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# 15.3E: Conservative Vector Fields (Exercises)

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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 $$D$$.

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 $$\vecs b$$.

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.

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 $$(3, 4).$$

$$\displaystyle \int _C \vecs F·d\vecs r=24$$ 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 $$0≤t≤1$$.

$$\displaystyle \int _C\vecs F·d\vecs r=\left(e−\frac{3π}{2}\right)$$ 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. $$\vecs{F}(x,y)=(−y+e^x\sin y)\,\mathbf{\hat i}+((x+2)e^x\cos y)\,\mathbf{\hat j}$$

Not conservative

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

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

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

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 $$(4, 4)$$

$$\displaystyle ∮_C(2y\,dx+2x\,dy)=32$$ 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 $$f(x,y)=5x^2+3xy+10y^2.$$

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

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

$$\vecs F$$ is not conservative.

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

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

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

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

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

$$\vecs F$$ 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 $$(2,1)$$.

$$\displaystyle \int _C\vecs F·d\vecs r=\left(e^2+1\right)$$ 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 $$(3, 5, 2).$$

$$\displaystyle \int _C\vecs F·d\vecs r=38$$ 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 $$C_1$$.

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

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

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

$$\displaystyle ∮_{C_2}\vecs F·d\vecs r=7$$ 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 $$\vecs r(t)=(\frac{\ln t}{\ln 2})\,\mathbf{\hat i}+t^{3/2}\,\mathbf{\hat j}+t\cos(πt),1≤t≤4.$$

$$\displaystyle \int _C\vecs F·d\vecs r=150$$ 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 $$0≤t≤1.$$

$$\displaystyle \int _C\vecs F·d\vecs r=−1$$ 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 $$1\;cm$$.

$$4×10^{31}$$ 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 $$\vecs{F}(x,y)=(2x\cos y)\,\mathbf{\hat i}−(x^2\sin y)\,\mathbf{\hat j}.$$

$$\displaystyle \int _C\vecs F·d\vecs s=0.4687$$ 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 $$\vecs r_2(t)=t\,\mathbf{\hat i},\quad −a≤t≤a.$$

$$\text{circulation}=πa^2$$ 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 $$f_y=Q(x,y).$$