Calculus Early Transcendentals 2

James Stewart

Chapter 16

Vector Calculus - all with Video Answers

Educators

Section 1

Vector Fields

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

Yujian Zeng

Numerade Educator

Problem 2

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$
$$
\mathbf{F}(x, y)=\frac{1}{2} x \mathbf{i}+y \mathbf{j}
$$

Frank Lin

Numerade Educator

Problem 3

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$
$$
\mathbf{F}(x, y)=-\frac{1}{2} \mathbf{i}+(y-x) \mathbf{j}
$$

Frank Lin

Numerade Educator

Problem 4

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$
$$
\mathbf{F}(x, y)=y \mathbf{i}+(x+y) \mathbf{j}
$$

Frank Lin

Numerade Educator

Problem 5

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$
$$
\mathbf{F}(x, y)=\frac{y \mathbf{i}+x \mathbf{j}}{\sqrt{x^{2}+y^{2}}}
$$

Frank Lin

Numerade Educator

Problem 6

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$
$$
\mathbf{F}(x, y)=\frac{y \mathbf{i}-x \mathbf{j}}{\sqrt{x^{2}+y^{2}}}
$$

Frank Lin

Numerade Educator

Problem 7

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$
$$
\mathbf{F}(x, y, z)=\mathbf{i}
$$

Elliott Walker

Numerade Educator

Problem 8

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

Problem 9

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$
$$
\mathbf{F}(x, y, z)=-y \mathbf{i}
$$

Frank Lin

Numerade Educator

Problem 10

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure $9 .$
$$
\mathbf{F}(x, y, z)=\mathbf{i}+\mathbf{k}
$$

Frank Lin

Numerade Educator

Problem 11

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

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

Cameron Bunney

Numerade Educator

Problem 12

Match the vector fields $F$ with the plots labeled I-IV. Give reasons for your choices.
$$
\mathbf{F}(x, y)=\langle y, x-y\rangle
$$

Frank Lin

Numerade Educator

Problem 13

Match the vector fields $F$ with the plots labeled I-IV. Give reasons for your choices.
$$
\mathbf{F}(x, y)=\langle y, y+2\rangle
$$

Frank Lin

Numerade Educator

Problem 14

Match the vector fields $F$ with the plots labeled I-IV. Give reasons for your choices.
$$
\mathbf{F}(x, y)=\langle\cos (x+y), x\rangle
$$

Frank Lin

Numerade Educator

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

Problem 16

Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled I-IV. Give reasons for your choices.
$$
\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+z \mathbf{k}
$$

Frank Lin

Numerade Educator

Problem 17

Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled I-IV. Give reasons for your choices.
$$
\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+3 \mathbf{k}
$$

Frank Lin

Numerade Educator

Problem 18

Match the vector fields $\mathbf{F}$ on $\mathbb{R}^{3}$ with the plots labeled I-IV. Give reasons for your choices.
$$
\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}
$$

Frank Lin

Numerade Educator

Problem 19

If you have a CAS that plots vector fields (the command is fie $1 \mathrm{d}$ p 1 ot in Maple and PlotvectorField or vectorP1ot in Mathematica), use it to plot
$$
\mathbf{F}(x, y)=\left(y^{2}-2 x y\right) \mathbf{i}+\left(3 x y-6 x^{2}\right) \mathbf{j}
$$
$$
\begin{array}{l}{\text { Explain the appearance by finding the set of points }(x, y)} \\ {\text { such that } \mathbf{F}(x, y)=\mathbf{0} .}\end{array}
$$

Shu Naito

Numerade Educator

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

Problem 21

$21-24$ Find the gradient vector field of $f$
$$
f(x, y)=y \sin (x y)
$$

Shu Naito

Numerade Educator

Problem 22

Find the gradient vector field of $f$
$$
f(s, t)=\sqrt{2 s+3 t}
$$

Frank Lin

Numerade Educator

Problem 23

Find the gradient vector field of $f$
$$
f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}
$$

Frank Lin

Numerade Educator

Problem 24

Find the gradient vector field of $f$
$$
f(x, y, z)=x^{2} y e^{y / x}
$$

Frank Lin

Numerade Educator

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

Problem 26

Find the gradient vector field $\nabla f$ of $f$ and sketch it.
$$
f(x, y)=\frac{1}{2}\left(x^{2}-y^{2}\right)
$$

Frank Lin

Numerade Educator

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

Problem 28

Plot the gradient vector field of $f$ together with a contour map of $f$. Explain how they are related to each other.
$$
f(x, y)=\cos x-2 \sin y
$$

Frank Lin

Numerade Educator

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

Problem 30

Match the functions $f$ with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices.
$$
f(x, y)=x(x+y)
$$

Frank Lin

Numerade Educator

Problem 31

Match the functions $f$ with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices.
$$
f(x, y)=(x+y)^{2}
$$

Frank Lin

Numerade Educator

Problem 32

Match the functions $f$ with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices.
$$
f(x, y)=\sin \sqrt{x^{2}+y^{2}}
$$

Frank Lin

Numerade Educator

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

Problem 34

At time $t=1,$ a particle is located at position $(1,3) .$ If it moves in a velocity field
$$
\mathbf{F}(x, y)=\left\langle x y-2, y^{2}-10\right\rangle
$$
$$
\text { find its approximate location at time } t=1.05
$$

Cameron Bunney

Numerade Educator

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.
$$
\begin{array}{l}{\text { (a) Use a sketch of the vector field } \mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j} \text { to }} \\ {\text { draw some flow lines. From your sketches, can you }} \\ {\text { guess the equations of the flow lines? }} \\ {\text { (b) If parametric equations of a flow line are } x=x(t)} \\ {y=y(t), \text { explain why these functions satisfy the differ- }} \\ {\text { ential equations } d x / d t=x \text { and } d y / d t=-y . \text { Then solve }} \\ {\text { the differential equations to find an equation of the flow }} \\ {\text { line that passes through the point }(1,1) .}\end{array}
$$

Yujian Zeng

Numerade Educator

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