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Calculus

James Stewart

Chapter 16

Vector Calculus - all with Video Answers

Educators

Section 1

16.1 Exercises

01:21

Problem 1

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.
$\mathbf{F}(x, y)=0.3 \mathbf{i}-0.4 \mathbf{j}$

Frank Lin
Frank Lin
Numerade Educator

Problem 2

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.
$\mathbf{F}(x, y)=\frac{1}{2} x \mathbf{i}+y \mathbf{j}$

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Problem 3

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.
$\mathbf{F}(x, y)=-\frac{1}{2} \mathbf{i}+(y-x) \mathbf{j}$

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Problem 4

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.
$\mathbf{F}(x, y)=y \mathbf{i}+(x+y) \mathbf{j}$

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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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01:55

Problem 8

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.
$\mathbf{F}(x, y, z)=-y \mathbf{k}$

Frank Lin
Frank Lin
Numerade Educator

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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03:18

Problem 10

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.
$\mathbf{F}(x, y, z)=\mathbf{j}-\mathbf{i}$

Yujian Zeng
Yujian Zeng
Numerade Educator

Problem 11

Match the vector fields $\mathbf{F}$ with the plots labeled I-IV. Give reasons for your choices. GRAPH CAN'T COPY.
$\mathbf{F}(x, y)=\langle x,-y\rangle$

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01:22

Problem 12

Match the vector fields $\mathbf{F}$ with the plots labeled I-IV. Give reasons for your choices. GRAPH CAN'T COPY.
$\mathbf{F}(x, y)=\langle y, x-y\rangle$

WM
William Mead
Numerade Educator

Problem 13

Match the vector fields $\mathbf{F}$ with the plots labeled I-IV. Give reasons for your choices. GRAPH CAN'T COPY.
$\mathbf{F}(x, y)=\langle y, y+2\rangle$

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01:34

Problem 14

Match the vector fields $\mathbf{F}$ with the plots labeled I-IV. Give reasons for your choices. GRAPH CAN'T COPY.
$\mathbf{F}(x, y)=\langle\cos (x+y), x\rangle$

WM
William Mead
Numerade Educator

Problem 15

Match the vector fields $\mathbf{F}$ on $\mathbb{R}^3$ with the plots labeled I-IV. Give reasons for your choices. GRAPH CAN'T COPY.
$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}$

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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. GRAPH CAN'T COPY.
$\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+z \mathbf{k}$

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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. GRAPH CAN'T COPY.
$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+3 \mathbf{k}$

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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. GRAPH CAN'T COPY.
$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$

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01:23

Problem 19

If you have a CAS that plots vector fields (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 $\mathbf{F}(x, y)=\mathbf{0}$.

Frank Lin
Frank Lin
Numerade Educator
04:14

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 $\mathbf{F}(\mathbf{x})=\mathbf{0}$.

Cameron Bunney
Cameron Bunney
Numerade Educator
01:24

Problem 21

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

Cameron Bunney
Cameron Bunney
Numerade Educator
01:29

Problem 22

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

Cameron Bunney
Cameron Bunney
Numerade Educator
01:23

Problem 23

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

Elliott Walker
Elliott Walker
Numerade Educator
01:29

Problem 24

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

Cameron Bunney
Cameron Bunney
Numerade Educator
02:39

Problem 25

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

Cameron Bunney
Cameron Bunney
Numerade Educator
02:58

Problem 26

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

Yujian Zeng
Yujian Zeng
Numerade Educator
03:09

Problem 27

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

Frank Lin
Frank Lin
Numerade Educator
03:23

Problem 28

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

Cameron Bunney
Cameron Bunney
Numerade Educator

Problem 29

Match the functions $f$ with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices. GRAPH CAN'T COPY.
$f(x, y)=x^2+y^2$

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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. GRAPH CAN'T COPY.
$f(x, y)=x(x+y)$

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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. GRAPH CAN'T COPY.
$f(x, y)=(x+y)^2$

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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. GRAPH CAN'T COPY.
$f(x, y)=\sin \sqrt{x^2+y^2}$

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01:48

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 $t=3.01$.

Cameron Bunney
Cameron Bunney
Numerade Educator
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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

$$
\mathbf{F}(x, y)=\left\langle x y-2, y^2-10\right\rangle
$$

find its approximate location at time $t=1.05$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:43

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 $(1,1)$.

Nick Johnson
Nick Johnson
Numerade Educator
01:33

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 $\mathbf{F}$, find an equation of the path it follows.

Frank Lin
Frank Lin
Numerade Educator