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# 15.1E: Vector Fields (Exercises)

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1. The domain of vector field $$\vecs{F}=\vecs{F}(x,y)$$ is a set of points $$(x,y)$$ in a plane, and the range of $$\vecs F$$ is a set of what in the plane?

Solution: Vectors

For the following exercises, determine whether the statement is true or false.

2. Vector field $$\vecs{F}=⟨3x^2,1⟩$$ is a gradient field for both $$ϕ_1(x,y)=x^3+y$$ and $$ϕ_2(x,y)=y+x^3+100.$$

3. Vector field $$\vecs{F}=\dfrac{⟨y,x⟩}{\sqrt{x^2+y^2}}$$ is constant in direction and magnitude on a unit circle

Solution: False

4. Vector field $$\vecs{F}=\dfrac{⟨y,x⟩}{\sqrt{x^2+y^2}}$$ is neither a radial field nor a rotation.

For the following exercies, describe each vector field by drawing some of its vectors.

5. [T] $$\vecs{F}(x,y)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}$$

Solution:

6. [T] $$\vecs{F}(x,y)=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}$$

7. [T] $$\vecs{F}(x,y)=x\,\hat{\mathbf i}−y\,\hat{\mathbf j}$$

Solution:

8. [T] $$\vecs{F}(x,y)=\,\hat{\mathbf i}+\,\hat{\mathbf j}$$

9. [T] $$\vecs{F}(x,y)=2x\,\hat{\mathbf i}+3y\,\hat{\mathbf j}$$

Solution:

10. [T] $$\vecs{F}(x,y)=3\,\hat{\mathbf i}+x\,\hat{\mathbf j}$$

11. [T] $$\vecs{F}(x,y)=y\,\hat{\mathbf i}+\sin x\,\hat{\mathbf j}$$

Solution:

12. [T] $$\vecs F(x,y,z)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}+z\,\hat{\mathbf k}$$

13. [T] $$\vecs F(x,y,z)=2x\,\hat{\mathbf i}−2y\,\hat{\mathbf j}−2z\,\hat{\mathbf k}$$

14. [T] $$\vecs F(x,y,z)=yz\,\hat{\mathbf i}−xz\,\hat{\mathbf j}$$

For the following exercises, find the gradient vector field of each function $$f$$.

15. $$f(x,y)=x\sin y+\cos y$$

Solution: $$\vecs{F}(x,y)=\sin(y)\,\hat{\mathbf i}+(x\cos y−\sin y)\,\hat{\mathbf j}$$

16. $$f(x,y,z)=ze^{−xy}$$

17. $$f(x,y,z)=x^2y+xy+y^2z$$

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

18. $$\vecs{F}(x,y)=x^2\sin(5y)$$

19. $$\vecs{F}(x,y)=\ln(1+x^2+2y^2)$$

Solution: $$\vecs{F}(x,y)=(\dfrac{2x}{1+x^2+2y^2})\,\hat{\mathbf i}+(\dfrac{4y}{1+x^2+2y^2})\,\hat{\mathbf j}$$

20. $$f(x,y,z)=x\cos(\dfrac{y}{z})$$

21. What is vector field $$\vecs{F}(x,y)$$ with a value at $$(x,y)$$ that is of unit length and points toward $$(1,0)$$?

Solution: $$\vecs{F}(x,y)=\dfrac{(1−x)\,\hat{\mathbf i}−y\,\hat{\mathbf j}}{\sqrt{(1−x)^2+y^2}}$$

For the following exercises, write formulas for the vector fields with the given properties.

22. All vectors are parallel to the $$x$$-axis and all vectors on a vertical line have the same magnitude.

23. All vectors point toward the origin and have constant length.

Solution: $$\vecs{F}(x,y)=\dfrac{(y\,\hat{\mathbf i}−x\,\hat{\mathbf j})}{\sqrt{x^2+y^2}}$$

24. All vectors are of unit length and are perpendicular to the position vector at that point.

25. Give a formula $$\vecs{F}(x,y)=M(x,y)\,\hat{\mathbf i}+N(x,y)\,\hat{\mathbf j}$$ for the vector field in a plane that has the properties that $$\vecs{F}=\vecs 0$$ at $$(0,0)$$ and that at any other point $$(a,b), \vecs F$$ is tangent to circle $$x^2+y^2=a^2+b^2$$ and points in the clockwise direction with magnitude $$\|\vecs F\|=\sqrt{a^2+b^2}$$.

Solution: $$\vecs{F}(x,y)=y\,\hat{\mathbf i}−x\,\hat{\mathbf j}$$

26. Is vector field $$\vecs{F}(x,y)=(P(x,y),Q(x,y))=(\sin x+y)\,\hat{\mathbf i}+(\cos y+x)\,\hat{\mathbf j}$$ a gradient field?

27. Find a formula for vector field $$\vecs{F}(x,y)=M(x,y)\,\hat{\mathbf i}+N(x,y)\,\hat{\mathbf j}$$ given the fact that for all points $$(x,y)$$, $$\vecs F$$ points toward the origin and $$\|\vecs F\|=\dfrac{10}{x^2+y^2}$$.

Solution: $$\vecs{F}(x,y)=\dfrac{−10}{(x^2+y^2)^{3/2}}(x\,\hat{\mathbf i}+y\,\hat{\mathbf j})$$

For the following exercises, assume that an electric field in the $$xy$$-plane caused by an infinite line of charge along the $$x$$-axis is a gradient field with potential function $$V(x,y)=c\ln(\dfrac{r_0}{\sqrt{x^2+y^2}})$$, where $$c>0$$ is a constant and $$r_0$$ is a reference distance at which the potential is assumed to be zero.

28. Find the components of the electric field in the $$x$$- and $$y$$-directions, where $$\vecs E(x,y)=−\vecs ∇V(x,y).$$

29. Show that the electric field at a point in the $$xy$$-plane is directed outward from the origin and has magnitude $$\|\vecs E\|=\dfrac{c}{r}$$, where $$r=\sqrt{x^2+y^2}$$.

Solution: $$\|\vecs E\|=\dfrac{c}{|r|^2}r=\dfrac{c}{|r|}\dfrac{r}{|r|}$$

flow line (or streamline) of a vector field $$\vecs F$$ is a curve $$\vecs r(t)$$ such that $$d\vecs{r}/dt=\vecs F(\vecs r(t))$$. If $$\vecs F$$ represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field. For the following exercises, show that the given curve $$\vecs c(t)$$ is a flow line of the given velocity vector field $$\vecs F(x,y,z)$$.

30. $$\vecs c(t)=⟨ e^{2t},\ln|t|,\dfrac{1}{t} ⟩,\,t≠0;\quad \vecs F(x,y,z)=⟨2x,z,−z^2⟩$$

31. $$\vecs c(t)=⟨ \sin t,\cos t,e^t⟩;\quad \vecs F(x,y,z) =〈y,−x,z〉$$

Solution: $$\vecs c′(t)=⟨ \cos t,−\sin t,e^{−t}⟩=\vecs F(\vecs c(t))$$

For the following exercises, let $$\vecs{F}=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}$$, $$\vecs G=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}$$, and $$\vecs H=x\,\hat{\mathbf i}−y\,\hat{\mathbf j}$$. Match $$\vecs F$$, $$\vecs G$$, and $$\vecs H$$ with their graphs.

32.

33.

Solution: $$\vecs H$$

34.

For the following exercises, let $$\vecs{F}=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}$$, $$\vecs G=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}$$, and $$\vecs H=−x\,\hat{\mathbf i}+y\,\hat{\mathbf j}$$. Match the vector fields with their graphs in (I)−(IV).

1. $$\vecs F+\vecs G$$
2. $$\vecs F+\vecs H$$
3. $$\vecs G+\vecs H$$
4. $$−\vecs F+\vecs G$$

35.

Solution: d. $$−\vecs F+\vecs G$$

36.

37.

Solution: a. $$\vecs F+\vecs G$$

38.