Calculus: Early Transcendentals

Jon Rogawski, Colin Adams, Robert Franzosa

Chapter 16

Line and Surface Integrals - all with Video Answers

Educators

Section 1

Vector Fields

Problem 1

Compute and sketch the vector assigned to the points $P=(1,2)$ and $Q=(-1,-1)$ by the vector field $\mathbf{F}=\left\langle x^{2}, x\right\rangle .$

Linh Vu

Linh Vu

Numerade Educator

Problem 2

Compute and sketch the vector assigned to the points $P=(1,2)$ and $Q=(-1,-1)$ by the vector field $\mathbf{F}=\langle-y, x\rangle .$

Linh Vu

Numerade Educator

Problem 3

Compute and sketch the vector assigned to the points $P=(0,1,1)$ and $Q=(2,1,0)$ by the vector field $\mathbf{F}=\left\langle x y, z^{2}, x\right) .$

Linh Vu

Numerade Educator

Problem 4

Compute the vector assigned to the points $P=(1,1,0)$ and $Q=(2,1,2)$ by the vector fields e $_{r}, \frac{e_{r}}{r},$ and $\frac{e_{r}}{r^{2}}$.

Linh Vu

Numerade Educator

Problem 5

In Exercises $5-12,$ sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle $-3 \leq x \leq 3,-3 \leq y \leq 3$, Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
$$
\mathbf{F}=\langle 1,0\rangle
$$

Linh Vu

Numerade Educator

Problem 6

In Exercises $5-12,$ sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle $-3 \leq x \leq 3,-3 \leq y \leq 3$, Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
$$
\mathbf{F}=(1,1)
$$

Linh Vu

Numerade Educator

Problem 7

In Exercises $5-12,$ sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle $-3 \leq x \leq 3,-3 \leq y \leq 3$, Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
$$
\mathbf{F}=\mathbf{x i}
$$

Linh Vu

Numerade Educator

Problem 8

In Exercises $5-12,$ sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle $-3 \leq x \leq 3,-3 \leq y \leq 3$, Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
$$
\mathbf{F}=y \mathbf{i}
$$

Linh Vu

Numerade Educator

Problem 9

In Exercises $5-12,$ sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle $-3 \leq x \leq 3,-3 \leq y \leq 3$, Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
$$
\mathbf{F}=\langle 0, x\rangle
$$

Linh Vu

Numerade Educator

Problem 10

In Exercises $5-12,$ sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle $-3 \leq x \leq 3,-3 \leq y \leq 3$, Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
$$
\mathbf{F}=x^{2} \mathbf{i}+y \mathbf{j}
$$

Linh Vu

Numerade Educator

Problem 11

In Exercises $5-12,$ sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle $-3 \leq x \leq 3,-3 \leq y \leq 3$, Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
$$
\mathbf{F}=\left\langle\frac{x}{x^{2}+y^{2}}, \frac{y}{x^{2}+y^{2}}\right\rangle
$$

Linh Vu

Numerade Educator

Problem 12

In Exercises $5-12,$ sketch the following planar vector fields by drawing the vectors attached to points with integer coordinates in the rectangle $-3 \leq x \leq 3,-3 \leq y \leq 3$, Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
$$
\mathbf{F}=\left\langle\frac{-y}{\sqrt{x^{2}+y^{2}}}, \frac{x}{\sqrt{x^{2}+y^{2}}}\right\rangle
$$

Linh Vu

Numerade Educator

Problem 13

In Exercises $13-16,$ match each of the following planar vector fields with the corresponding plot in Figure $10 .$
$$
\mathbf{F}=(2, x)
$$

Linh Vu

Numerade Educator

Problem 14

In Exercises $13-16,$ match each of the following planar vector fields with the corresponding plot in Figure $10 .$
$$
\mathbf{F}=(2 x+2, y)
$$

Linh Vu

Numerade Educator

Problem 15

In Exercises $13-16,$ match each of the following planar vector fields with the corresponding plot in Figure $10 .$
$$
\mathbf{F}=(y, \cos x)
$$

Linh Vu

Numerade Educator

Problem 16

In Exercises $13-16,$ match each of the following planar vector fields with the corresponding plot in Figure $10 .$
$$
\mathbf{F}=\langle x+y, x-y\rangle
$$

Linh Vu

Numerade Educator

Problem 17

In Exencises $17-20,$ match each three-dimensional vector field with the corresponding plot in Figure $11 .$
$$
\mathbf{F}=\langle 1,1,1\rangle
$$

Linh Vu

Numerade Educator

Problem 18

In Exencises $17-20,$ match each three-dimensional vector field with the corresponding plot in Figure $11 .$
$$
\mathbf{F}=\langle x, 0, z)
$$

Linh Vu

Numerade Educator

Problem 19

In Exencises $17-20,$ match each three-dimensional vector field with the corresponding plot in Figure $11 .$
$$
\mathbf{F}=\langle x, y, z)
$$

Linh Vu

Numerade Educator

Problem 20

In Exencises $17-20,$ match each three-dimensional vector field with the corresponding plot in Figure $11 .$
$$
\mathbf{F}=\mathbf{e}_{r}
$$

Linh Vu

Numerade Educator

Problem 21

A river 200 meters wide is modeled by the region in the $x y$ -plane given by $-100 \leq x \leq 100 .$ The velocity vector field on the surface of the river is given by $\mathbf{F}=\left\langle-0.05 x, 20-0.0001 x^{2}\right\rangle$ in meters per second. Determine the coordinates of those points that have the maximum speed.

Victor Salazar

Numerade Educator

Problem 22

The velocity vectors in kilometers per hour for the wind speed of a tornado near the ground are given by the vector field $\mathbf{F}=\left\langle\frac{-y}{e^{\left(x^{2}+y^{2}-1\right)^{2}}}, \frac{x}{e^{\left(x^{2}+y^{2}-1\right)^{2}}}\right\rangle .$ Determine the coordinates of those points where the wind speed is the highest.

Ahmad Reda

Numerade Educator

Problem 23

In Exercises $23-30,$ calculate $\operatorname{div}(\mathbf{F})$ and $\operatorname{curl}(\mathbf{F})$.
$$
\mathbf{F}=\langle x, y, z\rangle
$$

Linh Vu

Numerade Educator

Problem 24

In Exercises $23-30,$ calculate $\operatorname{div}(\mathbf{F})$ and $\operatorname{curl}(\mathbf{F})$.
$$
\mathbf{F}=\langle y, z, x\rangle
$$

Linh Vu

Numerade Educator

Problem 25

In Exercises $23-30,$ calculate $\operatorname{div}(\mathbf{F})$ and $\operatorname{curl}(\mathbf{F})$.
$$
\mathbf{F}=\left\langle x-2 z x^{2}, z-x y, z^{2} x^{2}\right\rangle
$$

Linh Vu

Numerade Educator

Problem 26

In Exercises $23-30,$ calculate $\operatorname{div}(\mathbf{F})$ and $\operatorname{curl}(\mathbf{F})$.
$$
\sin (x+z) \mathbf{i}-y e^{x z} \mathbf{k}
$$

Linh Vu

Numerade Educator

Problem 27

In Exercises $23-30,$ calculate $\operatorname{div}(\mathbf{F})$ and $\operatorname{curl}(\mathbf{F})$.
$$
\mathbf{F}=\langle y z, x z, x y\rangle
$$

Linh Vu

Numerade Educator

Problem 28

In Exercises $23-30,$ calculate $\operatorname{div}(\mathbf{F})$ and $\operatorname{curl}(\mathbf{F})$.
$$
\mathbf{F}=\left\langle\frac{y}{x}, \frac{y}{z}, \frac{z}{x}\right)
$$

Linh Vu

Numerade Educator

Problem 29

In Exercises $23-30,$ calculate $\operatorname{div}(\mathbf{F})$ and $\operatorname{curl}(\mathbf{F})$.
$$
\mathbf{F}=\left\langle e^{y}, \sin x, \cos x\right\rangle
$$

Linh Vu

Numerade Educator

Problem 30

In Exercises $23-30,$ calculate $\operatorname{div}(\mathbf{F})$ and $\operatorname{curl}(\mathbf{F})$.
$$
\mathbf{F}=\left\langle\frac{x}{x^{2}+y^{2}}, \frac{y}{x^{2}+y^{2}}, 0\right\rangle
$$

Linh Vu

Numerade Educator

Problem 31

In Exercises $3 I-37,$ prove the identities assuming that the appropriate partial derivatives exist and are continuous.
$$
\operatorname{div}(\mathbf{F}+\mathbf{G})=\operatorname{div}(\mathbf{F})+\operatorname{div}(\mathbf{G})
$$

Linh Vu

Numerade Educator

Problem 32

In Exercises $3 I-37,$ prove the identities assuming that the appropriate partial derivatives exist and are continuous.
$$
\operatorname{curl}(\mathbf{F}+\mathbf{G})=\operatorname{curl}(\mathbf{F})+\operatorname{curl}(\mathbf{G})
$$

Linh Vu

Numerade Educator

Problem 33

In Exercises $3 I-37,$ prove the identities assuming that the appropriate partial derivatives exist and are continuous.
$$
\operatorname{div} \operatorname{curl}(\mathbf{F})=0
$$

Linh Vu

Numerade Educator

Problem 34

In Exercises $3 I-37,$ prove the identities assuming that the appropriate partial derivatives exist and are continuous.
$$
\operatorname{div}(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot \operatorname{curl}(\mathbf{F})-\mathbf{F}+\operatorname{curl}(\mathbf{G})
$$

Linh Vu

Numerade Educator

Problem 35

In Exercises $3 I-37,$ prove the identities assuming that the appropriate partial derivatives exist and are continuous.
If $f$ is a scalar function, then div $(f \mathbf{F})=f \operatorname{div}(\mathbf{F})+\mathbf{F} \cdot \nabla f$.

Linh Vu

Numerade Educator

Problem 36

In Exercises $3 I-37,$ prove the identities assuming that the appropriate partial derivatives exist and are continuous.
$$
\operatorname{curl}(f \mathbf{F})=f \operatorname{curl}(\mathbf{F})+(\nabla f) \times \mathbf{F}
$$

Linh Vu

Numerade Educator

Problem 37

In Exercises $3 I-37,$ prove the identities assuming that the appropriate partial derivatives exist and are continuous.
$$
\operatorname{div}(\nabla f \times \nabla g)=0
$$

Linh Vu

Numerade Educator

Problem 38

Find (by inspection) a potential function for $\mathbf{F}=(x, 0)$ and prove that $\mathbf{G}=(y, 0)$ is not conservative.

Linh Vu

Numerade Educator

Problem 39

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=\langle x, y\rangle
$$

Linh Vu

Numerade Educator

Problem 40

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=\langle y, x\rangle
$$

Linh Vu

Numerade Educator

Problem 41

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=\left\langle y^{2} z, 1+2 x y z, x y^{2}\right\rangle
$$

Linh Vu

Numerade Educator

Problem 42

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=(y z, x z, y)
$$

Linh Vu

Numerade Educator

Problem 43

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=\left\langle y e^{x y}, x e^{x y}\right\rangle
$$

Linh Vu

Numerade Educator

Problem 44

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=\left(2 x y z, x^{2} z, x^{2} y z\right)
$$

Linh Vu

Numerade Educator

Problem 45

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=\left\langle y z^{2}, x z^{2}, 2 x y z\right\rangle
$$

Linh Vu

Numerade Educator

Problem 46

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=\left\langle 2 x z e^{x^{2}}, 0, e^{x^{2}}\right\rangle
$$

Linh Vu

Numerade Educator

Problem 47

In Exercises 39 . 47 , find a potential function for the vector field $\mathbf{F}$ by inspection or show that one does not exist.
$$
\mathbf{F}=\langle y z \cos (x y z), x z \cos (x y z), x y \cos (x y z)\rangle
$$

Linh Vu

Numerade Educator

Problem 48

Find potential functions for $\mathbf{F}=\frac{\mathbf{e}_{r}}{r^{3}}$ and $\mathbf{G}=\frac{\mathbf{e}_{r}}{r^{4}}$ in $\mathbf{R}^{3}$. Hint: See Example $8 .$

Chris Trentman

Numerade Educator

Problem 49

Show that $\mathbf{F}=(3,1,2)$ is conservative. Then prove more generally that any constant vector field $\mathbf{F}=\langle a, b, c\rangle$ is conservative.

Linh Vu

Numerade Educator

Problem 50

Let $\varphi=\ln r,$ where $r=\sqrt{x^{2}+y^{2}}$. Express $\nabla \varphi$ in terms of the unit radial vector $\mathbf{e}_{r}$ in $\mathbf{R}^{2}$.

Linh Vu

Numerade Educator

Problem 51

For $P=(a, b),$ we define the unit radial vector field based at $P$ :
$$
\mathbf{e}_{P}=\frac{(x-a, y-b)}{\sqrt{(x-a)^{2}+(y-b)^{2}}}
$$
(a) Verify that ep is a unit vector field.
(b) Calculate e $_{P}(1,1)$ for $P=(3,2)$.
(c) Find a potential function for $\mathbf{e}_{\mathrm{P}}$.

Harshita Goel

Numerade Educator

Problem 52

Which of (A) or (B) in Figure 12 is the contour plot of a potential function for the vector field $\mathbf{F}$ ? Recall that the gradient vectors are perpendicular to the level curves.

Jeffrey Utley

Numerade Educator

Problem 53

Which of (A) or (B) in Figure 13 is the contour plot of a potential function for the vector field $\mathbf{F} ?$

Linh Vu

Numerade Educator

Problem 54

Match each of these descriptions with a vector field in Figure 14 .
(a) The gravitational field created by two planets of equal mass located at $P$ and $Q$
(b) The electrostatic field created by two equal and opposite charges located at $P$ and $Q$ (representing the force on a negative test charge:
opposite charges attract and like charges repel)

Mayukh Banik

Numerade Educator

Problem 55

In this exercise, we show that the vector field $\mathbf{F}$ in Figure 15 is not conservative. Explain the following statements:
(a) If a potential function $f$ for $\mathbf{F}$ exists, then the level curves of $f$ must be vertical lines.
(b) If a potential function $f$ for $\mathbf{F}$ exists, then the level curves of $f$ must grow farther apart as $y$ increases.
(c) Explain why
(a) and (b) are incompatible, and hence $f$ cannot exist.

Chris Trentman

Numerade Educator

Problem 56

Show that any vector field of the form
$$
\mathbf{F}=\langle f(x), g(y), h(z))
$$
has a potential function. Assume that $f, g,$ and $h$ are continuous.

Linh Vu

Numerade Educator

Problem 57

Let $\mathcal{D}$ be a disk in $\mathbf{R}^{2}$. This exercise shows that if $$ \nabla f(x, y)=0 $$ for all $(x, y)$ in $\mathcal{D},$ then $f$ is constant. Consider points $P=(a, b)$, $Q=(c, d),$ and $R=(c, b)$ as in Figure $16 .$
(a) Use single-variable calculus to show that $f$ is constant along the segments $\overline{P R}$ and $\overline{R Q}$
(b) Conclude that $f(P)=f(Q)$ for any two points $P, Q \in \mathcal{D}$.

Hoan Nguyen

Numerade Educator