Problem 1

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(X, y)=\frac{1}{2}(\mathbf{i}+\mathbf{j})$$

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Problem 2

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y)=\mathbf{i}+x \mathbf{j}$$

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Problem 3

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y)=y \mathbf{i}+\frac{1}{2} \mathbf{j}$$

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Problem 4

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y)=(x-y) \mathbf{i}+x \mathbf{j}$$

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Problem 5

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y)=\frac{y \mathbf{i}+x \mathbf{j}}{\sqrt{x^{2}+y^{2}}}$$

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Problem 6

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y)=\frac{y \mathbf{i}-x \mathbf{j}}{\sqrt{x^{2}+y^{2}}}$$

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Problem 7

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y, z)=\mathbf{k}$$

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Problem 8

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y, z)=-y \mathbf{k}$$

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Problem 9

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y, z)=x \mathbf{k}$$

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Problem 9

$$\mathbf{F}(x, y, z)=x \mathbf{k}$$

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Problem 10

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y, z)=\mathbf{j}-\mathbf{i}$$

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Problem 11

Match the vector fields $\mathbf{F}$ with the plots labeled I-IV. Give reasons for your choices.

$$\mathbf{F}(x, y)=\langle y, x\rangle$$

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Problem 12

Match the vector fields $\mathbf{F}$ with the plots labeled I-IV. Give reasons for your choices.

$$\mathbf{F}(x, y)=\langle 1, \sin y\rangle$$

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Problem 13

Match the vector fields $\mathbf{F}$ with the plots labeled I-IV. Give reasons for your choices.

$$\mathbf{F}(x, y)=\langle x-2, x+1\rangle$$

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Problem 14

Match the vector fields $\mathbf{F}$ with the plots labeled I-IV. Give reasons for your choices.

$$\mathbf{F}(x, y)=\langle y, 1 / x\rangle$$

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Problem 15

Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled I-IV. Give reasons for your choices.

$$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}$$

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Problem 16

Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled I-IV. Give reasons for your choices.

$$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+z \mat

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Problem 16

$$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+z \mat

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Problem 17

Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled I-IV. Give reasons for your choices.

$$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+3 \mathbf{k}$$

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Problem 18

Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled I-IV. Give reasons for your choices.

$$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$$

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Problem 19

If you have a CAS that plots vector fields (the command is fieldplot in Maple and PlotVectorField in

Mathematica), use it to plot

$$\mathbf{F}(x, y)=\left(y^{2}-2 x y\right) \mathbf{i}+\left(3 x y-6 x^{2}\right) \mathbf{j}$$

Explain the appearance by finding the set of points $(x, y)$ such that $\mathbf{F}(x, y)=\mathbf{0} .$

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Problem 20

Let $\mathbf{F}(\mathbf{x})=\left(r^{2}-2 r\right) \mathbf{x},$ where $\mathbf{x}=\langle x, y\rangle$ and $ r=|\mathbf{x}| .$ Use a CAS to plot this vector field in various domains until you can

see what is happening. Describe the appearance of the plot and explain it by finding the points where $\mathbf{F}(\mathbf{x})=\mathbf{0}$ .

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Problem 23

Find the gradient vector field of $f$

$$f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}$$

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Problem 25

Find the gradient vector field $\nabla$ of $f$ and sketch it.

$$f(x, y)=x^{2}-y$$

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Problem 26

Find the gradient vector field $\nabla$ of $f$ and sketch it.

$$f(x, y)=\sqrt{x^{2}+y^{2}}$$

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Problem 27

Plot the gradient vector field of together with a contour map of . Explain how they are related to each other.

$$f(x, y)=\sin x+\sin y$$

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Problem 28

Plot the gradient vector field of together with a contour map of . Explain how they are related to each other.

$$f(x, y)=\sin (x+y)$$

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Problem 29

Match the functions $f$ with the plots of their gradient vector fields (labeled I-IV). Give reasons for your choices.

$$f(x, y)=x^{2}+y^{2}$$

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Problem 30

Match the functions $f$ with the plots of their gradient vector fields (labeled I-IV). Give reasons for your choices.

$$f(x, y)=x(x+y)$$

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Problem 31

Match the functions $f$ with the plots of their gradient vector fields (labeled I-IV). Give reasons for your choices.

$$f(x, y)=(x+y)^{2}$$

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Problem 32

Match the functions $f$ with the plots of their gradient vector fields (labeled I-IV). Give reasons for your choices.

$$f(x, y)=\sin \sqrt{x^{2}+y^{2}}$$

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Problem 33

A particle moves in a velocity field $\mathbf{V}(x, y)=\left\langle x^{2}, x+y^{2}\right\rangle$

If it is at position $(2,1)$ at time $t=3,$ estimate its location at time $t=3.01$

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Problem 34

At time $t=1,$ a particle is located at position $(1,3) .$ If it moves in a velocity field

$\quad \mathbf{F}(x, y)=\left\langle x y-2, y^{2}-10\right\rangle$ find its approximate location at time $ t=1.05$

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Problem 35

The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines.

(a) Use a sketch of the vector field $\mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j}$ to draw some flow lines. From your sketches, can you guess the equations of the flow lines?

(b) If parametric equations of a flow line are $x=x(t)$ , $y=y(t),$ explain why these functions satisfy the differential equations $d x / d t=x$ and $d y / d t=-y .$ Then solve the differential equations to find an equation of the flow line that passes through the point (1, 1).

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Problem 36

(a) Sketch the vector field $\mathbf{F}(x, y)=\mathbf{i}+x \mathbf{j}$ and then sketch some flow lines. What shape do these flow lines appear to have?

(b) If parametric equations of the flow lines are $x=x(t)$ , $y=y(t),$ what differential equations do these functions satisfy? Deduce that $d y / d x=x$ .

(c) If a particle starts at the origin in the velocity field given by F, find an equation of the path it follows.

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