Deck 20: The Curl and Stokes Theorem

Question 1

True or false? If $\operatorname { div } \vec { F } = 4$ for all x, y, z and if S is a surface enclosing a volume V, then $\int _ { S } \vec { F } \cdot \vec { d{ V } } = 4 V$

A)Not possible to decide
B)True
C)False

Answer 1

False

Question 2

Let $\vec { F }$ be a smooth vector field.The flux of $\vec { F}$ out of a small sphere of radius r centered at the point (1, $\pi$ , 2 $\pi$ )is 12 $\pi$ r³. Find $\operatorname { div } \vec { F }$ at (1, $\pi$ , 2 $\pi$ )and use the result to estimate the flux of $\vec {F }$ out of the small cube of side 0.025 centered around the point (1, $\pi$ , 2 $\pi$ ).

A)0.000141
B)0.005625
C)0.000125
D)0.00625
E)0.25

Answer 2

0.000141

Question 3

Use the Divergence Theorem to find the flux of the vector field through the sphere x² + y² + z² = .

Answer 3

Question 4

An oceanographic vessel suspends a paraboloid-shaped net below the ocean at depth of $1200$ feet, held open at the top by a circular metal ring of radius $20$ feet, with bottom $90$ feet below the ring and just touching the ocean floor.Set up coordinates with the origin at the point where the net touches the ocean floor and with z measured upward.

A) $\int _ { - 20 } ^ { 20 } \int _ { - \sqrt { 400 - y } ^ { 2 } } ^ { \sqrt { 400 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { i } + \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) \vec { j } - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { k } \right) \cdot \left( \frac { x } { 2 } \vec { i } + \frac { y } { 2 } \vec { j } - \vec { k } \right) d x d y$
B) $\int _ { - 20 } ^ { 20 } \int _ { - y ^ { 400 - y ^ { 2 } } } ^ { \sqrt { 400 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { i } + \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) \vec { j } - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { k } \right) d x d y$
C) $\int _ { - 20 } ^ { 20 } \int _ { - \sqrt { 400 - y } ^ { 2 } } ^ { \sqrt { 400 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { i } - \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) \vec { j } + \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { k } \right) \cdot \left( \frac { x } { 2 } \vec { i } + \frac { y } { 2 } \vec { j } - \vec { k } \right) d x d y$
D) $\int _ { - 20 } ^ { 20 } \int _ { - \sqrt { 20 - y ^ { 2 } } } ^ { \sqrt { 20 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { i } - \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) \vec { j } + \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { k } \right) \cdot \left( \frac { x } { 2 } \vec { i } + \frac { y } { 2 } \vec { j } - \vec { k } \right) d x d y$
E) $\int _ { - 20 } ^ { 20 } \int _ { - \sqrt { 400 - y ^ { 2 } } } ^ { \sqrt { 400 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) - \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) + \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) d x d y \right.$

Question 5

Let Calculate div

Question 6

An oceanographic vessel suspends a paraboloid-shaped net below the ocean at depth of feet, held open at the top by a circular metal ring of radius feet, with bottom feet below the ring and just touching the ocean floor.Set up coordinates with the origin at the point where the net touches the ocean floor and with z measured upward.

Question 7

Let be a smooth velocity vector field describing the flow of a fluid.Suppose that Will there be an inflow or outflow of fluid at the point (1, 2,-1)?

Question 8

Let Calculate the flux , where S is the sphere (x-2)² + (y-3)² + z² = oriented inward.

Question 9

If ${ \vec { F } } = 5 y e ^ { x ^ { 2 } }{\vec { i } } + 4 x y e ^ { y} \vec { j } + 4 z \cos ( x y ) \vec { k }$ find $\operatorname { div } \vec { F} .$

A) $10 x y e ^ { x ^ { 2 } } \vec { i } + \left( 4 x e ^ { y } + 4 x y e ^ { y } \right) \vec { j } + 4 \cos ( x y ) \vec { k }$
B) $10 x y e ^ { x ^ { 2 } } + 4 x e ^ { y } + 4 x y e ^ { y } - 4 \cos ( x y )$
C) $10 x y e ^ { x ^ { 2 } } + 4 x y e ^ { y } + 4 \cos ( x y )$
D) $10 x y e ^ { x ^ { 2 } } + 4 x e ^ { y } + 4 x y e ^ { y } + 4 \cos ( x y )$
E) $5 x y e ^ { x ^ { 2 } } \vec { i } + \left( 4 x e ^ { y } + 4 x y e ^ { y } \right) \vec { j } + 4 \cos ( x y ) \vec { k }$

Question 10

Evaluate the flux integral , where S is the surface of the cylinder x² + y² = 1 bounded by the planes z = -1, z = 1 (including the ends of the cylinder).

Question 11

Consider the two-dimensional fluid flow given by where a is a constant.
(We allow a to be negative, so may or may not be defined at (0, 0).)
(a)Is the fluid flowing away from the origin, toward it, or neither?
(b)Calculate the divergence of .Simplify your answer.
(c)For what values of a is div positive? Zero? Negative?
(d)What does your answer to (c)mean in terms of flow? How does this fit in with your answer to (a)?

Question 12

Let Given that find the value of a.

Question 13

(a)Is a divergence free vector field?
(b)Do all divergence free vector fields have the form of the vector field in (a)?
(c)If has the form given in (a)can we conclude that for any closed surface S?

Question 14

Let $\vec { F } = \| \vec { r } \| ^ { p } \vec { r }$ , where p is a positive constant.Is there a value of p such that $\vec {F}$ is a divergence free vector field?

Question 15

Let What is

Question 16

Suppose is a vector field with the property that at every point of the surface If where c is a constant, find c.

Question 17

Let ,
What is , where S is the cube with corners at (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1,0,1), (0,1,1), and (1, 1, 1)?

Question 18

Let $\vec { a } = a _ { 1 } \vec { i } + \alpha _ { 2 } \vec { i } + \alpha _ { 3 } \vec { i }$ be a constant vector and f(x, y, z)be a smooth function.Which statement is true?

A)If $\operatorname { div } f \vec { a }$ is a divergence free vector field then $\nabla$ f is parallel to
$\vec { \alpha }$
B)If $\operatorname { div } f \vec { a }$ is not a divergence free vector field then $\nabla$ f is perpendicular to
$\vec { \alpha }$
C)If $\operatorname { div } f \vec { a }$ is a divergence free vector field then $\nabla$ f is perpendicular to
$\vec { \alpha }$
D)If $\operatorname { div } f \vec { a }$ is not a divergence free vector field then $\nabla$ f is parallel to
$\vec { \alpha }$
E)If $\operatorname { div } f \vec { a }$ is a divergence free vector field then $\nabla$ f is constant.

Question 19

State the Divergence Theorem.

A)If W is a solid region whose boundary S is a piecewise smooth surface, then $\int _ { S } \vec { H } \cdot \vec { d A } = \int _ { W } { div } { \vec { F} } d V$
B)If W is a solid region whose boundary S is a smooth surface oriented outward, then $\int _ { S } \vec { H } \cdot \vec { d A } = \int _ { W } { div } { \vec { F} } d V$
C)If W is a solid region whose boundary S is a piecewise smooth surface oriented outward, and if $\vec { F }$ is a smooth vector field on an open region containing W and S, then
$\int_{S} \vec{F} \cdot \overrightarrow{d A}=\int_{W} \operatorname{div} \vec{F} d V$
D)If W is a solid region whose boundary S is a piecewise smooth surface oriented outward, then $\int _ { S } \vec { F } \cdot \vec { d A } = \int _ { W } { div } \vec { F } \cdot \overrightarrow { d \bar { V } }$
E)If S is a solid region whose boundary W is a piecewise smooth surface oriented outward, then $\int _ { S } \vec { F } \cdot \vec { d A } = \int _ { W } { div } \vec { F } { d { V } }$

Question 20

Let be a smooth velocity vector field describing the flow of a fluid.Suppose that Estimate the value of where S is a sphere of radius 0.25 centered at (1, 2,-1)oriented outward.Give your answer to 4 decimal places.

Question 21

Let Find the curl of

Question 22

Find

Question 23

Let Use the Divergence Theorem to calculate where S is the sphere of radius a centered at the point ( a, 0, 0).

Question 24

Let Is a divergence free vector field?

Question 25

Let Let S₁ be the sphere of radius centered at the origin, oriented outward and let S₂ be the sphere of radius centered at the origin, also oriented outward.
Do you expect the value of to be larger, smaller or the same as ?

Question 26

Use the Divergence Theorem to find the flux of the vector field $\vec { F } = \left( 4 x ^ { 3 } + 4 y z \right) \vec { i } + \left( 4 y ^ { 3 } + 8 x z \right) \vec { j } + \left( 4 z ^ { 3 } + 16 y x \right) \vec { k }$ through the cube 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1, 0 $\le$ z $\le$ 1.

Question 27

Let be a smooth vector field with curl Estimate the circulation around a circle of radius 0.01 in the yz plane, oriented counterclockwise when viewed from the positive x-axis.

Question 28

Suppose that $\vec { F }$ is a vector field defined everywhere with constant negative divergence C.Decide if the following statement is true.
There is $\text { no oriented }$ surface S for which $\hat { \mathrm { Q } } { \vec { F } \times \vec{ d A } } = 0$

Question 29

Let Find the divergence of

Question 30

Calculate $\operatorname { curl } \overline { \bar { H } }$ where ${ \vec{ H } } = 9 x \vec { i } - 8 x y \vec { j } + x z ^ { 5 } \vec { k }$

A) $\operatorname { cur } 1 { \vec { H } } = z ^ { 5 } \vec { j } - 8 y \vec { k }$
B) $\operatorname { curl } { \vec { H } } = - z ^ { 5 } \vec { j } - 8 y \vec { k }$
C) $\operatorname { curl } { \vec{ H } } = z ^ { 5 } \vec { i } - 8 y \vec { k }$
D) $\operatorname { curl } { \vec { H } } = - z ^ { 5 } \vec { j } + 8 y \vec { k }$

Question 31

If $\operatorname { div } \vec { F } = - 7$ everywhere and S is a smooth surface (oriented outward)enclosing a volume W of size V find $\int _ { S } \vec { F } \cdot \vec { d A }$ .

A) $7 V$
B) $- 7$
C) $- 7 V$
D) $- 7 W$
E)0

Question 32

Below is a picture of the vector field from above. Match the value of curl with one of the following values:

Question 33

The figure below shows a vector field of the form

Question 34

How is curl defined? Select all that apply.

A)curl is a vector in the direction
for which
is the greatest that has magnitude equal to the circulation density of
around that direction.
B)If then

C)curl is a vector in the direction
for which the circulation is greatest and has magnitude equal to the circulation of
around that direction.
D)If then

E)If then

Question 35

Suppose is a vector field with the property that at every point of the surface If is not a constant, estimate the value of

Question 36

What is meant by curl $\vec { F }$ where $\vec { F }$ is a vector field? Is curl $\vec { F}$ a vector or a scalar?

A)curl $\vec { F}$ is a scalar which tells you how a vector field is rotating.Its direction gives the axis of rotation; its magnitude gives the strength of rotation about the axis.
B)curl $\vec { F }$ is a vector which tells you how a scalar field is rotating.Its direction gives the axis of rotation; its magnitude gives the strength of rotation about the axis.
C)curl $\vec { F }$ is a vector which tells you how a vector field is rotating.Its direction gives the axis of rotation; its magnitude gives the strength of rotation about the axis.
D)curl $\vec { F}$ is a vector which tells you how a vector field is rotating.Its magnitude gives the axis of rotation; its direction gives the strength of rotation about the axis.
E)curl $\vec { F }$ is a scalar which tells you how a scalar field is rotating.Its magnitude gives the strength of rotation about the axis.

Question 37

Let

A)The answer in (a)is bigger than (b).
B)The answer in (a)is smaller than (b).
C)They are the same.

Question 38

Suppose that where C_a is the circle for any a > 0.
Does knowing this tell you anything about curl Is so, what? If not, why not?

Question 39

Let be a smooth vector field with Determine the values of a, b and c.

Question 40

Suppose that is a vector field defined everywhere with constant negative divergence C.Decide if the following statement is true and explain your answer. for every surface S.

Question 41

Show that the vector field $\vec { F } = - 3 x e ^ { x } \vec { i } - y ( 3 z - 5 ) e ^ { x } \vec{ j } + ( 3 z - 5 ) e ^ { x } \vec { k }$ is a divergence free vector field.
Use this result to calculate $\hat { \mathrm {O } } _ {S}{ \vec { F } \times \vec { dA } }$ where S is the open surface which is the graph of $f ( x , y ) = - x ^ { 2 } - y ^ { 2 } + 4$ with f(x, y) $\ge$ 0.

Question 42

Suppose a vector field is always perpendicular to the normal vector at each point of a surface S.What is the value of

Question 43

Let Use Stokes' Theorem to find where C is a circle in the xz-plane of radius , centered at oriented when viewed from the positive y-axis.

Question 44

Let (a)Evaluate the line integral , where C is the circle on the xy-plane, oriented in a counter-clockwise direction when viewed from above.
(b)Without any computation, explain why the answer in part (a)is also equal to the flux integral where S₁ is lower hemisphere oriented inward.

Question 45

Suppose that curl $\vec {F }$ is not zero.True or false? If curl $\vec { F }$ is parallel to the $\mathcal { X }$ -axis for all x, y, and z and if C is a circle in the xy-plane, then the circulation of $\vec { F }$ around C is zero.

Question 46

Given , find the values of the constants a and b, without knowing the expression of .

Question 47

Given that use the curl test to find the values of the constants a, b and c.

Question 48

Use the curl test to determine whether the following vector field is a gradient field. $\vec { F } = y ^ { 3 } z ^ { 2 } \vec { i } + \left( 4 y ^ { 2 } x z ^ { 2 } + 4 y z ^ { 3 } \right) \vec { j } + \left( 2 x y ^ { 3 } z + 3 y ^ { 2 } z ^ { 2 } \right) \vec { k }$

A)Only at the origin
B)No
C)Yes

Question 49

Let ${ \vec { F } } = - 2 x z \vec { i } + 5 ( y - x ) \vec { j } + 5 x \vec { k }$ (a)Use Stokes' Theorem to find the circulation of $\vec { F}$ around the circle of radius a, $\vec { r } ( t ) = a \cos t \vec { i } + \alpha \sin t \vec { j }$ for 0 $\le$ t $\le$ 2 $\pi$ .
(b)Use this result to find the $\vec { k }$ component of $\operatorname { curl } \vec { F } ( 0,0,0 )$

Question 50

If $\operatorname { cur } 1 { \vec { F} } = \vec { 0 }$ then by Stokes' Theorem the line integral $\dot { \mathrm { Q } } { \bar { F } \times \vec { d r } }$ is equal to zero, where C is the curve y = x², for 0 $\le$ x $\le$ 2.

Question 51

Suppose that the flux of a smooth vector field out of a sphere of radius r centered at the origin is where a and b are constants.If for a smooth vector field find the values of a and b.

Question 52

Suppose that curl curl and curl Estimate the following line integrals.
(a) where C₁ is given by (b) where C₂ is given by (c) where C₃ is given by

Question 53

You want to build a windmill at the origin that maximizes the circulation of the wind.The wind vector field at any point (x, y, z)in your coordinate world is given by
(a)In which direction should you face the windmill to get maximum use from the wind?
(b)What will be the strength of the circulation of the wind when you face it in this direction?

Question 54

Let ${ \vec { F } } = - 3 x z \vec { i } + 4 ( y - x ) \vec { j } + 4 x \vec { k }$
(a)By direct computation, find the circulation of $\vec { F }$ around the circle of radius a, $\vec { r } ( t ) = \alpha \cos t \vec { i } + \alpha \sin t \vec { j }$ for 0 $\le$ t $\le$ 2 $\pi$ .
(b)Use this result to find the $\vec { k }$ component of $\operatorname { curl } \vec { F } ( 0,0,0 )$

Question 55

Let be a vector field such that (a)Is a curl field?
(b)Use spherical coordinates to evaluate where W is the solid ball of radius R centered at the origin.
(c)Use the result of part (b)to find the radius of a sphere centered at the origin, such that the flux of out of this sphere is zero.

Question 56

State Stokes' Theorem.

A)If S is a smooth oriented surface with smooth, oriented boundary C, then $\int _ { C } \vec { F } \cdot \vec { d r } = \int _ { S } \operatorname { curl } \vec { F } \cdot \vec { d A }$
B)If S is a smooth oriented surface with piecewise smooth, oriented boundary C, then $\int _ { S } \vec { F } \cdot \vec { d r } = \int _ { C } \operatorname { curl } \vec { F } \cdot \vec { d A }$
C)If S is a smooth oriented surface with piecewise smooth, oriented boundary C, then $\int _ { C } \vec { F } \cdot \vec { d r } = \int _ { S } \operatorname { cur } \vec { F } d V$
D)If S is a smooth oriented surface with piecewise smooth, oriented boundary C, and if $\vec { F }$ is a smooth vector field on an open region containing S and C, then
$\int _ { C } \vec { F } \cdot \vec{ d r } = \int _ { S } \operatorname { curl } \vec { F } \cdot \vec { d A }$
E)If S is a smooth oriented surface with piecewise smooth, oriented boundary C, then $\int _ { C } { \vec { F } } \cdot \vec { d r } = \int _ { S } d i v { \vec { F } } d V$

Question 57

The trumpet surface, S, is given parametrically by If S has outward pointing normal, use an appropriate line integral to calculate

Question 58

(a)The vector field has the property that the flux of through any closed surface is 0.What is the value of the constant a?
(b)The vector field has the property that the circulation of around any closed curve is 0.What are the values of the constants a, b and c?

Question 59

Let $\vec { F } = 4 z \vec { i } - 4 x \vec { k }$ Let C be the circle of radius a parameterized by x = a cos t, y = 0, z = a sin t, 0 $\le$ t $\le$ 2 $\pi$ and let S be the disk in the xz-plane enclosed by C, oriented in the positive y-direction.
(a)Evaluate directly $\int _ { C } \vec { F } \cdot d \vec { r }$ (b)Evaluate directly $\int _ { S } \operatorname { curl } { \vec { F } } \cdot d \vec { A }$ (c)Do these results contradict Stokes' Theorem?

Question 60

Suppose that the flux of a smooth vector field out of a sphere of radius r centered at the origin is ar³ + br⁴, where a and b are constants.Calculate , where W is the solid

Question 61

Let Use Stokes' Theorem to find where C is a circle in the xy-plane of radius , centered at oriented counter-clockwise when viewed from above.

Question 62

Let P be a plane through the origin with equation Let be a vector field with curl Suppose for any closed curve on the plane Using Stokes' Theorem, determine the value of a.

Question 63

Let $\vec { G }$ be a smooth vector field with $\operatorname { div } \vec { G } = 3$ at every point in space and let S₁ and S₂ be spheres of radius r, oriented outward, centered at (0,0,0)and at (1,2,1), respectively.
$\int _ { s _ { 1 } } \vec { G } \cdot d \vec { A } = \int _ { S _ { 2 } } \vec { G } \cdot d \vec { A }$ .

Question 64

Let S be the closed surface which is the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ with $- 2 \leq \boldsymbol { z } \leq 1$ topped by the disk in the plane $\bar { z } = 1$ , oriented outward.Then the flux of $\vec { r } = x \vec { i } + y \vec { j } + z \vec { k }$ through S is:

A) $9 \pi$
B) $24 \pi$
C) $27 \pi$
D) $18 \pi$
E) $36 \pi$

Question 65

Let S be the surface of the upper part of the cylinder 4x² + z² = 1, z $\ge$ 0, between the planes y = -1, y = 1, with an upward-pointing normal.
(a)Evaluate the flux integral $\int _ { S } \left( - 3 x y ^ { 2 } z \vec { i } + 7 \cos z \vec { j } + 3 \vec { k } \right) \cdot \vec { d A }$ (b)Consider W, the solid region described by -1 $\le$ y $\le$ 1, 4x² + z² $\le$ 1, z $\ge$ 0.Evaluate $\int _ { W } \operatorname { div } \left( - 3 x y ^ { 2 } z \vec { i } + 7 \cos z \vec { j } + 3 \vec { k } \right) d V$ Does this contradict the Divergence Theorem? Explain.

Question 66

For the following integral, say whether Stokes' Theorem, the Divergence Theorem, or neither applies. $\hat { Q } \left( 2 x \vec { i } + z \vec { j } + 5 y ^ { 3 } \vec { k } \right) \times \vec{ d r }$ where S is a triangular plane in space oriented upward.

A)Stokes' Theorem
B)Divergence Theorem.
C)Neither

Question 67

Using either Stokes' theorem or the Divergence theorem (whichever is appropriate), evaluate where S is the sphere of radius 2 oriented outward and centered at the point

Question 68

Let be a smooth vector field with at every point in space.Find the exponent p in the following:

Question 69

For the following integral, say whether Stokes' Theorem, the Divergence Theorem, or neither applies. $\dot{\mathrm { Q }} \left( 6 y \vec { i } + 9 z ^ { 2 } \vec { j } + 2 \vec { k } \right) \times \vec { d r }$ where C is a closed loop in space.

A)Stokes' Theorem
B)Divergence Theorem.
C)Neither

Question 70

On an exam, students are asked to find the line integral of ${ \vec { F } } = 2 x y \vec { i } + y z \vec { j } + y ^ { 2 } \vec { k }$ over the curve C which is the boundary of the upper hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 ^ { 2 } , z \geq 0$ oriented in a counter-clockwise direction when viewed from above.One student wrote: " $\operatorname { curl } { \vec { F } } = - 2 x \vec { j } + 2 \vec { k }$ By Stokes' Theorem $\int _ { C } \vec { F } \cdot \vec{ d r } = \int _ { S } ( - 2 x \vec { j } + 2 \vec { k } ) \cdot \vec { d A }$ where S is the hemisphere.Since $\operatorname { div } ( - 2 x \vec { j } + 2 \vec { k } ) = 0$ by the Divergence Theorem $\int _ { S } ( - 2 x \vec { j } + 2 \vec { k } ) \cdot \vec { d A } = \int _ { W } 0 d V = 0$ where W is the solid hemisphere.Hence we have $\int _ { C } \vec { F } \cdot \vec{ d r } = 0$ "
This answer is wrong.Which part of the student's argument is wrong? Select all that apply.

A)The student has been careless with the orientations of the curve and surface.
B)The student has to be careful with the orientations of the curve and surface.However, Stokes' Theorem has been applied correctly.
C)The student has used Divergence Theorem incorrectly.The upper hemisphere S does not include the bottom, hence it does not enclose any region in space, and we cannot apply the Divergence Theorem to this surface.
D)The student has to be careful with the orientations of the curve and surface.However, the Divergence Theorem has been applied correctly.
E)The student has the correct orientations of the curve and surface but the student has used Divergence Theorem for the hemisphere incorrectly.The upper hemisphere S does not include the bottom, hence it does not enclose any region in space, and we cannot apply the Divergence Theorem to this surface.

Question 71

Let S be the boundary surface of a solid region W with outward-pointing normal.Using an appropriate theorem, change the following flux integral into volume integral over W.

Question 72

Suppose that is defined and differentiable everywhere and satisfies the differential equation .Let , where .Suppose that S is a closed surface and W is its interior.Find Q in the following equation: .

Question 73

Let
(a)Compute .
(b) By direct computation, find the flux of through a cube with edge length l, centered at the origin and edges parallel to the axes.
(c)Explain how your answers in parts (b)are related to that of part (a).

Question 74

Suppose for any closed surface S in space with outward-pointing normal.What does this tell you about

Question 75

Using either Stokes' theorem or the Divergence theorem (whichever is appropriate), evaluate the following: where C is a closed loop parameterized by

Question 76

Let $\vec { a } = \alpha _ { 1 } \vec { i } + \alpha _ { 2 } \vec { j } + \alpha _ { 3 } \vec { k }$ be a nonzero constant vector and let $\vec { r } = x \vec { i } + y \vec { j } + z \vec { k }$ .Suppose S is the sphere of radius one centered at the origin.There are two (related)reasons why $\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0$ .Select them both.

A) $\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0$ because $\vec { a } \times \vec { r } = \overrightarrow { 0 }$ .
B) $\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0$ because $\operatorname { div } ( \vec { a } \times \vec { r } ) = 0$ .
C) $\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0$ because $\vec { r }$ is parallel to the
$d \vec { A }$ element everywhere on S and so
$\vec { a } \times \vec { r }$ is perpendicular to
$d \vec { A }$ on S.
D) $\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0$ because $\vec { a } \times \vec { r }$ is a constant vector field.
E) $\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0$ because $\int _ { H } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = Q > 0$ and
$\int _ { L } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = - Q$ , where H is the upper unit hemisphere and L is the lower unit hemisphere.

Question 77

Suppose that $\vec { F }$ is a smooth vector field, defined everywhere.
It is possible that $\int _ { S } \vec { F } \cdot d \vec { A } = 3 r ^ { 2 } + 2 r$ , where S is a sphere of radius r centered at the origin.

Question 78

A physicist is asked to find the flux of a magnetic field $\vec { F } = 4 \vec { k }$ through the curved surface of a right circular cone.The cone is of height 4 and has a circular base of radius 2.He says immediately that, by the Divergence Theorem, the answer is 16 $\pi$ .How did he know?

Question 79

Let .Calculate curl .

Question 80

For the following integral, say whether Stokes' Theorem, the Divergence Theorem, or neither applies. $\dot { \mathrm { Q } } ^ { \operatorname { curl } } \left( 2 x \vec { i } + z \vec { j } + 2 y ^ { 3 } \vec { k } \right) \times \vec{ d A }$ where S is a triangular plane in space oriented $\text { dow nw ard. }$

A)Stokes' Theorem
B)Divergence Theorem.
C)Neither