential function

\(\vecs{F}(x,y)=5x^2+3xy+10y^2.\)

Answer

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

Exercise \(\PageIndex{5}\)

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

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

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

Answer

F is not conservative.

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

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

Answer

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

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

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

Answer

F is conservative and a potential function is \(f(x,y,z)=z.\)

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

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

Answer

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

Exercise \(\PageIndex{6}\)

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

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

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

Answer

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

Exercise \(\PageIndex{7}\)

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

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

Solution: \(\int _C \vecs{F} · \vecs{dr}=e^2+1\)

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

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

Answer

\(\int _C \vecs{F} · \vecs{dr}=41\)

Exercise \(\PageIndex{8}\)

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

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

Solution: \(\int _C \vecs{F} · \vecs{dr}=e^2+1\)

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

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

Answer

\(\int _C \vecs{F} · \vecs{dr}=41\)

Exercise \(\PageIndex{9}\)

For the following exercises, let \(\vecs{F}(x,y)=2xy^2 \, \hat{ \mathbf i} +(2yx^2+2y)\, \hat{ \mathbf j}\) and \(G(x,y)=(y+x)\, \hat{ \mathbf i}+(y−x)\, \hat{ \mathbf j}\), 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 F over \(C_1\).

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

Solution: \(∮_{C_1}\bf{G}· \bf{dr}=−8π\)

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

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

Solution: \(∮_{C_2}F·dr=7\)

36. [T] Let \(F(x,y,z)=x^2 \, \hat{ \mathbf i} +zsin(yz)j+ysin(yz)k\). 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 \(F(x,y,z)=3x^2z \, \hat{ \mathbf i} +z^2\, \hat{ \mathbf j}+(x^3+2yz)\, \hat{ \mathbf k}\) along curve \(C\) parameterized by \(r(t)=(\dfrac{lnt}{ln2}) \, \hat{ \mathbf i} +t^{3/2}\, \hat{ \mathbf j}+tcos(πt) \, \hat{ \mathbf k},1≤t≤4.\)

Solution: \(\int _C \vecs{F} · \vecs{dr}=150\)

Answer

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Exercise \(\PageIndex{10}\)

For the following exercises, show that the following vector fields are conservative by using a computer. Calculate \(\int _C \vecs{F} · \vecs{dr}\) for the given curve.

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

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

Solution: \(\int _C \vecs{F} · \vecs{dr}=−1\)

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

Answer

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Exercise \(\PageIndex{11}\)

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 1cm.

Solution: \(4×10^{31}erg\)

42. [T] Let \(\vecs{F}=(x,y,z)=(e^xsiny)\, \hat{ \mathbf i}+(e^xcosy)\, \hat{ \mathbf j}+z^2\, \hat{ \mathbf k}\). Evaluate the integral \(\int _C \vecs{F}·\vecs{ds}\),where \(C(t)=(\sqrt{t},t^3,e^{\sqrt{t}}),0≤t≤1.\)

43. [T] Let \(c:[1,2]→ℝ^2\) be given by \(x=e^{t−1},y=sin(\dfrac{π}{t})\). Use a computer to compute the integral \(\int _CF·ds=\int _C2xcosydx−x^2sinydy\), where \(\vecs{F}=(2xcosy)i−(x^2siny)j.\)

Solution: \(\int _CF·ds=0.4687\)

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

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

Solution: \(circulation=πa^2\) and \(flux=0\)

46. Compute \(\int _C cosxcosydx−sinxsinydy,\) where \(C(t)=(t,t^2),0≤t≤1.\)

47. Complete the proof of Note by showing that \(f_y=Q(x,y).\)

Answer

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