Chapter 16

Vector Calculus

Educators

FL

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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}} $

FL
Frank L.
Numerade Educator

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}} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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 $

FL
Frank L.
Numerade Educator

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 $

FL
Frank L.
Numerade Educator

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 $

FL
Frank L.
Numerade Educator

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 $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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 $.

FL
Frank L.
Numerade Educator

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} $.

FL
Frank L.
Numerade Educator

Problem 21

Find the gradient vector field of $ f $.

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

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Frank L.
Numerade Educator

Problem 22

Find the gradient vector field of $ f $.

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

FL
Frank L.
Numerade Educator

Problem 23

Find the gradient vector field of $ f $.

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

FL
Frank L.
Numerade Educator

Problem 24

Find the gradient vector field of $ f $.

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

FL
Frank L.
Numerade Educator

Problem 25

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

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

FL
Frank L.
Numerade Educator

Problem 26

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

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

FL
Frank L.
Numerade Educator

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) $

FL
Frank L.
Numerade Educator

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 $

FL
Frank L.
Numerade Educator

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 $

FL
Frank L.
Numerade Educator

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) $

FL
Frank L.
Numerade Educator

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 $

FL
Frank L.
Numerade Educator

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} $

FL
Frank L.
Numerade Educator

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 $.

FL
Frank L.
Numerade Educator

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 $.

FL
Frank L.
Numerade Educator

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) $.

FL
Frank L.
Numerade Educator

Problem 36

(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.

FL
Frank L.
Numerade Educator