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Calculus Early Transcendentals

Howard Anton, Irl Bivens, Stephen Davis

Chapter 15

Topics in Vector Calculus - all with Video Answers

Educators

Section 1

Vector Fields

01:46

Problem 1

Match the vector field $\mathbf{F}(x, y)$ with one of the plots, and explain your reasoning.

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

R M
R M
Numerade Educator
01:01

Problem 2

Match the vector field $\mathbf{F}(x, y)$ with one of the plots, and explain your reasoning.

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

R M
R M
Numerade Educator
00:39

Problem 3

Determine whether the statement about the vector field $\mathbf{F}(x, y)$ is true or false. If false, explain why.

$\mathbf{F}(x, y)=x^{2} \mathbf{i}-y \mathbf{j}$.
(a) $\|\mathbf{F}(x, y)\| \rightarrow 0$ as $(x, y) \rightarrow(0,0)$.
(b) If $(x, y)$ is on the positive $y$ -axis, then the vector points in the negative $y$ -direction.
(c) If $(x, y)$ is in the first quadrant, then the vector points down and to the right.

R M
R M
Numerade Educator
01:03

Problem 4

Determine whether the statement about the vector field $\mathbf{F}(x, y)$ is true or false. If false, explain why.

$\mathbf{F}(x, y)=\frac{x}{\sqrt{x^{2}+y^{2}}} \mathbf{i}-\frac{y}{\sqrt{x^{2}+y^{2}}} \mathbf{j}$.
(a) As $(x, y)$ moves away from the origin, the lengths of the vectors decrease.
(b) If $(x, y)$ is a point on the positive $x$ -axis, then the vector points up.
(c) If $(x, y)$ is a point on the positive $y$ -axis, the vector points to the right.

R M
R M
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01:05

Problem 5

Sketch the vector field by drawing some representative nonintersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other.

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

Carl David Cepeda
Carl David Cepeda
Numerade Educator
01:18

Problem 6

Sketch the vector field by drawing some representative nonintersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other.

$$\mathbf{F}(x, y)=y \mathbf{j}, \quad y>0$$

Carl David Cepeda
Carl David Cepeda
Numerade Educator
00:54

Problem 7

Sketch the vector field by drawing some representative nonintersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other.

$\mathbf{F}(x, y)=y \mathbf{i}-x \mathbf{j} .$ [Note: Each vector in the field is perpendicular to the position vector $\mathbf{r}=x \mathbf{i}+y \mathbf{j} .]$

Carl David Cepeda
Carl David Cepeda
Numerade Educator
01:22

Problem 8

Sketch the vector field by drawing some representative nonintersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other.

$\mathbf{F}(x, y)=\frac{x \mathbf{i}+y \mathbf{j}}{\sqrt{x^{2}+y^{2}}} .$ [Note: Each vector in the field is
a unit vector in the same direction as the position vector $\mathbf{r}=x \mathbf{i}+y \mathbf{j} \cdot]$

Carl David Cepeda
Carl David Cepeda
Numerade Educator
01:21

Problem 9

Use a graphing utility to generate a plot of the vector field.

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

Carl David Cepeda
Carl David Cepeda
Numerade Educator
00:47

Problem 10

Use a graphing utility to generate a plot of the vector field.

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

Carl David Cepeda
Carl David Cepeda
Numerade Educator
00:32

Problem 11

Determine whether the statement is true or false. Explain your answer.

The vector-valued function
$$\mathbf{F}(x, y)=y \mathbf{i}+x^{2} \mathbf{j}+x y \mathbf{k}$$
is an example of a vector field in the $x y$ -plane.

R M
R M
Numerade Educator
00:30

Problem 12

Determine whether the statement is true or false. Explain your answer.

If $\mathbf{r}$ is a radius vector in 3 -space, then a vector field of the form
$$\mathbf{F}(\mathbf{r})=\frac{1}{\|\mathbf{r}\|^{2}} \mathbf{r}$$
is an example of an inverse-square field.

R M
R M
Numerade Educator
00:25

Problem 13

Determine whether the statement is true or false. Explain your answer.

If $\mathbf{F}$ is a vector field, then so is $\nabla \times \mathbf{F}$.

R M
R M
Numerade Educator
00:30

Problem 14

Determine whether the statement is true or false. Explain your answer.

If $\mathbf{F}$ is a vector field and $\nabla \cdot \mathbf{F}=\phi,$ then $\phi$ is a potential function for $\mathbf{F}$.

R M
R M
Numerade Educator
03:28

Problem 15

Confirm that $\phi$ is a potential function for $\mathbf{F}(\mathbf{r})$ on some region, and state the region.

(a) $\phi(x, y)=\tan ^{-1} x y$
$\mathbf{F}(x, y)=\frac{y}{1+x^{2} y^{2}} \mathbf{i}+\frac{x}{1+x^{2} y^{2}} \mathbf{j}$
(b) $\phi(x, y, z)=x^{2}-3 y^{2}+4 z^{2}$
$\mathbf{F}(x, y, z)=2 x \mathbf{i}-6 y \mathbf{j}+8 z \mathbf{k}$

Aman Gupta
Aman Gupta
Numerade Educator
03:02

Problem 16

Confirm that $\phi$ is a potential function for $\mathbf{F}(\mathbf{r})$ on some region, and state the region.

(a) $\phi(x, y)=2 y^{2}+3 x^{2} y-x y^{3}$
F $(x, y)=\left(6 x y-y^{3}\right) \mathbf{i}+\left(4 y+3 x^{2}-3 x y^{2}\right) \mathbf{j}$
(b) $\phi(x, y, z)=x \sin z+y \sin x+z \sin y$
$\mathbf{F}(x, y, z)=(\sin z+y \cos x) \mathbf{i}+(\sin x+z \cos y) \mathbf{j}$
$\quad+(\sin y+x \cos z) \mathbf{k}$

Aman Gupta
Aman Gupta
Numerade Educator
02:30

Problem 17

Find div $\mathbf{F}$ and curl $\mathbf{F}$.

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

Aman Gupta
Aman Gupta
Numerade Educator
02:17

Problem 18

Find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=x z^{3} \mathbf{i}+2 y^{4} x^{2} \mathbf{j}+5 z^{2} y \mathbf{k}$$

R M
R M
Numerade Educator
02:20

Problem 19

Find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=7 y^{3} z^{2} \mathbf{i}-8 x^{2} z^{5} \mathbf{j}-3 x y^{4} \mathbf{k}$$

R M
R M
Numerade Educator
01:52

Problem 20

Find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=e^{x y} \mathbf{i}-\cos y \mathbf{j}+\sin ^{2} z \mathbf{k}$$

R M
R M
Numerade Educator
07:24

Problem 21

Find div $\mathbf{F}$ and curl $\mathbf{F}$.

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

Aman Gupta
Aman Gupta
Numerade Educator
02:36

Problem 22

Find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=\ln x \mathbf{i}+e^{x y z} \mathbf{j}+\tan ^{-1}(z / x) \mathbf{k}$$

R M
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Numerade Educator
02:09

Problem 23

Find $\nabla \cdot(\mathbf{F} \times \mathbf{G})$

$$\begin{array}{l}
\mathbf{F}(x, y, z)=2 x \mathbf{i}+\mathbf{j}+4 y \mathbf{k} \\
\mathbf{G}(x, y, z)=x \mathbf{i}+y \mathbf{j}-z \mathbf{k}
\end{array}$$

R M
R M
Numerade Educator
01:15

Problem 24

Find $\nabla \cdot(\mathbf{F} \times \mathbf{G})$

$$\begin{array}{l}
\mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k} \\
\mathbf{G}(x, y, z)=x y \mathbf{j}+x y z \mathbf{k}
\end{array}$$

R M
R M
Numerade Educator
01:28

Problem 25

Find $\nabla \cdot(\nabla \times \mathbf{F})$.

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

R M
R M
Numerade Educator
01:47

Problem 26

Find $\nabla \cdot(\nabla \times \mathbf{F})$.

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

R M
R M
Numerade Educator
01:49

Problem 27

Find $\nabla \times(\nabla \times \mathbf{F})$.

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

R M
R M
Numerade Educator
01:50

Problem 28

Find $\nabla \times(\nabla \times \mathbf{F})$.

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

R M
R M
Numerade Educator
01:04

Problem 29

Use a CAS to check the calculations in Exercises 23,25 , and 27 .

Carson Merrill
Carson Merrill
Numerade Educator
01:04

Problem 30

Use a CAS to check the calculations in Exercises 24,26 , and 28 .

Carson Merrill
Carson Merrill
Numerade Educator
00:49

Problem 31

Let $k$ be a constant, $\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),$ and
$\phi=\phi(x, y, z) .$ Prove the following identities, assuming that all derivatives involved exist and are continuous.

$$\operatorname{div}(k \mathbf{F})=k \operatorname{div} \mathbf{F}$$

R M
R M
Numerade Educator
00:48

Problem 32

Let $k$ be a constant, $\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),$ and
$\phi=\phi(x, y, z) .$ Prove the following identities, assuming that all derivatives involved exist and are continuous.

$$\operatorname{curl}(k \mathbf{F})=k$ curl $\mathbf{F}$$

R M
R M
Numerade Educator
01:38

Problem 33

Let $k$ be a constant, $\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),$ and
$\phi=\phi(x, y, z) .$ Prove the following identities, assuming that all derivatives involved exist and are continuous.

$$\operatorname{div}(\mathbf{F}+\mathbf{G})=\operatorname{div} \mathbf{F}+\operatorname{div} \mathbf{G}$$

R M
R M
Numerade Educator
03:22

Problem 34

Let $k$ be a constant, $\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),$ and
$\phi=\phi(x, y, z) .$ Prove the following identities, assuming that all derivatives involved exist and are continuous.

$$\operatorname{curl}(\mathbf{F}+\mathbf{G})=\operatorname{curl} \mathbf{F}+\operatorname{curl} \mathbf{G}$$

R M
R M
Numerade Educator
01:39

Problem 35

Let $k$ be a constant, $\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),$ and
$\phi=\phi(x, y, z) .$ Prove the following identities, assuming that all derivatives involved exist and are continuous.

$$\operatorname{div}(\phi \mathbf{F})=\phi \operatorname{div} \mathbf{F}+\nabla \phi \cdot \mathbf{F}$$

R M
R M
Numerade Educator
03:25

Problem 36

Let $k$ be a constant, $\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),$ and
$\phi=\phi(x, y, z) .$ Prove the following identities, assuming that all derivatives involved exist and are continuous.

$$\operatorname{curl}(\phi \mathbf{F})=\phi \operatorname{curl} \mathbf{F}+\nabla \phi \times \mathbf{F}$$

R M
R M
Numerade Educator
03:45

Problem 37

Let $k$ be a constant, $\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),$ and
$\phi=\phi(x, y, z) .$ Prove the following identities, assuming that all derivatives involved exist and are continuous.

$$\operatorname{div}(\operatorname{curl} \mathbf{F})=0$$

R M
R M
Numerade Educator
02:58

Problem 38

Let $k$ be a constant, $\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),$ and
$\phi=\phi(x, y, z) .$ Prove the following identities, assuming that all derivatives involved exist and are continuous.

$$\operatorname{curl}(\nabla \phi)=\mathbf{0}$$

R M
R M
Numerade Educator
01:07

Problem 39

Rewrite the identities in Exercises $31,33,35,$ and 37 in an equivalent form using the notation $\nabla \cdot$ for divergence and $\nabla \times$ for curl.

R M
R M
Numerade Educator
00:47

Problem 40

Rewrite the identities in Exercises $32,34,36,$ and 38 in an equivalent form using the notation $\nabla \cdot$ for divergence and $\nabla \times$ for curl.

R M
R M
Numerade Educator
03:34

Problem 41

Verify that the radius vector $\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ has the stated property.

(a) curl $\mathbf{r}=\mathbf{0}$
(b) $\nabla\|\mathbf{r}\|=\frac{\mathbf{r}}{\|\mathbf{r}\|}$

Aman Gupta
Aman Gupta
Numerade Educator
02:28

Problem 42

Verify that the radius vector $\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ has the stated property.

(a) $\operatorname{div} \mathbf{r}=3$
(b) $\nabla \frac{1}{\|\mathbf{r}\|}=-\frac{\mathbf{r}}{\|\mathbf{r}\|^{3}}$$

Aman Gupta
Aman Gupta
Numerade Educator
03:43

Problem 43

Let $\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},$ let $r=\|\mathbf{r}\|,$ let $f$ be a differentiable function of one variable, and let $\mathbf{F}(\mathbf{r})=f(r) \mathbf{r}$.

(a) Use the chain rule and Exercise $41(\mathrm{~b})$ to show that
$$\nabla f(r)=\frac{f^{\prime}(r)}{r} \mathbf{r}$$
(b) Use the result in part (a) and Exercises 35 and 42 (a) to show that $\operatorname{div} \mathbf{F}=3 f(r)+r f^{\prime}(r)$

Aman Gupta
Aman Gupta
Numerade Educator
01:48

Problem 44

Let $\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},$ let $r=\|\mathbf{r}\|,$ let $f$ be a differentiable function of one variable, and let $\mathbf{F}(\mathbf{r})=f(r) \mathbf{r}$.

(a) Use part (a) of Exercise $43,$ Exercise $36,$ and Exercise $41($ a) to show that curl $\mathbf{F}=\mathbf{0}$.
(b) Use the result in part (a) of Exercise 43 and Exercises 35 and $42(\mathrm{a})$ to show that
$$\nabla^{2} f(r)=2 \frac{f^{\prime}(r)}{r}+f^{\prime \prime}(r)$$

R M
R M
Numerade Educator
00:30

Problem 45

Use the result in Exercise $43(\mathrm{~b})$ to show that the divergence of the inverse-square field $\mathbf{F}=\mathbf{r} /\|\mathbf{r}\|^{3}$ is zero.

R M
R M
Numerade Educator
01:18

Problem 46

Use the result of Exercise $43(\mathrm{~b})$ to show that if $\mathbf{F}$ is a vector field of the form $\mathbf{F}=f(\|\mathbf{r}\|) \mathbf{r}$ and if $\operatorname{div} \mathbf{F}=0,$ then $\mathbf{F}$
is an inverse-square field. [Suggestion: Let $r=\|\mathbf{r}\|$ and multiply $3 f(r)+r f^{\prime}(r)=0$ through by $r^{2}$. Then write the result as a derivative of a product.]

R M
R M
Numerade Educator
02:53

Problem 47

A curve $C$ is called a flow line of a vector field $\mathbf{F}$ if $\mathbf{F}$ is a tangent vector to $C$ at each point along $C$ (see the accompanying figure).
(a) Let $C$ be a flow line for $\mathbf{F}(x, y)=-y \mathbf{i}+x \mathbf{j},$ and let $(x, y)$ be a point on $C$ for which $y \neq 0 .$ Show that the flow lines satisfy the differential equation
$$\frac{d y}{d x}=-\frac{x}{y}$$
(b) Solve the differential equation in part (a) by separation of variables, and show that the flow lines are concentric circles centered at the origin.

Aman Gupta
Aman Gupta
Numerade Educator
00:22

Problem 48

Find a differential equation satisfied by the flow lines of $\mathbf{F}$ (see Exercise 47 ), and solve it to find equations for the flow lines of $\mathbf{F}$. Sketch some typical flow lines and tangent vectors.

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

R M
R M
Numerade Educator
00:20

Problem 49

Find a differential equation satisfied by the flow lines of $\mathbf{F}$ (see Exercise 47 ), and solve it to find equations for the flow lines of $\mathbf{F}$. Sketch some typical flow lines and tangent vectors.

$$\mathbf{F}(x, y)=x \mathbf{i}+\mathbf{j}, \quad x>0$$

R M
R M
Numerade Educator
00:55

Problem 50

Find a differential equation satisfied by the flow lines of $\mathbf{F}$ (see Exercise 47 ), and solve it to find equations for the flow lines of $\mathbf{F}$. Sketch some typical flow lines and tangent vectors.

$$\mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j}, \quad x>0 \text { and } y>0$$

R M
R M
Numerade Educator
01:17

Problem 51

Discuss the similarities and differences between the concepts "vector field" and "slope field."

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
02:39

Problem 52

In physical applications it is often necessary to deal with vector quantities that depend not only on position in space but also on time. Give some examples and discuss how the concept of a vector field would need to be modified to apply to such situations.

Harshita Goel
Harshita Goel
Numerade Educator