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## Educators

### Problem 1

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

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

Frank L.

### Problem 2

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

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

Frank L.

### Problem 3

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

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

Frank L.

### Problem 4

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

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

Frank L.

### Problem 5

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

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

Frank L.

### Problem 6

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

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

Frank L.

### Problem 7

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

$\textbf{F} (x, y, z) = \textbf{i}$

Frank L.

### Problem 8

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

$\textbf{F} (x, y, z) = z \textbf{i}$

Frank L.

### Problem 9

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

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

Frank L.

### Problem 10

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

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

Frank L.

### Problem 11

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

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

Frank L.

### Problem 12

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

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

Frank L.

### Problem 13

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

$\textbf{F} (x, y) = \langle y, y + 2 \rangle$

Frank L.

### Problem 14

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

$\textbf{F} (x, y) = \langle \cos (x + y), x \rangle$

Frank L.

### Problem 15

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

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

Frank L.

### Problem 16

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

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

Frank L.

### Problem 17

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

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

Frank L.

### Problem 18

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

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

Frank L.

### Problem 19

If you have a CAS that plots vector fields (the command is $\large \text{fieldplot}$ in Maple and $\large \text{PlotVectorField}$ or $\large \text{VectorPlot}$ in Mathematica), use it to plot $$\textbf{F} (x, y) = (y^2 - 2xy) \textbf{i} + (3xy - 6x^2) \textbf{j}$$ Explain the appearance by finding the set of points $(x, y)$ such that $\textbf{F} (x, y) = 0$.

Frank L.

### Problem 20

Let $\textbf{F(x)} = (r^2 - 2r)\textbf{x}$, where $\textbf{x} = \langle x, y \rangle$ and $r = | \textbf{x} |$. 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 $\textbf{F(x) = 0}$.

Frank L.

### Problem 21

Find the gradient vector field of $f$.

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

Bobby B.
University of North Texas

### Problem 22

Find the gradient vector field of $f$.

$f(s, t) = \sqrt{2s + 3t}$

Frank L.

### Problem 23

Find the gradient vector field of $f$.

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

Frank L.

### Problem 24

Find the gradient vector field of $f$.

$f(x, y, z) = x^2 y e^{y/z}$

Jimmy Y.

### Problem 25

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

$f(x, y) = \frac{1}{2}(x - y)^2$

Frank L.

### Problem 26

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

$f(x, y) = \frac{1}{2}(x^2 - y^2)$

Frank L.

### Problem 27

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

$f(x, y) = \ln (1 + x^2 + 2y^2)$

Frank L.

### Problem 28

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

$f(x, y) = \cos x - 2 \sin y$

Carson M.

### Problem 29

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

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

Frank L.

### Problem 30

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

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

Frank L.

### Problem 31

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

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

Frank L.

### Problem 32

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

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

Frank L.

### Problem 33

A particle moves in a velocity field $\textbf{V}(x, y) = \langle x^2, x + y^2 \rangle$. If it is at position $(2, 1)$ at time $t = 3$, estimate its location at time $t = 3.01$.

Frank L.

### Problem 34

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

Frank L.

### Problem 35

The $\textbf{flow lines}$ (or $\textbf{streamlines}$) 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 $\textbf{F}(x, y) = x \textbf{i} - y \textbf{j}$ 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 $dx/dt = x$ and $dy/dt = -y$. Then solve the differential equations to find an equation of the flow line that passes through the point $(1, 1)$.

Frank L.
(a) Sketch the vector field $\textbf{F}(x, y) = \textbf{i} + x \textbf{j}$ 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 $dy/dx = x$.
(c) If a particle starts at the origin in the velocity field given by $\textbf{F}$, find an equation of the path it follows.