Calculus Early Transcendentals

Howard Anton, Irl C. Bivens, Stephen Davis

Chapter 15

Topics In Vector Calculus - all with Video Answers

Educators

Section 1

Vector Fields

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

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

Numerade Educator

Problem 3

$3-4$ Determine whether the statement about the vector field $\mathbf{F}(x, y)$ is true or false. If false, explain why.
$\mathbf{F}(x, y)=x^{2} \mathbf{i}-y \mathbf{j}$
(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

Numerade Educator

Problem 4

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

Ravindra Yadav

Numerade Educator

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

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

Problem 7

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

Aman Gupta

Numerade Educator

Problem 8

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

Aman Gupta

Numerade Educator

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

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

Problem 11

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.

Aman Gupta

Numerade Educator

Problem 12

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.

Aman Gupta

Numerade Educator

Problem 13

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

Vysakh M

Numerade Educator

Problem 14

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

Aman Gupta

Numerade Educator

Problem 15

Confirm that $\phi$ is a potential function for $\mathbf{F}(\mathbf{r})$ on some region, and state the region.(a) $\phi(x, y)=\tan ^{-1} x y$
$\quad \mathbf{F}(x, y)=\frac{y}{1+x^{2} y^{2}} \mathbf{i}+\frac{x}{1+x^{2} y^{2}} \mathbf{j}$
(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

Numerade Educator

Problem 16

Confirm that $\phi$ is a potential function for $\mathbf{F}(\mathbf{r})$ on some region, and state the region.(a) $\phi(x, y)=2 y^{2}+3 x^{2} y-x y^{3}$
$\mathbf{F}(x, y)=\left(6 x y-y^{3}\right) \mathbf{i}+\left(4 y+3 x^{2}-3 x y^{2}\right) \mathbf{j}$
(b) $\phi(x, y, z)=x \sin z+y \sin x+z \sin y$
$\mathbf{F}(x, y, z)=(\sin z+y \cos x) \mathbf{i}+(\sin x+z \cos y) \mathbf{j}$
$+(\sin y+x \cos z) \mathbf{k}$

Aman Gupta

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

Problem 23

Find $\nabla \cdot(\mathbf{F} \times \mathbf{G})$
$\mathbf{F}(x, y, z)=2 x \mathbf{i}+\mathbf{j}+4 y \mathbf{k}$
$\mathbf{G}(x, y, z)=x \mathbf{i}+y \mathbf{j}-z \mathbf{k}$

Aman Gupta

Numerade Educator

Problem 24

Find $\nabla \cdot(\mathbf{F} \times \mathbf{G})$$\mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}$
$\mathbf{G}(x, y, z)=x y \mathbf{j}+x y z \mathbf{k}$

Aman Gupta

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

Problem 29

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

Carson Merrill

Numerade Educator

Problem 30

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

Carson Merrill

Numerade Educator

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

Numerade Educator

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

R M

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

Problem 41

Verify that the radius vector $\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ has the stated proper.
(a) $\operatorname{curl} \mathbf{r}=\mathbf{0}$
(b) $\nabla\|\mathbf{r}\|=\frac{\mathbf{r}}{\|\mathbf{r}\|}$

Aman Gupta

Numerade Educator

Problem 42

Verify that the radius vector $\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ has the stated proper.(a) $\operatorname{div} \mathbf{r}=3$
(b) $\nabla \frac{1}{\|\mathbf{r}\|}=-\frac{\mathbf{r}}{\|\mathbf{r}\|^{3}}$

Aman Gupta

Numerade Educator

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

Numerade Educator

Problem 44

(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 (a) to show that
$$
\nabla^{2} f(r)=2 \frac{f^{\prime}(r)}{r}+f^{\prime \prime}(r)
$$

Aman Gupta

Numerade Educator

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

Numerade Educator

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

Numerade Educator

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

Numerade Educator

Problem 48

$48-50$ 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.$\mathbf{F}(x, y)=\mathbf{i}+x \mathbf{j}$

R M

Numerade Educator

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

Numerade Educator

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

R M

Numerade Educator

Problem 51

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

Carl David Cepeda

Carl David Cepeda

Numerade Educator

Problem 52

Writing 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.

Carl David Cepeda

Carl David Cepeda

Numerade Educator