Calculus Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillet

Chapter 17

Vector Calculus - all with Video Answers

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Section 1

Vector Fields

Problem 1

How is a vector field $\mathbf{F}=\langle f, g, h\rangle$ used to describe the motion of air at one instant in time?

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Problem 2

Sketch the vector field $\mathbf{F}=\langle x, y\rangle.$

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Problem 3

How do you graph the vector field $\mathbf{F}=\langle f(x, y), g(x, y)\rangle ?$

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Problem 4

Given a differentiable, scalar-valued function $\varphi,$ why is the gradient of $\varphi$ a vector field?

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Problem 5

Interpret the gradient field of the temperature function $T=f(x, y)$

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Problem 6

Show that all the vectors in vector field $\mathbf{F}=\frac{\sqrt{2}\langle x, y\rangle}{\sqrt{x^{2}+y^{2}}}$ have the same length, and state the length of the vectors.

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Problem 7

Sketch a few representative vectors of vector field $\mathbf{F}=\langle 0,1\rangle$ along the line $y=2.$

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Problem 8

Sketch the following vector fields.
$$\mathbf{F}=\langle 1,0\rangle$$

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Problem 9

Sketch the following vector fields.
$$\mathbf{F}=\langle-1,1\rangle$$

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Problem 10

Sketch the following vector fields.
$$\mathbf{F}=\langle 1, y\rangle$$

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Problem 11

Sketch the following vector fields.
$$\mathbf{F}=\langle x, 0\rangle$$

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Problem 12

Sketch the following vector fields.
$$\mathbf{F}=\langle-x,-y\rangle$$

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Problem 13

Sketch the following vector fields.
$$\mathbf{F}=\langle x,-y\rangle$$

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Problem 14

Sketch the following vector fields.
$$\mathbf{F}=\langle 2 x, 3 y\rangle$$

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Problem 15

Sketch the following vector fields.
$$\mathbf{F}=\langle y,-x\rangle$$

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Problem 16

Sketch the following vector fields.
$$\mathbf{F}=\langle x+y, y\rangle$$

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Problem 17

Sketch the following vector fields.
$$\mathbf{F}=\langle x, y-x\rangle$$

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Problem 18

Sketch the following vector fields.
$$\mathbf{F}=\left\langle\frac{x}{\sqrt{x^{2}+y^{2}}}, \frac{y}{\sqrt{x^{2}+y^{2}}}\right\rangle$$

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Problem 19

Sketch the following vector fields.
$$\mathbf{F}=\left\langle e^{-x}, 0\right\rangle$$

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Problem 20

Sketch the following vector fields.
$$\mathbf{F}=\langle 0,0,1\rangle$$

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Problem 21

Sketch the following vector fields.
$$\mathbf{F}=\langle 1,0, z\rangle$$

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Problem 22

Sketch the following vector fields.
$$\mathbf{F}=\frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$$

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Problem 23

Sketch the following vector fields.
$$\mathbf{F}=\langle y,-x, 0\rangle$$

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Problem 24

Matching vector fields with graphs Match vector fields a-d with graphs A-D.
a. $\mathbf{F}=\left\langle 0, x^{2}\right\rangle$
b. $\mathbf{F}=\langle x-y, x\rangle$
c. $\mathbf{F}=\langle 2 x,-y\rangle$
d. $\mathbf{F}=\langle y, x\rangle$
a. (Graph cant copy)
b. (Graph cant copy)
c. (Graph cant copy)
d. (Graph cant copy)

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Problem 25

For the vector field $\mathbf{F}$ and curve $C$, complete the following:
a. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is tangent to $C$.
b. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is normal to $C$
c. Sketch $C$ and a few representative vectors of $\mathbf{F}$ on $C$.
$$\mathbf{F}=\left\langle\frac{1}{2}, 0\right\rangle ; C=\left\{(x, y): y-x^{2}=1\right\}$$

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Problem 26

For the vector field $\mathbf{F}$ and curve $C$, complete the following:
a. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is tangent to $C$.
b. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is normal to $C$
c. Sketch $C$ and a few representative vectors of $\mathbf{F}$ on $C$.

$$\mathbf{F}=\left\langle\frac{y}{2},-\frac{x}{2}\right\rangle ; C=\left\{(x, y): y-x^{2}=1\right\}$$

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Problem 27

For the vector field $\mathbf{F}$ and curve $C$, complete the following:
a. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is tangent to $C$.
b. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is normal to $C.$
c. Sketch $C$ and a few representative vectors of $\mathbf{F}$ on $C$.

$$\mathbf{F}=\langle x, y\rangle ; C=\left\{(x, y): x^{2}+y^{2}=4\right\}$$

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Problem 28

For the vector field $\mathbf{F}$ and curve $C$, complete the following:
a. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is tangent to $C$.
b. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is normal to $C$
c. Sketch $C$ and a few representative vectors of $\mathbf{F}$ on $C$.

$$\mathbf{F}=\langle y,-x\rangle ; C=\left\{(x, y): x^{2}+y^{2}=1\right\}$$

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Problem 29

For the vector field $\mathbf{F}$ and curve $C$, complete the following:
a. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is tangent to $C$.
b. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is normal to $C$
c. Sketch $C$ and a few representative vectors of $\mathbf{F}$ on $C$.

$$\mathbf{F}=\langle x, y\rangle ; C=\{(x, y) ; x=1\}$$

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Problem 30

For the vector field $\mathbf{F}$ and curve $C$, complete the following:
a. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is tangent to $C$.
b. Determine the points (if any) along the curve C at which the vector field $\mathbf{F}$ is normal to $C$
c. Sketch $C$ and a few representative vectors of $\mathbf{F}$ on $C$.

$$\mathbf{F}=\langle y, x\rangle ; C=\left\{(x, y): x^{2}+y^{2}=1\right\}$$

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Problem 31

Specify the component functions of a vector field $\mathbf{F}$ in $\mathbb{R}^{2}$ with the following properties. Solutions are not unique.
$\mathbf{F}$ is everywhere normal to the line $y=x.$

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Problem 32

Specify the component functions of a vector field $\mathbf{F}$ in $\mathbb{R}^{2}$ with the following properties. Solutions are not unique.
$\mathbf{F}$ is everywhere normal to the line $x=2.$

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Problem 33

Specify the component functions of a vector field $\mathbf{F}$ in $\mathbb{R}^{2}$ with the following properties. Solutions are not unique.

At all points except $(0,0),$ F has unit magnitude and points away from the origin along radial lines.

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Problem 34

Specify the component functions of a vector field $\mathbf{F}$ in $\mathbb{R}^{2}$ with the following properties. Solutions are not unique.

The flow of $\mathbf{F}$ is counterclockwise around the origin, increasing in magnitude with distance from the origin.

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Problem 35

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the follow. ing potential functions $\varphi$.
$$\varphi(x, y)=x^{2} y-y^{2} x$$

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Problem 36

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the follow. ing potential functions $\varphi$.
$$\varphi(x, y)=\sqrt{x y}$$

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Problem 37

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the follow. ing potential functions $\varphi$.
$$\varphi(x, y)=x / y$$

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Problem 38

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the follow. ing potential functions $\varphi$.
$$\varphi(x, y)=\tan ^{-1}(x / y)$$

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Problem 39

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the follow. ing potential functions $\varphi$.
$$\varphi(x, y, z)=\frac{x^{2}+y^{2}+z^{2}}{2}$$

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Problem 40

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the follow. ing potential functions $\varphi$.
$$\varphi(x, y, z)=\ln \left(1+x^{2}+y^{2}+z^{2}\right)$$

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Problem 41

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the follow. ing potential functions $\varphi$.
$$\varphi(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}$$

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Problem 42

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the follow. ing potential functions $\varphi$.
$$\varphi(x, y, z)=e^{-z} \sin (x+y)$$

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Problem 43

For the potential function $\varphi$ and points $A, B, C,$ and D on the level curve $\varphi(x, y)=0,$ complete the following steps.
a. Find the gradient field $\mathbf{F}=\nabla \varphi.$
b. Evaluate $\mathbf{F}$ at the points $A, B, C,$ and $D.$
c. Plot the level curve $\varphi(x, y)=0$ and the vectors $\mathbf{F}$ at the points $A$ $B, C,$ and $D.$

$$\varphi(x, y)=y-2 x ; A(-1,-2), B(0,0), C(1,2), \text { and } D(2,4)$$

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Problem 44

For the potential function $\varphi$ and points $A, B, C,$ and D on the level curve $\varphi(x, y)=0,$ complete the following steps.
a. Find the gradient field $\mathbf{F}=\nabla \varphi.$
b. Evaluate $\mathbf{F}$ at the points $A, B, C,$ and $D.$
c. Plot the level curve $\varphi(x, y)=0$ and the vectors $\mathbf{F}$ at the points $A$ $B, C,$ and $D.$

$$\begin{aligned}
&\varphi(x, y)=\frac{1}{2} x^{2}-y ; A(-2,2), B(-1,1 / 2), C(1,1 / 2), \text { and } D(2,2)
\end{aligned}$$

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Problem 45

For the potential function $\varphi$ and points $A, B, C,$ and D on the level curve $\varphi(x, y)=0,$ complete the following steps.
a. Find the gradient field $\mathbf{F}=\nabla \varphi.$
b. Evaluate $\mathbf{F}$ at the points $A, B, C,$ and $D.$
c. Plot the level curve $\varphi(x, y)=0$ and the vectors $\mathbf{F}$ at the points $A$ $B, C,$ and $D.$

$$\begin{aligned}
&\varphi(x, y)=-y+\sin x ; A(\pi / 2,1), B(\pi, 0), C(3 \pi / 2,-1)\text { and } D(2 \pi, 0)
\end{aligned}$$

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Problem 46

For the potential function $\varphi$ and points $A, B, C,$ and D on the level curve $\varphi(x, y)=0,$ complete the following steps.
a. Find the gradient field $\mathbf{F}=\nabla \varphi.$
b. Evaluate $\mathbf{F}$ at the points $A, B, C,$ and $D.$
c. Plot the level curve $\varphi(x, y)=0$ and the vectors $\mathbf{F}$ at the points $A$ $B, C,$ and $D.$

$$\begin{aligned}
&\varphi(x, y)=\frac{32-x^{4}-y^{4}}{32} ; A(2,2), B(-2,2), C(-2,-2) \text { and } D(2,-2)
\end{aligned}$$

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Problem 47

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the potential function $\varphi .$ Sketch a few level curves of $\varphi$ and a few vectors of $\mathbf{F}$.
$$\varphi(x, y)=x^{2}+y^{2}, \text { for } x^{2}+y^{2} \leq 16$$

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Problem 48

Find the gradient field $\mathbf{F}=\nabla \varphi$ for the potential function $\varphi .$ Sketch a few level curves of $\varphi$ and a few vectors of $\mathbf{F}$.
$$\varphi(x, y)=x+y, \text { for }|x| \leq 2,|y| \leq 2$$

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Problem 49

Consider the following potential functions and the graphs of their equipotential curves.
a. Find the associated gradient field $\mathbf{F}=\nabla \varphi.$
b. Show that the vector field is orthogonal to the equipotential curve at the point $(1,1) .$ Illustrate this result on the figure.
c. Show that the vector field is orthogonal to the equipotential curve at all points $(x, y).$
d. Sketch two flow curves representing $\mathbf{F}$ that are everywhere orthogonal to the equipotential curves.
$$\varphi(x, y)=2 x+3 y$$
(Graph cant copy)

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Problem 50

Consider the following potential functions and the graphs of their equipotential curves.
a. Find the associated gradient field $\mathbf{F}=\nabla \varphi.$
b. Show that the vector field is orthogonal to the equipotential curve at the point $(1,1) .$ Illustrate this result on the figure.
c. Show that the vector field is orthogonal to the equipotential curve at all points $(x, y).$
d. Sketch two flow curves representing $\mathbf{F}$ that are everywhere orthogonal to the equipotential curves.
$$\varphi(x, y)=x+y^{2}$$
(Graph cant copy)

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Problem 51

Consider the following potential functions and the graphs of their equipotential curves.
a. Find the associated gradient field $\mathbf{F}=\nabla \varphi.$
b. Show that the vector field is orthogonal to the equipotential curve at the point $(1,1) .$ Illustrate this result on the figure.
c. Show that the vector field is orthogonal to the equipotential curve at all points $(x, y).$
d. Sketch two flow curves representing $\mathbf{F}$ that are everywhere orthogonal to the equipotential curves.
$$\varphi(x, y)=e^{x-y}$$
(Graph cant copy)

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Problem 52

Consider the following potential functions and the graphs of their equipotential curves.
a. Find the associated gradient field $\mathbf{F}=\nabla \varphi.$
b. Show that the vector field is orthogonal to the equipotential curve at the point $(1,1) .$ Illustrate this result on the figure.
c. Show that the vector field is orthogonal to the equipotential curve at all points $(x, y).$
d. Sketch two flow curves representing $\mathbf{F}$ that are everywhere orthogonal to the equipotential curves.
$$\varphi(x, y)=x^{2}+2 y^{2}$$
(Graph cant copy)

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Problem 53

Determine whether the following statements are true and give an explanation or counterexample.
a. The vector field $\mathbf{F}=\left\langle 3 x^{2}, 1\right\rangle$ is a gradient field for both $\varphi_{1}(x, y)=x^{3}+y$ and $\varphi_{2}(x, y)=y+x^{3}+100.$
b. The vector field $\mathbf{F}=\frac{\langle y, x\rangle}{\sqrt{x^{2}+y^{2}}}$ is constant in direction and magnitude on the unit circle.
c. The vector field $\mathbf{F}=\frac{\langle y, x\rangle}{\sqrt{x^{2}+y^{2}}}$ is neither a radial field nor a rotation field.

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Problem 54

The electric field in the xy-plane due to a point charge at (0,0) is a gradient field with a potential function $V(x, y)=\frac{k}{\sqrt{x^{2}+y^{2}}},$ where $k>0$ is a physical constant.
a. Find the components of the electric field in the $x$ - and $y$ -directions, where $\mathbf{E}(x, y)=-\nabla V(x, y).$
b. Show that the vectors of the electric field point in the radial direction (outward from the origin) and the radial component of $\mathbf{E}$ can be expressed as $E_{r}=\frac{k}{r^{2}},$ where $r=\sqrt{x^{2}+y^{2}}.$
c. Show that the vector field is orthogonal to the equipotential curves at all points in the domain of $V$

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Problem 55

The electric field in the xy-plane due to an infinite line of charge along the z-axis is a gradient field with a potential function $V(x, y)=c \ln \left(\frac{r_{0}}{\sqrt{x^{2}+y^{2}}}\right)$ where $c>0$ is a constant and $r_{0}$ is a reference distance at which the potential is assumed to be 0 (see figure).
a. Find the components of the electric field in the $x$ - and $y$ -directions, where $\mathbf{E}(x, y)=-\nabla V(x, y).$
b. Show that the electric field at a point in the $x y$ -plane is directed outward from the origin and has magnitude $|\mathbf{E}|=c / r,$ where $r=\sqrt{x^{2}+y^{2}}.$
c. Show that the vector field is orthogonal to the equipotential curves at all points in the domain of $V$

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Problem 56

The gravitational force on a point mass $m$ due to a point mass $M$ is a gradient field with potential $U(r)=G M m / r,$ where $G$ is the gravitational constant and $r=\sqrt{x^{2}+y^{2}+z^{2}}$ is the distance between the masses.
a. Find the components of the gravitational force in the $x$ -, $y$ - and z-directions, where $\mathbf{F}(x, y, z)=-\nabla U(x, y, z).$
b. Show that the gravitational force points in the radial direction (outward from point mass $M$ ) and the radial component is $F(r)=G M m / r^{2}.$
c. Show that the vector field is orthogonal to the equipotential surfaces at all points in the domain of $U.$

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Problem 57

Let $\mathbf{F}(x, y)=\langle f(x, y), g(x, y)\rangle b e$ defined on $\mathbb{R}^{2}.$
Explain why the flow curves or streamlines of $\mathbf{F}$ satisfy $y^{\prime}=\frac{g(x, y)}{f(x, y)}$ and are everywhere tangent to the vector field.

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Problem 58

Let $\mathbf{F}(x, y)=\langle f(x, y), g(x, y)\rangle b e$ defined on $\mathbb{R}^{2}.$
Find and graph the flow curves for the vector field $\mathbf{F}=\langle 1, x\rangle.$

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Problem 59

Let $\mathbf{F}(x, y)=\langle f(x, y), g(x, y)\rangle b e$ defined on $\mathbb{R}^{2}.$
Find and graph the flow curves for the vector field $\mathbf{F}=\langle x, x\rangle.$

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Problem 60

Let $\mathbf{F}(x, y)=\langle f(x, y), g(x, y)\rangle b e$ defined on $\mathbb{R}^{2}.$
Find and graph the flow curves for the vector field $\mathbf{F}=\langle y, x\rangle.$
Note that $\frac{d}{d x}\left(y^{2}\right)=2 y y^{\prime}(x).$

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Problem 61

Let $\mathbf{F}(x, y)=\langle f(x, y), g(x, y)\rangle b e$ defined on $\mathbb{R}^{2}.$
Find and graph the flow curves for the vector field $\mathbf{F}=\langle-y, x\rangle.$

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Problem 62

Vectors in $\mathbb{R}^{2}$ may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted $\mathbf{u}_{r}$ and $\mathbf{u}_{\theta}$ (see figure). Unlike the coordinate unit vectors in Cartesian coordinates, $\mathbf{u}_{r}$ and $\mathbf{u}_{0}$ change their direction depending on the point $(r, \theta) .$ Use the figure to show that for $r>0,$ the following relationships among the unit vectors in Cartesian and polar coordinates hold:
$$\mathbf{u}_{r}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j} \quad \mathbf{i}=\mathbf{u}_{r} \cos \theta-\mathbf{u}_{\theta} \sin \theta$$
$$\mathbf{u}_{\theta}=-\sin \theta \mathbf{i}+\cos \theta \mathbf{j} \quad \mathbf{j}=\mathbf{u}_{r} \sin \theta+\mathbf{u}_{\theta} \cos \theta$$

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Problem 63

Verify that the relationships in Exercise 62 are consistent when $\theta=0, \pi / 2, \pi,$ and $3 \pi / 2.$

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Problem 64

A vector field in polar coordinates has the form $\mathbf{F}(r, \theta)=f(r, \theta) \mathbf{u}_{r}+g(r, \theta) \mathbf{u}_{\theta},$ where the unit vectors are defined in Exercise 62. Sketch the following vector fields and express them in Cartesian coordinates.
$$\mathbf{F}=\mathbf{u}_{r}$$

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Problem 65

A vector field in polar coordinates has the form $\mathbf{F}(r, \theta)=f(r, \theta) \mathbf{u}_{r}+g(r, \theta) \mathbf{u}_{\theta},$ where the unit vectors are defined in Exercise 62. Sketch the following vector fields and express them in Cartesian coordinates.
$$\mathbf{F}=\mathbf{u}_{\theta}$$

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Problem 66

A vector field in polar coordinates has the form $\mathbf{F}(r, \theta)=f(r, \theta) \mathbf{u}_{r}+g(r, \theta) \mathbf{u}_{\theta},$ where the unit vectors are defined in Exercise 62. Sketch the following vector fields and express them in Cartesian coordinates.
$$\mathbf{F}=r \mathbf{u}_{\theta}$$

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Problem 67

Cartesian vector field to polar vector field Write the vector field $\mathbf{F}=\langle-y, x\rangle$ in polar coordinates and sketch the field.

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