9.3E: EXERCISES

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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 $0≤t≤1$.

$A vector field in three dimensions. The arrows are roughly the same length and all point up into the z-plane. A curve is drawn seemingly parallel to the (x,y)-plane. In the (x,y)-plane, it would look like a decreasing concave down curve in quadrant 1.$

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).

$A vector fields in two dimensions is shown. It has short arrows close to the origin. Longer arrows are in the upper right corner of quadrant 1 and somewhat in the bottom right of quadrant 4, upper left of quadrant 2, and lower left of quadrant 3. The arrows all point away from the origin at about 90-degrees in their respective quadrants. A unit circle with center at the origin is drawn as C_1. Curve C_2 connects the origin, (1,1), and (3,1) with arrowheads pointing in that order.$ $A vector field has the same curves C_1 and C_2. However, the arrows are different. Here, the arrows spiral out from the origin in a clockwise manner. The further away they are from the origin, the longer they become. They are largely horizontal in quadrants 1 and 3 and largely vertical in quadrants 2 and 4.$

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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

$A vector field in two dimensions. The arrows are shorter the closer they are to the origin. They surround the origin in a counterclockwise radial pattern. The upper half of a circle with radius 2 and center at the origin is drawn. (-2,0) and (2,0) are labeled as –a and a, respectively, and the curve is labeled r_1.$

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: Add texts here. Do not delete this text first.