Thomas' Calculus, Multivariable (12th Edition)

George B. Thomas, Maurice D. Weir, Joel Hass

Chapter 16 INTEGRATION IN VECTOR FIELDS - all with Video Answers

Educators

Chapter Questions

04:01

Problem 1

Match the vector equations with the graphs (a)-(h) given here.
(a To h Figure can't copy)

$\mathbf{r}(t)=t \mathbf{1}+(1-t) \mathbf{j}, \quad 0 \leq t \leq 1$

Jack Hou

Numerade Educator

01:58

Problem 1

Find the gradient fields of the functions.

$f(x, y, z)=\left(x^2+y^2+z^2\right)^{-1 / 2}$

Vikash Ranjan

Numerade Educator

02:09

Problem 1

Which fields are conservative, and which are not?

$\mathbf{F}=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}$

Jillian Rae Villa

Numerade Educator

04:17

Verify the conchsion of Green's Theorem by evaluating both sides of Equations (3) and (4) for the field $F=M \mathbf{i}+N \mathbf{j}$. Take the domains of integration in each case to be the disk $R: x^2+y^2 \leq a^2$ and its bounding circle $C: \mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi$.

$\mathbf{F}=-y \mathbf{i}+x \mathbf{j}$

Yuou Sun

Numerade Educator

02:48

Problem 1

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.)

The paraboloid $z=x^2+y^2, z \leq 4$

Dorcas Attuabea Addo

Numerade Educator

05:19

Problem 1

Integrate the given function over the given surface.

Parabolic cylinder $G(x, y, z)=x$, over the parabolic cylinder $y=x^2, 0 \leq x \leq 2,0 \leq z \leq 3$

Dorcas Attuabea Addo

Numerade Educator

03:27

Problem 1

Use the surface integral in Stokes' Theorem to calculate the circulation of the field $\mathbf{F}$ around the curve $C$ in the indicated direction.

$\mathbf{F}=x^2 \mathbf{i}+2 x \mathbf{j}+z^2 \mathbf{k}$
C: The ellipse $4 x^2+y^2=4$ in the $x y$-plane, counterclockwise when viewed from above

Vikash Ranjan

Numerade Educator

03:26

Problem 1

Find the divergence of the field.

The spin field in Figure 16.12

Kevin Harmer

Numerade Educator

03:05

Problem 2

Match the vector equations with the graphs (a)-(h) given here.
(a To h Figure can't copy)

$\mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad-1 \leq t \leq 1$

Jack Hou

Numerade Educator

02:07

Problem 2

Find the gradient fields of the functions.

$f(x, y, z)=\ln \sqrt{x^2+y^2+z^2}$

Vikash Ranjan

Numerade Educator

02:44

Problem 2

Which fields are conservative, and which are not?

$\mathbf{F}=(y \sin z) \mathbf{i}+(x \sin z) \mathbf{j}+(x y \cos z) \mathbf{k}$

Jillian Rae Villa

Numerade Educator

01:29

Problem 2

$\mathrm{F}=y \mathbf{i}$

Yuou Sun

Numerade Educator

05:08

Problem 2

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.)

The paraboloid $z=9-x^2-y^2, z \geq 0$

Dorcas Attuabea Addo

Numerade Educator

View

Problem 2

Integrate the given function over the given surface.

Which fields are conservative, and which are not?

$\mathbf{F}=-y \mathbf{i}+x \mathbf{j}$

Jillian Rae Villa

Numerade Educator

04:17

Numerade Educator

02:27

Problem 6

Which fields are conservative, and which are not?

$\mathbf{F}=\left(e^x \cos y\right) \mathbf{i}-\left(e^x \sin y\right) \mathbf{j}+z \mathbf{k}$

Jillian Rae Villa

Numerade Educator

05:27

Problem 6

Use Green's Theorem to find the counterclockwise circulation and outward flux for the field $\mathrm{F}$ and curve $C$.

Let $\mathbf{n}$ be the outer unit normal (normal away from the origin) of the parabolic shell
$$
S: 4 x^2+y+z^2=4, \quad y \geq 0,
$$
and let
$$
\mathbf{F}=\left(-z+\frac{1}{2+x}\right) \mathbf{1}+\left(\tan ^{-1} y\right) \mathbf{j}+\left(x+\frac{1}{4+z}\right) \mathbf{k} .
$$

Find the value of
$$
\iint_S \nabla \times \mathbf{F} \cdot \mathbf{n} d \sigma .
$$

Chris Trentman

Numerade Educator

14:44

Problem 8

Use the Divergence Theorem to find the outward flux of $\mathrm{F}$ across the boundary of the region $D$.

$\mathbf{F}=x^2 \mathbf{i}+x z \mathbf{j}+3 z \mathbf{k}$
$D$; The solid sphere $x^2+y^2+z^2 \leq 4$

Kevin Harmer

Numerade Educator

02:20

Problem 9

Evaluate $\int_C(x+y) d s$ where $C$ is the straight-line segment $x=t, y=(1-t), z=0$, from $(0,1,0)$ to $(1,0,0)$.

Ekaveera Kumar

Numerade Educator

07:07

Integrate $G(x, y, z)=x+y+z$ over the surface of the cube cut from the first octant by the planes $x=a, y=a, z=a$.

Carlos Pinilla

Numerade Educator

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Problem 9

Let $S$ be the cylinder $x^2+y^2=a^2, 0 \leq z \leq h$, together with its top, $x^2+y^2 \leq a^2, z=h$. Let $\mathbf{F}=-y \mathbf{i}+x \mathbf{j}+x^2 \mathbf{k}$. Use Stokes' Theorem to find the flux of $\nabla \times \mathrm{F}$ outward through $S$.

Victor Salazar

Numerade Educator

17:47

Problem 9

Use the Divergence Theorem to find the outward flux of $\mathrm{F}$ across the boundary of the region $D$.

$\mathbf{F}=x^2 \mathbf{i}-2 x y \mathbf{j}+3 x z k$
$D$ : The region cut from the first octant by the sphere $x^2+y^2+$ $z^2=4$

Kevin Harmer

Numerade Educator

04:26

Problem 10

Evaluate $\int_C(x-y+z-2) d s$ where $C$ is the straight-line segment $x=t, y=(1-t), z=1$, from $(0,1,1)$ to $(1,0,1)$.

Jack Hou

Numerade Educator

06:25

Integrute $G(x, y, z)=y+z$ over the surface of the wedge in the first octant bounded by the coordinate planes and the planes $x=2$ and $y+z=1$.

Nick Johnson

Numerade Educator

06:04

Problem 10

Evaluate
$$
\iint_S \nabla \times(y \mathbf{i}) \cdot \mathbf{n} d \sigma,
$$
where $S$ is the hemisphere $x^2+y^2+z^2=1, z \geq 0$.

Carlos Pinilla

Numerade Educator

06:07

Problem 10

Use the Divergence Theorem to find the outward flux of $\mathrm{F}$ across the boundary of the region $D$.

$\mathbf{F}=\left(6 x^2+2 x y\right) \mathbf{i}+\left(2 y+x^2 z\right) \mathbf{j}+4 x^2 y^3 \mathbf{k}$
$D$ : The region cut from the first octant by the cylinder $x^2+y^2=$ 4 and the plane $z=3$

Alaa Ragai

Numerade Educator

03:51

Problem 11

Evaluate $\int_c(x y+y+z) d s$ along the curve $\mathbf{r}(t)=2 t \mathbf{i}+$ $t \mathbf{j}+(2-2 t) \mathbf{k}, 0 \leq t \leq 1$.

Jack Hou

Numerade Educator

07:42

Integrute $G(x, y, z)=x y z$ over the surface of the rectangular solid cut from the first octant by the planes $x=a, y=b$, and $\boldsymbol{z}=\boldsymbol{c}$.

Carlos Pinilla

Numerade Educator

02:20

Problem 11

Flux of curl $F$ Show that
$$
\iint_S \nabla \times \mathbf{F} \cdot \mathbf{n} d \sigma
$$
has the same value for all oriented surfaces $S$ that $\operatorname{span} C$ and that induce the same positive direction on $C$.

Chris Trentman

Numerade Educator

01:43

Problem 11

Use the Divergence Theorem to find the outward flux of $\mathrm{F}$ across the boundary of the region $D$.

$\mathbf{F}=2 x \mathrm{i}-x y \mathbf{-}-z^2 \mathbf{k}$
$D$ : The wedge cut from the first octant by the plane $y+z=4$ and the elliptical cylinder $4 x^2+y^2=16$

Alaa Ragai

Numerade Educator

03:50

Problem 12

Evaluate $\int_C \sqrt{x^2+y^2} d s$ along the curve $\mathbf{r}(t)=(4 \cos t) \mathbf{i}+$ $(4 \sin t) \mathbf{j}+3 t \mathbf{k},-2 \pi \leq t \leq 2 \pi$.

Jack Hou

Numerade Educator

06:56

Integrate $G(x, y, z)=x y z$ over the surface of the rectangular solid bounded by the planes $x= \pm a, y= \pm b$, and $z= \pm c$.

Carlos Pinilla

Numerade Educator

03:59

Problem 12

Let $\mathbf{F}$ be a differentiable vector field defined on a region containing a smooth closed oriented surface $S$ and its interiot. Let $\mathbf{n}$ be the unit normal vector field on $S$. Suppose that $S$ is the union of two surfaces $S_1$ and $S_2$ joined along a smooth simple closed curve C. Can anything be said about
$$
\iint_S \nabla \times \mathbf{F} \cdot \mathbf{n} d \sigma ?
$$
Give reasons for your answer.

Chris Trentman

Numerade Educator

10:34

Problem 12

Use the Divergence Theorem to find the outward flux of $\mathrm{F}$ across the boundary of the region $D$.

$\mathbf{F}=x^3 \mathbf{i}+y^3 \mathbf{j}+z^3 \mathbf{k}$
$D$ : The solid sphere $x^2+y^2+z^2 \leq a^2$

Kevin Harmer

Numerade Educator

08:34

Integrate $G(x, y, z)=x+y+z$ over the portion of the plane $2 x+2 y+z=2$ that lies in the first octant,

Carlos Pinilla

Numerade Educator

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Problem 13

Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field $\mathbf{F}$ across the surface $S$ in the direction of the outward unit normal $\mathbf{n}$.

$$
\begin{aligned}
& \mathbf{F}=2 x \mathbf{i}+3 x \mathbf{j}+5 y \mathbf{k} \\
& S: \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(4-r^2\right) \mathbf{k}, \\
& 0 \leq r \leq 2, \quad 0 \leq \theta \leq 2 \pi
\end{aligned}
$$

Victor Salazar

Numerade Educator

07:12

Problem 13

Use the Divergence Theorem to find the outward flux of $\mathrm{F}$ across the boundary of the region $D$.

$\mathbf{F}=\sqrt{x^2+y^2+z^2}(x \mathbf{i}+y \mathbf{j}+z \mathbf{k})$
D: The region $1 \leq x^2+y^2+z^2 \leq 2$

Alaa Ragai

Numerade Educator

10:01

Integrate $G(x, y, z)=x \sqrt{y^2+4}$ over the surface cut from the parabolic cylinder $y^2+4 z=16$ by the planes $x=0, x=1$, and $z=0$.

Carlos Pinilla

Numerade Educator

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Problem 14

Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field $\mathbf{F}$ across the surface $S$ in the direction of the outward unit normal $\mathbf{n}$.

$$
\begin{aligned}
& \mathbf{F}=(y-z) \mathbf{i}+(z-x) \mathbf{j}+(x+z) \mathbf{k} \\
& S ; \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(9-r^2\right) \mathbf{k} \\
& 0 \leq r \leq 3, \quad 0 \leq \theta \leq 2 \pi
\end{aligned}
$$

Victor Salazar

Numerade Educator

04:50

Problem 14

Use the Divergence Theorem to find the outward flux of $\mathrm{F}$ across the boundary of the region $D$.

$\mathbf{F}=(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) / \sqrt{x^2+y^2+z^2}$
D: The region $1 \leq x^2+y^2+z^2 \leq 4$

Alaa Ragai

Numerade Educator

View

Find the counterclockwise circulation and outward flux of the field $\mathrm{F}=x y \mathbf{I}+y^2 \mathrm{~J}$ around and over the boundary of the region enclosed by the curves $y=x^2$ and $y=x$ in the first quadrant.

Yuou Sun

Numerade Educator

01:46

Problem 15

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.)

The portion of the cylinder $y^2+$ $(z-5)^2=25$ between the planes $x=0$ and $x=10$

Dorcas Attuabea Addo

Numerade Educator

21:23

Integrate $f(x, y, z)=x+\sqrt{y}-z^2$ over the path from $(0,0,0)$ to $(1,1,1)$ (see accompanying figure) given by
$$
\begin{array}{ll}
C_1: & \mathbf{r}(t)=t \mathbf{k}, \quad 0 \leq t \leq 1 \\
C_2: & \mathbf{r}(t)=t \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1 \\
C_3: & \mathbf{r}(t)=t \mathbf{i}+\mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1
\end{array}
$$

Rebecca Polasek

Numerade Educator

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Problem 16

Find the line integrals along the given path $C$.

$\int_C \sqrt{x+y} d x$, where $C$ is given in the accompanying figure.

Victor Salazar

Numerade Educator

12:47

Problem 16

Show that the differential forms in the integrals are exact. Then evaluate the integrals.

$\int_{(0,0,0)}^{(3,3,1)} 2 x d x-y^2 d y-\frac{4}{1+z^2} d z$

Carlos Pinilla

Numerade Educator

07:20

Problem 16

Find the counterclockwise circulation and the outward flux of the field $\mathbf{F}=(-\sin y) \mathbf{I}+(x \cos y) \mathbf{J}$ around and over the square cut from the first quadrant by the lines $x=\pi / 2$ and $y=\pi / 2$.

Carlos Pinilla

Numerade Educator

01:46

Problem 16

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.)

The portion of the cylinder $y^2+$ $(z-5)^2=25$ between the planes $x=0$ and $x=10$

Dorcas Attuabea Addo

Numerade Educator

10:52

Problem 16

Integrate $G(x, y, z)=x$ over the surface given by
$$
z=x^2+y \text { for } 0 \leq x \leq 1,-1 \leq y \leq 1 .
$$

Carlos Pinilla

Numerade Educator

15:00

Problem 16

Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field $\mathbf{F}$ across the surface $S$ in the direction of the outward unit normal $\mathbf{n}$.

$$
\begin{aligned}
& \mathbf{F}=(x-y) \mathbf{i}+(y-z) \mathbf{j}+(z-x) \mathbf{k} \\
& S: \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+(5-r) \mathbf{k} \\
& 0 \leq r \leq 5, \quad 0 \leq \theta \leq 2 \pi
\end{aligned}
$$

Joel Mueller

Numerade Educator

07:12

Problem 16

Use the Divergence Theorem to find the outward flux of $\mathrm{F}$ across the boundary of the region $D$.

div $(\operatorname{curl} G)$ is zero
a. Show that if the necessary partial derivatives of the components of the field $\mathbf{G}=M \mathbf{i}+N \mathbf{j}+P \mathbf{k}$ are continuous, then $\nabla \cdot \nabla \times \mathbf{G}=0$.
b. What, if anything, can you conclude about the flux of the field $\nabla \times \mathbf{G}$ across a closed surface? Give reasons for your answer.

James Kiss

Numerade Educator

05:34

Problem 18

Integrate $f(x, y, z)=-\sqrt{x^2+z^2}$ over the circle
$$
\mathbf{r}(t)=(a \cos t) \mathbf{j}+(a \sin t) \mathbf{k}, \quad 0 \leq t \leq 2 \pi .
$$

Chris Trentman

Numerade Educator

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Problem 18

Along the curve $\mathrm{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}-(\cos t) \mathbf{k}, 0 \leq t \leq \pi$, evaluate each of the following integrals.
a. $\int_C x z d x$
b. $\int_C x z d y$
c. $\int_C x y z d z$

Victor Salazar

Numerade Educator

03:23

Problem 18

Although they are not defined on all of space $R^3$, the fields associated with are simply connected and the Component Test can be used to show they are conservative. Find a potential function for each field and evaluate the integrals as in Example 6.

$\int_{(0,2,1)}^{(1, \pi / 2,2)} 2 \cos y d x+\left(\frac{1}{y}-2 x \sin y\right) d y+\frac{1}{z} d z$

William Semus

Numerade Educator

02:14

Problem 18

Find the counterclockwise circulation of $\mathbf{F}=\left(y+e^x \ln y\right) \mathbf{i}+$ $\left(e^x / y\right) \mathrm{j}$ around the boundary of the region that is bounded above by the curve $y=3-x^2$ and below by the curve $y=x^4+1$.

Yuou Sun

Numerade Educator

08:44

Problem 18

The portion of the plane $z=-x$ inside the cylinder $x^2+y^2-4$

Dorcas Attuabea Addo

Numerade Educator

10:57

Problem 18

Integrate $G(x, y, z)=x-y-z$ over the portion of the plane $x+y=1$ in the first octant between $z=0$ and $z=1$ (see the accompanying figure).
(Figure can't copy)

Carlos Pinilla

Numerade Educator

View

Problem 18

Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field $\mathbf{F}$ across the surface $S$ in the direction of the outward unit normal $\mathbf{n}$.

$$
\begin{aligned}
& \mathbf{F}=y^2 \mathbf{i}+z^2 \mathbf{j}+x \mathbf{k} \\
& S: \quad \mathbf{r}(\phi, \theta)=(2 \sin \phi \cos \theta) \mathbf{i}+(2 \sin \phi \sin \theta) \mathbf{j}+(2 \cos \phi) \mathbf{k}, \\
& 0 \leq \phi \leq \pi / 2, \quad 0 \leq \theta \leq 2 \pi
\end{aligned}
$$

Victor Salazar

Numerade Educator

02:23

Problem 18

Let $\mathbf{F}_1$ and $\mathbf{F}_2$ be differentiable vector fields and let $a$ and $b$ be arbitrary real constants. Verify the following identities.
a. $\nabla \cdot\left(a \mathbf{F}_1+b \mathbf{F}_2\right)=a \nabla \cdot \mathbf{F}_1+b \nabla \cdot \mathbf{F}_2$
b. $\nabla \times\left(a \mathbf{F}_1+b \mathbf{F}_2\right)=a \nabla \times \mathbf{F}_1+b \nabla \times \mathbf{F}_2$
c. $\nabla \cdot\left(\mathbf{F}_1 \times \mathbf{F}_2\right)=\mathbf{F}_2 \cdot \nabla \times \mathbf{F}_1-\mathbf{F}_1 \cdot \nabla \times \mathbf{F}_2$

Alaa Ragai

Numerade Educator

07:50

Problem 19

Evaluate $\int_C x d s$, where $C$ is
a. the straight-line segment $x=t, y=t / 2$, from $(0,0)$ to $(4,2)$.
b. the parabolic curve $x=t, y=t^2$, from $(0,0)$ to $(2,4)$.

Jack Hou

Numerade Educator

01:23

Problem 19

Find the work done by $\mathbf{F}$ over the curve in the direction of increasing $t$.

$$
\begin{aligned}
& \mathbf{F}=x y \mathbf{i}+y \mathbf{j}-y z \mathbf{k} \\
& \mathbf{r}(t)=t \mathbf{i}+t^2 \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1
\end{aligned}
$$

Vikash Ranjan

Numerade Educator

03:23

Problem 19

$\int_{(1,1,1)}^{(1,2,3)} 3 x^2 d x+\frac{z^2}{y} d y+2 z \ln y d z$

William Semus

Numerade Educator

01:21

Problem 19

Find the work done by $\mathbf{F}$ in moving a particle once counterclockwise around the given curve.

$\mathrm{F}=2 x y^3 \mathrm{i}+4 x^2 y^2 \mathrm{j}$
C: The boundary of the "triangular" region in the first quadrant enclosed by the $x$-axis, the line $x=1$, and the curve $y=x^3$

Yuou Sun

Numerade Educator

08:33

Problem 19

The portion of the cone $z=2 \sqrt{x^2+y^2}$ between the planes $z=2$ and $z=6$

Dorcas Attuabea Addo

Numerade Educator

View

Problem 19

Use a parametrization to find the flux $\iint_S \mathbf{F} \cdot \mathbf{n} d \sigma$ across the surface in the given direction.

Find the work done by $\mathbf{F}$ over the curve in the direction of increasing $t$.

$$
\begin{aligned}
& \mathbf{F}=z \mathbf{i}+x \mathbf{j}+y \mathbf{k} \\
& \mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2 \pi
\end{aligned}
$$

Vikash Ranjan

Numerade Educator

03:23

Problem 21

$\int_{(1,1,1)}^{(2,2,2)} \frac{1}{y} d x+\left(\frac{1}{z}-\frac{x}{y^2}\right) d y-\frac{y}{z^2} d z$

William Semus

Numerade Educator

02:37

Problem 21

Apply Green's Theorem to evaluate the integrals.

$\oint_C\left(y^2 d x+x^2 d y\right)$
C. The triangle bounded by $x=0, x+y=1, y=0$

Melissa Munoz

Numerade Educator

05:16

Problem 21

The portion of the cylinder $x^2+y^2=1$ between the planes $z=1$ and $z=4$

Dorcas Attuabea Addo

Numerade Educator

View

Problem 21

Use a parametrization to find the flux $\iint_S \mathbf{F} \cdot \mathbf{n} d \sigma$ across the surface in the given direction.

$\mathbf{F}=z \mathbf{k}$ across the portion of the sphere $x^2+y^2+$ $z^2=a^2$ in the first octant in the direction away from the origin

Victor Salazar

Numerade Educator

03:19

Problem 21

Let $C$ be a simple closed smooth curve in the plane $2 x+2 y+z=2$, oriented as shown here. Show that
$$
\oint_C 2 y d x+3 z d y-x d z
$$
(Figure can't copy)
depends only on the area of the region enclosed by $C$ and not on the position or shape of $C$.

Vikash Ranjan

Numerade Educator

03:59

Problem 21

Let $\mathbf{F}$ be a field whose components have continuous first partial derivatives throughout a portion of space containing a region $D$ bounded by a smooth closed surface $S$. If $|\mathbf{F}| \leq 1$, can any bound be placed on the size of
$$
\iiint_D \nabla \cdot \mathbf{F} d V ?
$$
Give reasons for your answer.

Chris Trentman

Numerade Educator

02:08

Problem 22

Find the line integral of $f(x, y)=x-y+3$ along the curve $\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{d}, 0 \leq t \leq 2 \pi$.

Diogo Caetano

Numerade Educator

03:34

Problem 22

Joseph Liao

Numerade Educator

07:20

Problem 23

Evaluate $\int_C \frac{x^2}{y^{4 / 3}} d s$, where $C$ is the curve $x=t^2, y=t^3$, for $1 \leq t \leq 2$.

Kyle Flanagan

Numerade Educator

01:39

Apply Green's Theorem to evaluate the integrals.

$\oint_C\left(2 x+y^2\right) d x+(2 x y+3 y) d y$
C: Any simple closed curve in the plane for which Green's Theorem holds

Yuou Sun

Numerade Educator

08:32

Problem 24

The portion of the paraboloid $z=x^2+y^2$ between the planes $z=1$ and $z=4$

Dorcas Attuabea Addo

Numerade Educator

View

Problem 24

Use a parametrization to find the flux $\iint_S \mathbf{F} \cdot \mathbf{n} d \sigma$ across the surface in the given direction.

$\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ outward through the portion of the cylinder $x^2+y^2=1$ cut by the planes $z=0$ and $z=a$

Victor Salazar

Numerade Educator

01:44

Problem 24

Does Stokes' Theorem say anything special about circulation in a field whose curl is zero? Give reasons for your answer.

Vikash Ranjan

Numerade Educator

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Problem 24

Among all rectangular solids defined by the inequalities $0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq 1$, find the one for which the total flux of $\mathbf{F}=\left(-x^2-4 x y\right) \mathbf{i}-6 y z \mathbf{j}+12 z \mathbf{k}$ outward through the six sides is greatest. What is the greatest flux?

Victor Salazar

Numerade Educator

09:04

Problem 25

Evaluate $\int_C(x+\sqrt{y}) d s$ where $C$ is given in the accompanying figure.

Skyler Noble

Numerade Educator

01:39

The lower portion cut from the sphere $x^2+y^2+z^2=2$ by the cone $z=\sqrt{x^2+y^2}$

Victor Salazar

Numerade Educator

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Problem 25

Use a parametrization to find the flux $\iint_S \mathbf{F} \cdot \mathbf{n} d \sigma$ across the surface in the given direction.

$\mathbf{F}=x y \mathbf{i}-z \mathbf{k}$ outward (normal away from the $z$-axis) through the cone $z=\sqrt{x^2+y^2}, 0 \leq z \leq 1$

Victor Salazar

Numerade Educator

06:44

Problem 25

Let $R$ be a region in the $x y$-plane that is bounded by a piecewise smooth simple closed curve $C$ and suppose that the moments of inertia of $R$ about the $x$ - and $y$-axes are known to be $I_x$ and $I_y$. Evaluate the integral
$$
\oint_C \nabla\left(r^4\right) \cdot \mathbf{n} d s,
$$
where $r=\sqrt{x^2+y^2}$, in terms of $I_x$ and $I_y$.

Chris Trentman

Numerade Educator

01:24

Problem 25

Let $\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ and suppose that the surface $S$ and region $D$ satisfy the hypotheses of the Divergence Theorem. Show that the volume of $D$ is given by the formula
$$
\text { Volume of } D=\frac{1}{3} \iint_S \mathbf{F} \cdot \mathbf{n} d \sigma .
$$

Arwa Ali

Numerade Educator

09:04

Problem 26

Evaluate $\int_C \frac{1}{x^2+y^2+1} d s$ where $C$ is given in the accompanying figure.

Skyler Noble

Numerade Educator

02:58

Problem 26

Evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$ for the vector field $\mathbf{F}=y \mathbf{i}-x \mathbf{j}$ counterclockwise along the unit circle $x^2+y^2=1$ from $(1,0)$ to $(0,1)$.

Chris Trentman

Numerade Educator

03:32

Problem 26

Independence of path Show that the values of the integrals do not depend on the path taken from $A$ to $B$.

$\int_A^B \frac{x d x+y d y+z d z}{\sqrt{x^2+y^2+z^2}}$

Melissa Munoz

Numerade Educator

02:25

Problem 26

The ellipse $\mathbf{r}(t)=(a \cos t) \mathbf{i}+(b \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi$

Melissa Munoz

Numerade Educator

08:33

Problem 26

The portion of the sphere $x^2+y^2+z^2=4$ between the planes $z=-1$ and $z=\sqrt{3}$

Dorcas Attuabea Addo

Numerade Educator

View

Problem 26

Use a parametrization to find the flux $\iint_S \mathbf{F} \cdot \mathbf{n} d \sigma$ across the surface in the given direction.

$\mathbf{F}=y^2 \mathbf{i}+x z \mathbf{j}-\mathbf{k}$ outward (normal away from the $z$-axis) through the cone $z=2 \sqrt{x^2+y^2}, 0 \leq z \leq 2$

Victor Salazar

Numerade Educator

01:51

A function $f(x, y, z)$ is said to be harmonic in a region $D$ in space if it satisfies the Laplace equation
$$
\nabla^2 f=\nabla \cdot \nabla f=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2}=0
$$
throughout $D$.
a. Suppose that $f$ is harmonic throughout a bounded region $D$ enclosed by a smooth surface $S$ and that $\mathbf{n}$ is the chosen unit normal vector on $S$. Show that the integral over $S$ of $\nabla f \cdot \mathbf{n}$, the derivative of $f$ in the direction of $\mathbf{n}$, is zero.
b. Show that if $f$ is harmonic on $D$, then
$$
\iint_S f \nabla f \cdot \mathbf{n} d \sigma=\iiint_D|\nabla f|^2 d V .
$$

Kevin Harmer

Numerade Educator

02:35

Problem 28

Integrate $f$ over the given curve.

$f(x, y)=\left(x+y^2\right) / \sqrt{1+x^2}, \quad C: y=x^2 / 2$ from $(1,1 / 2)$ to $(0,0)$

Jack Hou

Numerade Educator

04:30

Problem 28

Find the work done by the gradient of $f(x, y)=(x+y)^2$ counterclockwise around the circle $x^2+y^2=4$ from $(2,0)$ to itsclf.

Vikash Ranjan

Numerade Educator

06:59

Problem 28

Find a potential function for $\mathbf{F}$.

$\mathbf{F}=\left(e^x \ln y\right) \mathbf{I}+\left(\frac{e^x}{y}+\sin z\right) \mathfrak{J}+(y \cos z) \mathbf{k}$

Carlos Pinilla

Numerade Educator

02:42

Find the work done by $\mathbf{F}=$ $\left(x^2+y\right) \mathbf{i}+\left(y^2+x\right) \mathbf{j}+z e^x \mathbf{k}$ over the following paths from $(1,0,0)$ to $(1,0,1)$.
a. The line segment $x=1, y=0,0 \leq z \leq 1$
b. The helix $\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(t / 2 \pi) \mathbf{k}, 0 \leq t \leq 2 \pi$
c. The $x$-axis from $(1,0,0)$ to $(0,0,0)$ followed by the parabola $z=x^2, y=0$ from $(0,0,0)$ to $(1,0,1)$

Khoi V

Numerade Educator

02:06

Problem 29

Let $C$ be the boundary of a region on which Green's Theorem holds. Use Green's Theorem to calculate
a. $\oint_C f(x) d x+g(y) d y$
b. $\oint_C k y d x+h x d y \quad$ ( $k$ and $h$ constants).

Zack A

Numerade Educator

View

Problem 29

The circular cylinder $\mathbf{r}(\theta, z)=(3 \sin 2 \theta) \mathbf{i}+$ $\left(6 \sin ^2 \theta\right) \mathbf{j}+z \mathbf{k}, 0 \leq \theta \leq \pi$, at the point $P_0(3 \sqrt{3} / 2,9 / 2,0)$ corresponding to $(\theta, z)=(\pi / 3,0)$ (See Example 3.)

Victor Salazar

Numerade Educator

03:24

Problem 29

Find the flux of the field $\mathbf{F}$ across the portion of the given surface in the specified direction.

$\mathbf{F}(x, y, z)=-\mathbf{1}+2 \mathbf{j}+3 \mathbf{k}$
$S$ : rectangular surface $z=0, \quad 0 \leq x \leq 2, \quad 0 \leq y \leq 3$, direction $\mathbf{k}$

Arun Bana

Numerade Educator

06:00

Problem 29

Suppose that $f$ and $g$ are scalar functions with continuous first- and second-order partial derivatives throughout a region $D$ that is bounded by a closed piecewise smooth surface $S$. Show that
$$
\iint_S f \nabla g \cdot \mathbf{n} d \sigma=\iiint_D\left(f \nabla^2 g+\nabla f \cdot \nabla g\right) d V .
$$

Equation (9) is Green's first formula. (Hint: Apply the Divergence Theorem to the field $\mathbf{F}=f \nabla_g$.)

Arwa Ali

Numerade Educator

04:24

Problem 30

Integrate $f$ over the given curve.

$f(x, y)=x^2-y, \quad C: x^2+y^2=4$ in the first quadrant from $(0,2)$ to $(\sqrt{2}, \sqrt{2})$

Jack Hou

Numerade Educator

10:53

Problem 30

Find the flux of the fields
$$
\mathbf{F}_1=2 x \mathbf{i}-3 y \mathbf{j} \quad \text { and } \quad \mathbf{F}_2=2 x \mathbf{i}+(x-y) \mathbf{j}
$$
across the circle
$$
\mathbf{r}(t)=(a \cos t) \mathrm{i}+(a \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi .
$$

Vikash Ranjan

Numerade Educator

06:04

Problem 30

Find the work done by $\mathbf{F}=$ $e^{y z} \mathbf{i}+\left(x z e^{y z}+z \cos y\right) \mathbf{j}+\left(x y e^{y z}+\sin y\right) \mathbf{k}$ over the following paths from $(1,0,1)$ to $(1, \pi / 2,0)$.
a. The line segment $x=1, y=\pi t / 2, z=1-t, 0 \leq t \leq 1$
b. The line segment from $(1,0,1)$ to the origin followed by the line segment from the origin to $(1, \pi / 2,0)$
c. The line segment from $(1,0,1)$ to $(1,0,0)$, followed by the $x$-axis from $(1,0,0)$ to the origin, followed by the parabola $y=\pi x^2 / 2, z=0$ from there to $(1, \pi / 2,0)$

Khoi V

Numerade Educator

View

Problem 30

Show that the value of
$$
\oint_C x y^2 d x+\left(x^2 y+2 x\right) d y
$$
around any square depends only on the area of the square and not on its location in the plane.

Victor Salazar

Numerade Educator

View

Problem 30

The parabolic cylinder surface $\mathbf{r}(x, y)=$ $x \mathbf{i}+y \mathbf{j}-x^2 \mathbf{k},-\infty<x<\infty,-\infty<y<\infty$, at the point $P_0(1,2,-1)$ corresponding to $(x, y)=(1,2)$

Victor Salazar

Numerade Educator

03:24

Problem 30

Find the flux of the field $\mathbf{F}$ across the portion of the given surface in the specified direction.

$\mathbf{F}(x, y, z)=y x^2 \mathbf{i}-2 \mathbf{j}+x z \mathbf{k}$
$S$ : rectangular surface $y=0, \quad-1 \leq x \leq 2, \quad 2 \leq z \leq 7$, direction $-\mathbf{j}$

Arun Bana

Numerade Educator

05:16

Problem 30

(Continuation of Exercise 29.) Interchange $f$ and $g$ in Equation (9) to obtain a similar formula. Then subtract this formula from Equation (9) to show that
$$
\iint_S(f \nabla g-g \nabla f) \cdot \mathbf{n} d \sigma=\iiint_D\left(f \nabla^2 g-g \nabla^2 f\right) d V .
$$

This equation is Green's second formula.

Kevin Harmer

Numerade Educator

02:14

Problem 31

Find the area of one side of the "winding wall" standing orthogonally on the curve $y=x^2, 0 \leq x \leq 2$, and beneath the curve on the surface $f(x, y)=x+\sqrt{y}$.

Harrison Nascimento

Numerade Educator

09:12

Problem 31

Find the circulation and flux of the field $F$ around and across the closed semicircular path that consists of the semicircular arch $\mathbf{r}_1(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq \pi$, followed by the line segment $\mathbf{r}_2(t)=t,-a \leq t \leq a$.

$\mathrm{F}=x \mathbf{i}+y \mathbf{j}$

Vikash Ranjan

Numerade Educator

18:09

Problem 31

Let $\mathbf{F}=\nabla\left(x^3 y^2\right)$ and let $C$ be the path in the $x y$-plane from $(-1,1)$ to $(1,1)$ that consists of the line segment from $(-1,1)$ to $(0,0)$ followed by the line segment from $(0,0)$ to $(1,1)$. Evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$ in two ways.
a. Find parametrizations for the segments that make up $C$ and evaluate the integral.
b. Use $f(x, y)=x^3 y^2$ as a potential function for $\mathbf{F}$.

Carlos Pinilla

Numerade Educator

02:08

Problem 31

What is special about the integral
$$
\oint_C 4 x^3 y d x+x^4 d y ?
$$
Give reasons for your answer.

Chris Trentman

Numerade Educator

11:31

Problem 31

a. A torus of revolution (donghnut) is obtained by rotating a circle $C$ in the $x z$-plane about the $z$-axis in space. (See the accompanying figure.) If $C$ has radius $r>0$ and center $(R, 0,0)$, show that a parametrization of the torus is
$$
\begin{aligned}
\mathbf{r}(u, v)= & ((R+r \cos u) \cos v) \mathbf{i} \\
& +((R+r \cos u) \sin v) \mathbf{j}+(r \sin u) \mathbf{k},
\end{aligned}
$$
where $0 \leq u \leq 2 \pi$ and $0 \leq v \leq 2 \pi$ are the angles in the figure.
b. Show that the surface area of the torus is $A=4 \pi^2 R$.
(Figure can't copy)

Chris Trentman

Numerade Educator

05:26

Problem 31

Find the flux of the field $\mathbf{F}$ across the portion of the sphere $x^2+y^2+z^2=a^2$ in the first octant in the direction away from the origin.

$\mathbf{F}(x, y, z)=z \mathbf{k}$

Dorcas Attuabea Addo

Numerade Educator

11:33

Problem 31

Let $\mathbf{v}(t, x, y, z)$ be a continuously differentiable vector field over the region $D$ in space and let $p(t, x, y, z)$ be a continuously differentiable sealar function. The variable $t$ represents the time domain. The Law of Conservation of Mass asserts that
$$
\frac{d}{d t} \iiint_D p(t, x, y, z) d V=-\iint_S p \mathrm{v} \cdot \mathrm{n} d \sigma,
$$
where $S$ is the surface enclosing $D$.
a. Give a physical interpretation of the conservation of mass law if $\mathbf{v}$ is a velocity flow field and $p$ represents the density of the fluid at point $(x, y, z)$ at time $t$.
b. Use the Divergence Theorem and Leibniz's Rule,
$$
\frac{d}{d t} \iiint_D p(t, x, y, z) d V=\iiint_D \frac{\partial p}{\partial t} d V,
$$
to show that the Law of Conservation of Mass is equivalent to the continuity equation,
$$
\nabla \cdot p \mathbf{v}+\frac{\partial p}{\partial t}=0
$$
(In the first term $\nabla \cdot p v$, the variable $t$ is held fixed, and in the second term $\partial p / \partial t$, it is assumed that the point $(x, y, z)$ in $D$ is held fixed.)

Carlos Pinilla

Numerade Educator

02:51

Problem 32

Find the arca of one side of the "wall" standing orthogonally on the curve $2 x+3 y=6,0 \leq x \leq 6$, and beneath the curve on the surface $f(x, y)=4+3 x+2 y$.

Lucas Finney

Numerade Educator