## Educators

Problem 1

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

$$\mathbf{F}(x, y)=0.3 \mathbf{i}-0.4 \mathbf{j}$$

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Problem 2

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 3

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 4

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 5

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 6

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 7

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 8

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 9

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 10

$1-10$ " Sketch the vector ficld $\mathbf{F}$ by drawing a diagram like
Figure 4 or Figure $8 .$

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

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Problem 11

$11-14=$ Match the vector fields $\mathbf{F}$ with the plots labeled I-IV.
$$\mathbf{F}(x, y)=\langle x,-y\rangle$$

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Problem 12

$11-14=$ Match the vector fields $\mathbf{F}$ with the plots labeled I-IV.
$$\mathbf{F}(x, y)=\langle y, x-y\rangle$$

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Problem 13

$11-14=$ Match the vector fields $\mathbf{F}$ with the plots labeled I-IV.
$$\mathbf{F}(x, y)=\langle y, y+2\rangle$$

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Problem 14

$11-14=$ Match the vector fields $\mathbf{F}$ with the plots labeled I-IV.
$$\mathbf{F}(x, y)=\langle\cos (x+y), x\rangle$$

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Problem 15

$15-18=$ Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled
I-IV. Give reasons for your choices.
$$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}$$

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Problem 16

$15-18=$ Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled
I-IV. Give reasons for your choices.
$$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+z \mathbf{k}$$

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Problem 17

$15-18=$ Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled
I-IV. Give reasons for your choices.
$$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+3 \mathbf{k}$$

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Problem 18

$15-18=$ 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 filds (the command
is fieldplot in Maple and PlotvectorField or
vectorplot 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 21

$21-24=$ Find the gradient vector field of $f$
$$f(x, y)=x e^{x y}$$

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Problem 22

$21-24=$ Find the gradient vector field of $f$
$$f(x, y)=\tan (3 x-4 y)$$

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Problem 23

$21-24=$ Find the gradient vector field of $f$
$$f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}$$

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Problem 24

$21-24=$ Find the gradient vector field of $f$
$$f(x, y, z)=x \ln (y-2 z)$$

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Problem 25

$25-26=$ Find the gradient vector field $\nabla f$ of $f$ and sketch it.
$$f(x, y)=x^{2}-y$$

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Problem 26

$25-26=$ Find the gradient vector field $\nabla f$ of $f$ and sketch it.
$$f(x, y)=\sqrt{x^{2}+y^{2}}$$

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Problem 27

$27-28=$ Plot the gradient vector field of $f$ together with a
contour map of $f .$ Explain how they are related to each other.
$$f(x, y)=\ln \left(1+x^{2}+2 y^{2}\right)$$

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Problem 28

$27-28=$ Plot the gradient vector field of $f$ together with a
contour map of $f .$ Explain how they are related to each other.
$$f(x, y)=\cos x-2 \sin y$$

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Problem 29

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 30

At time $t=1,$ a particle is located at position $(1,3) .$ If it
moves in a velocity field
$$F(x, y)=\left\langle x y-2, y^{2}-10\right\rangle$$
find its approximate location at time $t=1.05$

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Problem 31

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 dif-
ferential 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 32

(a) Sketch the vector ficld $\mathbf{F}(x, y)=\mathbf{i}+x \mathbf{j}$ 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 cquations do these func-
tions satisfy? Deduce that $d y / d x=x$ .
(c) If a particle starts at the origin in the velocity field
given by $\mathbf{F},$ find an equation of the path it follows.

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