Calculus for AP

Jon Rogawski & Colin Adams

Chapter 16

LINE AND SURFACE INTEGRALS - all with Video Answers

Educators

GA

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 $\mathbf{F}=\left\langle x y, z^{2}, x\right\rangle$.

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 $\frac{\mathrm{er}_{r}}{r^{2}}$.

Linh Vu

Numerade Educator

Problem 5

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

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.
$\mathbf{F}=\langle 1,1\rangle$

Linh Vu

Numerade Educator

Problem 7

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.
$\mathbf{F}=x \mathbf{i}$

Linh Vu

Numerade Educator

Problem 8

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

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

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

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

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

Match each of the following planar vector fields with the corresponding plot in Figure 12.
$\mathbf{F}=\langle 2, x\rangle$

Linh Vu

Numerade Educator

Problem 14

Match each of the following planar vector fields with the corresponding plot in Figure 12.
$\mathbf{F}=\langle 2 x+2, y\rangle$

Linh Vu

Numerade Educator

Problem 15

Match each of the following planar vector fields with the corresponding plot in Figure 12.
$\mathbf{F}=\langle y, \cos x\rangle$

Linh Vu

Numerade Educator

Problem 16

Match each of the following planar vector fields with the corresponding plot in Figure 12.
$\mathbf{F}=\langle x+y, x-y\rangle$

Linh Vu

Numerade Educator

Problem 17

Match each three-dimensional vector field with the corresponding plot in Figure 13.
$\mathbf{F}=\langle 1,1,1\rangle$

Linh Vu

Numerade Educator

Problem 18

Match each three-dimensional vector field with the corresponding plot in Figure 13.
$\mathbf{F}=\langle x, 0, z\rangle$

Linh Vu

Numerade Educator

Problem 19

Match each three-dimensional vector field with the corresponding plot in Figure 13.
$\mathbf{F}=\langle x, y, z\rangle$

Linh Vu

Numerade Educator

Problem 20

Match each three-dimensional vector field with the corresponding plot in Figure 13.
$\mathbf{F}=\mathbf{e}_{r}$

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 $\mathbf{F}=\left\langle\frac{-x}{20}, 20-\frac{x^{2}}{1000}\right\rangle$ in meters per second $(\mathrm{m} / \mathrm{s}) .$ 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 $\mathbf{F}=\left\langle\frac{-y}{\left(x^{2}+y^{2}-1\right)^{2}}, \frac{x}{e^{\left(x^{2}+y^{2}-1\right)^{2}}}\right\rangle \cdot$ Determine the coordinates of those points where the wind speed is the highest.

Ahmad Reda

Numerade Educator

Problem 23

Calculate div(F) and curl(F).
$\mathbf{F}=\left\langle x y, y z, y^{2}-x^{3}\right\rangle$

Linh Vu

Numerade Educator

Problem 24

Calculate div(F) and curl(F).
$x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$

Linh Vu

Numerade Educator

Problem 25

Calculate div(F) and curl(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

Calculate div(F) and curl(F).
$\sin (x+z) \mathbf{i}-y e^{x z} \mathbf{k}$

Linh Vu

Numerade Educator

Problem 27

Calculate div(F) and curl(F).
$\mathbf{F}=\left\langle z-y^{2}, x+z^{3}, y+x^{2}\right\rangle$

Linh Vu

Numerade Educator

Problem 28

Calculate div(F) and curl(F).
$\mathbf{F}=\left\langle\frac{y}{x}, \frac{y}{z}, \frac{z}{x}\right\rangle$

Linh Vu

Numerade Educator

Problem 29

Calculate div(F) and curl(F).
$\mathbf{F}=\left\langle e^{y}, \sin x, \cos x\right\rangle$

Linh Vu

Numerade Educator

Problem 30

Calculate div(F) and curl(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

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

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

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

Prove the identities assuming that the appropriate partial derivatives exist and are continuous.
$\operatorname{div}(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot \operatorname{curl}(\mathbf{F})-\mathbf{F} \cdot \operatorname{curl}(\mathbf{G})$

Linh Vu

Numerade Educator

Problem 35

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

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

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 $\mathbf{F}=\langle x, 0\rangle$ and prove that $\mathbf{G}=\langle y, 0\rangle$ is not conservative.

Linh Vu

Numerade Educator

Problem 39

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

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

Linh Vu

Numerade Educator

Problem 41

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 42

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

Linh Vu

Numerade Educator

Problem 43

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 44

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 45

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 46

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 47

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

Linh Vu

Numerade Educator

Problem 48

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 49

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

Harshita Goel

Numerade Educator

Problem 50

Which of (A) or (B) in Figure 14 is the contour plot of a potential function for the vector field F? Recall that the gradient vectors are perpendicular to the level curves.

Jeffrey Utley

Numerade Educator

Problem 51

Which of $(\mathrm{A})$ or $(\mathrm{B})$ in Figure 15 is the contour plot of a potential function for the vector field $\mathbf{F}$ ?

Linh Vu

Numerade Educator

Problem 52

Match each of these descriptions with a vector field in Figure 16:
(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)

Prashant Bana

Numerade Educator

Problem 53

In this exercise, we show that the vector field F in Figure 17 is not conservative. Explain the following statements:
(a) If a potential function $f$ for $F$ exists, then the level curves of $f$ must be vertical lines.
(b) If a potential function $f$ for $F$ 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 54

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

Linh Vu

Numerade Educator

Problem 55

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 18.
(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