Deck 19: Flux Integrals and Divergence

Question 1

Let where a, b and c are constants.Suppose that the flux of through a surface of area 3 lying in the plane y = 3, oriented in the positive y-direction, is 45.Find the flux of through a surface of area 4 lying in the plane y = 3, oriented in the negative y-direction.

Answer 1

-60

Question 2

What is the flux of the vector field through a circle in the xy-plane of radius 2 oriented upward with center at the origin?

Answer 2

Question 3

$\int_{S} \vec{F} \cdot d \bar{A}$ is a vector.

Answer 3

True

Question 4

A circular disk, S, of radius 2 and centered on an axis, is perpendicular to the y-axis at y = -6 with normal in the direction of decreasing y. Consider the vector field $\vec { F } = x \vec { i } + y \vec { j } + ( z + x ) \vec { k }$ .Is the flux integral $\int_{S} \vec{F} \cdot \overrightarrow{d A}$ positive, negative or zero?

A)Positive
B)Zero
C)Negative

Question 5

Suppose S is a disk of radius 2 in the plane x + z = 0 centered at (0, 0, 0)oriented "upward".
Calculate the flux of through S.

Question 6

Let be the constant vector field .
Find a condition on a, b and c such that for any surface S lying on the plane -5x + 3y - 2z = 1.

Question 7

Compute the flux integral of the vector field through the square in the yz-plane, oriented so that the normal vector points in the direction of the x-axis.

Question 8

Calculate the flux of through a surface of area 3 lying in the plane 2x + 5y + 5z = 10, oriented away from the origin.

Question 9

Let S be the sphere of radius 3 centered at the origin, oriented outward.
Suppose is normal to at every point of S.Find the flux of out of S.

Question 10

Explain why it is impossible to give the Möbius strip a continuous orientation.

Question 11

Compute the flux of the vector field through the surface S, where S is the part of the plane z = x + 2y above the rectangle oriented upward.

Question 12

Let S be the sphere of radius 6 centered at the origin, oriented outward.
Let be a vector field such that at every point of S.Find the flux of out of S.

Question 13

Suppose S is a disk of radius 2 in the plane x + z = 0 centered at (0, 0, 0)oriented "upward".Write down an area vector for the surface S.

Question 14

If the surface area of surface S₁ is larger than the surface area of surface S₂, then $\int _ { S _ { 1 } } \vec { F } \cdot d \vec { A } \geq \int _ { S _ { 2 } } \vec { F } \cdot d \vec { A }$ .

Question 15

Suppose S is a disk of radius 4 in the plane x + z = 0 centered at (0, 0, 0)oriented "upward".Calculate the flux of through S.

Question 16

Let S be an oriented surface with surface area 6.Suppose $\vec{F}=a \vec{i}+b \vec{j}+c \vec{k}$ is a constant vector field with magnitude 3.If the angle between $\vec{F}$ and $\vec { n }$ is $\pi$ /6 at each point of the surface S, determine the value of the flux integral $\int_{S} \vec{F} \cdot \overrightarrow{d A}$ .

Question 17

Let S be the sphere of radius 4 centered at the origin, oriented outward. Find $\vec { n }$ , the unit normal vector to S in the direction of orientation.

A) $\vec { n } = \frac { x } { 4 } \vec { i } + \frac { y } { 4 } \vec { j } + \frac { z } { 4 } \vec { k }$
B) $\vec { n } = x \vec { i } + y \vec { j } + z \vec { k }$
C) $\vec { n } = \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \vec { i } + \frac { y ^ { 2 } } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \vec { j } + \frac { z ^ { 2 } } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \vec { k }$
D) $\vec { n } = \frac { x } { 16 } \vec { i } + \frac { y } { 16 } \vec { j } + \frac { z } { 16 } \vec { k }$

Question 18

Let , and let S₁ be a horizontal rectangle with corners at (0,0,1), (0,2,1), (3,0,1)and (3,2,1), oriented upward; S₂ a rectangle parallel to the xz-plane, with corners at (1,3,1), (2,3,1), (1,3,5)and (2,3,5), oriented in the positive y-direction, and S₃ a rectangle parallel to the yz-plane, with corners at (1,2,1), (1,4,1), (1,2,5)and (1,4,5), oriented in the negative x-direction.
Arrange , and in ascending order.

Question 19

Let $\vec { n }$ be the unit normal vector of S.If the angle between $\vec{F}$ and $\vec { n }$ is less than $\pi$ /2 at each point of the surface, then $\int _ { S } \vec { F } \cdot \overrightarrow { d A } \geq 0$

Question 20

If $\int_{S} \vec{F} \cdot \overrightarrow{d A}=0$ , then $\vec{F}$ is perpendicular to the surface S at every point.

Question 21

(a)Compute the flux of the vector field through S_a, the sphere of radius a, , oriented outward.
(b)Find .

Question 22

Suppose the surface S is the part of the surface x = g(y, z), for points (y, z)belonging to a region R in the yz-plane.If S is oriented in the positive x-direction, what will be the formula for computing the flux of through S?

Question 23

Let ${ \vec { F } } = x \vec { i } + y \vec { j }$ .Write down an iterated integral that computes the flux of $\vec { F }$ through S, where S is the part of the surface $z = x ^ { 2 } + y ^ { 2 }$ below the plane z = 16, oriented downward.

A) $\int _ { - 4 } ^ { 4 } \int _ { - \sqrt { 16 - x ^ { 2 } } } ^ { \sqrt { 16 - x ^ { 2 } } } 2 \left( x ^ { 2 } + y ^ { 2 } \right) d y d x$
B) $- \int _ { - 4 } ^ { 4 } \int _ { - \sqrt { 16 - x ^ { 2 } } } ^ { \sqrt { 6 - x ^ { 2 } } } 2 \left( x ^ { 2 } + y ^ { 2 } \right) d x d y$
C) $\int _ { 0 } ^ { 4 } \int _ { - \sqrt { 16 - x ^ { 2 } } } ^ { \sqrt { 16 - x ^ { 2 } } } 2 \left( x ^ { 2 } + y ^ { 2 } \right) d y d x$
D) $- \int _ { - 4 } ^ { 4 } \int _ { - \sqrt { 16 - x ^ { 2 } } } ^ { \sqrt { 16 - x ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) d y d x$
E) $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) d y d x$

Question 24

Compute the flux of through the cylindrical surface oriented away from the z-axis.

Question 25

Evaluate $\int _ { S } \vec { F } \cdot \vec { d A }$ , where ${ \vec { F } } = x \vec { i } + 6 y \vec { j }$ and S is the part of the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ with x $\ge$ 0, y $\le$ 0, 0 $\ge$ z $\le$ 2, oriented toward the z-axis.

Question 26

Compute the flux of the vector field through the surface S that is the part of the surface above the disk , oriented in the positive z-direction.

Question 27

Find the flux of through the disk of radius 5 in the xz-plane, centered at the origin, and oriented upward.Give an exact answer.

Question 28

Let S be the spherical region of radius R with $\pi$ /3 $\le$ $\varphi$ $\le$ 2 $\pi$ /3 and $\pi$ /3 $\le$$\theta$ $\le$ 2 $\pi$ /3.Find the value of R so that $\int _ { S } \vec { r } \cdot \vec { d A } = 40 \pi$ Give your answer to two decimal places.

Question 29

Let ${ \vec { F } } = y \vec { i } + x \vec { j }$ .Write down an iterated integral that computes the flux of $\vec { F }$ through S, where S is the part of the cylinder $x ^ { 2 } + y ^ { 2 } = 9$ with x $\ge$ 0, y $\ge$ 0, 0 $\le$ z $\le$ 4, bounded between the planes y = 0 and y = x, oriented outward.

A) $- 2 \int _ { 0 } ^ { 4 } \int _ { 0 } ^ { \pi / 4 } 9 \cos \theta \sin \theta d \theta d z$
B) $2 \int _ { 0 } ^ { 4 } \int _ { 0 } ^ { \pi / 2 } 9 \cos \theta \sin \theta d \theta d z$
C) $2 \int _ { 0 } ^ { 4 } \int _ { 0 } ^ { \pi / 4 } 9 \cos \theta \sin \theta d \theta d z$
D) $- 2 \int _ { 0 } ^ { 4 } \int _ { 0 } ^ { \pi / 4 } 9 \cos \theta \sin \theta d z d \theta$
E) $2 \int _ { 0 } ^ { 4 } \int _ { 0 } ^ { \pi / 4 } 3 \cos \theta \sin \theta d \theta d z$

Question 30

A greenhouse is in the shape of the graph with the floor at z = 0.Suppose the temperature around the greenhouse is given by .Let be the heat flux density field.
Calculate the total heat flux outward across the boundary wall of the greenhouse.

Question 31

If $\vec { F }$ is a constant vector field and S₁ and S₂ are oriented rectangles with areas 1 and 2 respectively, then $\int _ { S _ { 2 } } \vec { F } \cdot \vec { d A } = 2 \int _ { S _ { 1 } } \vec { F } \cdot \vec { d A }$ .

Question 32

Let S be the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$ with x $\ge$ 0, y $\ge$ 0, z $\ge$ 0, oriented outward.Evaluate $\int _ { S } ( x z \vec { i } + 4 y z \vec { j } ) \cdot \vec { d A }$ .

Question 33

Let .Calculate the flux of through the surface oriented upward and given by z = f(x, y)= xy, over the region in the xy-plane bounded by the curves and between the origin and the point (1, 1).

Question 34

Compute the flux of the vector field through the surface S, where S is the part of the plane z = x + 2y above the rectangle oriented upward.
What is the answer if the plane is oriented downward?

Question 35

Let C be the portion of the cylinder $x ^ { 2 } + y ^ { 2 } = R ^ { 2 }$ of fixed radius R with $\pi$ /3 $\le$ $\theta$ $\le$ 2 $\pi$ /3 and -a $\le$ z $\le$ a oriented outward for some positive number a.Let S be the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = R ^ { 2 }$ with $\pi$ /3 $\le$$\theta$$\le$ 2 $\pi$ /3 and $\pi$ /3 $\le$ $\varphi$ $\le$ 2 $\pi$ /3 oriented outward.Determine the value of a for which the flux of $\vec { r }$ through each of these surfaces is equal in magnitude but opposite in sign for any choice of R.

Question 36

If $\vec{F}=-4 \vec{G}$ and S is an oriented surface, then $\int _ { S } \vec { F } \cdot \vec { d A } = - 4 \int _ { S } \vec { G } \cdot \vec { d A }$ .

Question 37

Compute the flux of the vector field through the surface S, where S is the part of the plane z = x + 2y above the rectangle oriented downward.

Question 38

Let be a constant vector field and S be an oriented surface.
Show that

Question 39

Calculate $\int _ { C } ( 5 x \vec { i } + 3 y \vec { j } ) \cdot \vec { d A }$ where C is a cylinder of radius R with 0 $\le$ z $\le$ 1.

A) $8 \pi R ^ { 4 }$
B) $2 \pi R ^ { 3 }$
C) $8 \pi R ^ { 2 }$
D) $8 \pi R ^ { 3 }$
E) $8 R ^ { 3 }$

Question 40

If $\vec { F}$ is a constant vector field and S₁ and S₂ are oriented rectangles with areas 3 and 15 respectively, both orientated with the vector $\overrightarrow { { i } } ,$ is it true that $\int _ { S _ { 2 } } \vec { F } \cdot \vec { d A } = 5 \oint _ { S _ { 1 } } \vec { F } \cdot \vec { d A }$ ?

Question 41

Calculate the flux of through the disk on the plane , oriented in the positive y-direction.

Question 42

Calculate the flux of $\vec { F} = \left( x ^ { 2 } + z ^ { 2 } \right) y \vec { i }$ , through the plane rectangle z = 3, 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 5, oriented in the positive z-direction.

Question 43

Let S be the cylinder .Find Q if .

Question 44

Calculate the flux of ${ \vec { F } } = 5 \vec { i } + 2 \vec { j } - 2 x \vec { k }$ through the plane rectangle z= 1, 0 $\le$ x $\le$ 5, 0 $\le$ y $\le$ 2, oriented downward.

Question 45

Calculate the flux of through a disk of radius 5 in the plane x = 3, oriented away from the origin.

Question 46

Let be a constant vector field with , where a, b, c are constants satisfying the condition .Let S be a surface lying on the plane x + 4y - 5z = 10 oriented upward.
If the surface area of S is 10, what is the smallest possible value of , and what are the corresponding values of a, b, c?

Question 47

Suppose that S is the surface which is a portion of the graph of a smooth function over a region R in the xy-plane, oriented upward.Consider the vector field .
Find so that .

Question 48

Suppose T is the triangle with vertices and oriented upward.Calculate the flux of through T exactly, and then give an answer rounded to 3 decimal places.

Question 49

Find the flux of ${ \vec { F } } = x \vec { i } + y \vec { j } + 8 z \vec { k }$ over the sphere S_a, $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ , oriented outward, with a > 0.

A) $\frac { 40 } { 3 } a ^ { 3 } \pi$
B) $40 a ^ { 3 }$
C) $\frac { 40 } { 3 } a ^ { 2 } \pi$
D) $\frac { 40 } { 3 } a ^ { 3 }$

Question 50

Let $\vec { F } = \frac { x \vec { i } + y \vec { j } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } }$ .
True or False: The flux of $\vec { F }$ through any cylinder C (with $x ^ { 2 } + y ^ { 2 } = R ^ { 2 } , 0 \leq z \leq 1$ )does not depend on the radius R.

Question 51

Calculate the flux of ${ \vec { F } } = \left( x ^ { 2 } + z ^ { 2 } \right) y \vec { j }$ , through the plane rectangle y = 4, 0 $\le$ x $\le$ 4, 0 $\le$ z $\le$ 5, oriented in the positive y-direction.

Question 52

Calculate the flux of $\vec { F } = 5 \vec { i } + 5 \vec { j } - 2 x \vec { k }$ through the plane rectangle y = 1, 0 $\le$ x $\le$ 1, 0 $\le$ z $\le$ 3, oriented in the negative y-direction.