Calculus

James Stewart

Chapter 16 Vector Calculus - all with Video Answers

Educators

Section 1

Vector Fields

02:24

Problem 1

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Yujian Zeng

Numerade Educator

03:03

Problem 2

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Yujian Zeng

Numerade Educator

03:38

Problem 3

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Yujian Zeng

Numerade Educator

02:35

Problem 4

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Yujian Zeng

Numerade Educator

03:04

Problem 5

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Yujian Zeng

Numerade Educator

03:41

Problem 6

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

$$\mathbf{F}(x, y)=\frac{y \mathbf{1}-x \mathbf{J}}{\sqrt{x^{2}+y^{2}}}$$

Shu Naito

Numerade Educator

01:27

Problem 7

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Shu Naito

Numerade Educator

03:20

Problem 8

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Shu Naito

Numerade Educator

03:37

Problem 9

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Shu Naito

Numerade Educator

01:17

Problem 10

$1-10$ Sketch the vector field $\mathbf{F}$ by drawing a diagram like
Figure 5 or Figure $9 .$

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

Shu Naito

Numerade Educator

View

Problem 11

$11-14$ Match the vector fields $F$ with the plots labeled I-IV.
Give reasons for your choices.

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

Victor Salazar

Numerade Educator

03:11

Problem 12

$11-14$ Match the vector fields $F$ with the plots labeled I-IV.
Give reasons for your choices.

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

Cameron Bunney

Numerade Educator

01:43

Problem 13

$11-14$ Match the vector fields $F$ with the plots labeled I-IV.
Give reasons for your choices.

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

Cameron Bunney

Numerade Educator

03:58

Problem 14

$11-14$ Match the vector fields $F$ with the plots labeled I-IV.
Give reasons for your choices.

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

Cameron Bunney

Numerade Educator

01:56

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.

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

Cameron Bunney

Numerade Educator

02:17

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.

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

Cameron Bunney

Numerade Educator

02:01

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.

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

Cameron Bunney

Numerade Educator

03:41

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.

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

Cameron Bunney

Numerade Educator

02:33

If you have a CAS that plots vector fields (the command
is field plot in Maple and Plot Vector Field or
Vector Plot 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 $\mathbf{F}(x, y)=\mathbf{0} .$

Shu Naito

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

Numerade Educator

01:50

Problem 21

$21-24$ Find the gradient vector field of $f$

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

Shu Naito

Numerade Educator

02:03

Problem 22

$21-24$ Find the gradient vector field of $f$

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

Shu Naito

Numerade Educator

01:50

Problem 23

$21-24$ Find the gradient vector field of $f$

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

Cameron Bunney

Numerade Educator

03:23

Problem 24

$21-24$ Find the gradient vector field of $f$

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

Shu Naito

Numerade Educator

03:23

Problem 25

$25-26$ Find the gradient vector field $\nabla f$ of $f$ and sketch it.

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

Shu Naito

Numerade Educator

03:30

Problem 26

$25-26$ Find the gradient vector field $\nabla f$ of $f$ and sketch it.

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

Shu Naito

Numerade Educator

03:33

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.

$$f(x, y)=\ln \left(1+x^{2}+2 y^{2}\right)$$

Cameron Bunney

Numerade Educator

03:23

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.

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

Cameron Bunney

Numerade Educator

01:26

Problem 29

$29-32$ Match the functions $f$ with the plots of their gradient
vector fields labeled I-IV. Give reasons for your choices.

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

Yujian Zeng

Numerade Educator

02:19

Problem 30

$29-32$ Match the functions $f$ with the plots of their gradient
vector fields labeled I-IV. Give reasons for your choices.

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

Yujian Zeng

Numerade Educator

01:57

Problem 31

$29-32$ Match the functions $f$ with the plots of their gradient
vector fields labeled I-IV. Give reasons for your choices.

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

Yujian Zeng

Numerade Educator

04:30

Problem 32

$29-32$ Match the functions $f$ with the plots of their gradient
vector fields labeled I-IV. Give reasons for your choices.

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

Yujian Zeng

Numerade Educator

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

Numerade Educator

01:56

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 $t=1.05$

Cameron Bunney

Numerade Educator

13:03

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
(b) If parametric equations of a flow line are $x=x(t)$ ,
$y=y(t),$ explain why these functions satisfy the differ-
ential 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 $(1,1)$ .

Shu Naito

Numerade Educator

08:16

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.

Shu Naito

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36