Deck 16: Vector Calculus

Question 1

Use Stokes' Theorem to evaluate . ;
S is the part of the ellipsoid lying above the xy-plane and oriented with normal pointing upward.

Answer 1

0

Question 2

Use Stoke's theorem to calculate the surface integral where and S is the part of the cone

Answer 2

0

Question 3

Use Stoke's theorem to evaluate C is the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.

Answer 3

Question 4

Use the Divergence Theorem to calculate the surface integral $\iint _ { 5 } \mathbf { F } \cdot d \mathbf { S }$ ; that is, calculate the flux of $F$ across $S$ . $\mathbf { F } ( x , y , z ) = x y e ^ { x } \mathbf { i } + x y ^ { 2 } z ^ { 3 } \mathbf { j } - y e ^ { x } \mathbf { k }$ S is the surface of the box bounded by the coordinate planes and the planes $x = 4 , y = 2 \text { and } z = 1$ .

A) $\frac { 11 } { 3 }$
B) $\frac { 5 } { 3 }$
C) $\frac { 1 } { 8 }$
D) $8$
E) $\frac { 7 } { 3 }$

Question 5

Use Stokes' Theorem to evaluate $\oint _ { C } \mathbf { F } \cdot d \mathbf { r }$ . $\mathbf { F } ( x , y , z ) = 7 \cos x \mathbf { i } + 5 e ^ { y } \mathbf { j } + 2 x y \mathbf { k }$ ; C is the curve obtained by intersecting the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ with the hyperbolic paraboloid $z = x ^ { 2 } - y ^ { 2 }$ , oriented in a counterclockwise direction when viewed from above

A) $- 2 \pi$
B) 0
C) $\pi$
D) $1$

Question 6

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { cur } \mathbf { F } \cdot d \mathbf { S }$ . $\mathbf { F } ( x , y , z ) = 6 x y \mathbf { i } + 5 y z \mathbf { j } + 2 z ^ { 2 } \mathbf { k }$ ; S is the part of the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ lying below the plane $z = 6$ and oriented with normal pointing downward.

A) 0
B) $- 3$
C) $3$
D) $- 4$

Question 7

Use Stokes' Theorem to evaluate S consists of the top and the four sides (but not the bottom) of the cube with vertices oriented outward.

Question 8

Use a computer algebra system to compute the flux of F across S. S is the surface of the cube cut from the first octant by the planes $x = \frac { \pi } { 2 } , y = \frac { \pi } { 2 } , z = \frac { \pi } { 2 }$ $\mathbf { F } ( x , y , z ) = 3 \sin x \cos ^ { 2 } y \mathbf { i } + 3 \sin ^ { 3 } z \cos ^ { 4 } z \mathbf { j } + 3 \sin ^ { 5 } z \cos ^ { 6 } x \mathbf { k }$

A) 3
B) 4
C) $6$
D) 0.67
E) 1

Question 9

Evaluate the surface integral. $\iint _ { S } 8 x z d S$ S is the part of the plane $2 x + 2 y + z = 4$ that lies in the first octant.

A) $I = 37$
B) $I = 47$
C) $I = 32$
D) $I = 57$
E) $I = 67$

Question 10

Use Stoke's theorem to evaluate C is the curve of intersection of the plane z = x + 9 and the cylinder

Question 11

Use Stoke's theorem to evaluate C is the boundary of the part of the paraboloid in the first octant. C is oriented counterclockwise as viewed from above.

Question 12

Evaluate the surface integral. Round your answer to four decimal places. $\iint _ { S } 3 z d S$ S is surface $x = y ^ { 2 } + 2 z ^ { 2 } , 0 \leq y \leq 1,0 \leq z \leq 1$

A) $10.5963$
B) $13.5963$
C) $4.5963$
D) $23.5963$
E) $8.5963$

Question 13

Suppose that $f ( x , y , z ) = g \left( \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } \right)$ where g is a function of one variable such that $g ( 5 ) = 8$ . Evaluate $\iint _ { S } f ( x , y , z ) d S$ where S is the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$

A) $920 \pi$
B) $820 \pi$
C) $800 \pi$
D) $880 \pi$
E) None of these

Question 14

Use Stokes' Theorem to evaluate . ;
C is the boundary of the triangle with vertices , , and oriented in a counterclockwise direction when viewed from above

Question 15

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find $\iint _ { 5 } 3 \mathbf { a } \mathbf { n } d S$ , where a is the constant vector.

A) 6
B) 5
C) 7
D) 8
E) 3

Question 16

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ $\mathbf { F } ( x , y , z ) = 7 x y \mathbf { i } + 7 e ^ { x } \mathbf { j } + 7 x y ^ { 2 } \mathbf { k }$ S consists of the four sides of the pyramid with vertices (0, 0, 0), (3, 0, 0), (0, 0, 3), (3, 0,3) and (0, 3, 0) that lie to the right of the xz-plane, oriented in the direction of the positive y-axis.

A) 0
B) 12
C) 16
D) 49
E) 1

Question 17

The temperature at the point $( x , y , z )$ in a substance with conductivity $K$ is $u ( x , y , z ) = 6 x ^ { 2 } + 6 y ^ { 2 }$ Find the rate of heat flow inward across the cylindrical $y ^ { 2 } + z ^ { 2 } = 6 , \quad 0 \leq x \leq 5$

A) $2530 \pi$
B) $2560 \pi$
C) $2520$
D) $2540 \pi$
E) $2520 \pi$

Question 18

Use Stoke's theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = e ^ { - 7 x } \mathbf { i } + e ^ { 4 y } \mathbf { j } + e ^ { 5 x } \mathbf { k }$ and C is the boundary of the part of the plane $8 x + y + 8 z = 8$ in the first octant.

A) 69
B) 16
C) 49
D) 0
E) 23

Question 19

Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.

Question 20

Evaluate $\iint _ { S } f ( x , y , z ) d S$ . $f ( x , y , z ) = x + y$ ; S is the part of the plane $7 x + 3 y + z = 21$ in the first octant.

A) 0
B) $\sqrt { 327 }$
C) $35$ $\sqrt { 59 }$
D) $2 \sqrt { 139 }$

Question 21

Find the area of the surface. The part of the paraboloid $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + u ^ { 2 } \mathbf { k }$ ; $0 \leq u \leq 7$ , $0 \leq v \leq 2 \pi$

A) $\frac { \pi ( 197 \sqrt { 197 } - 1 ) } { 12 }$
B) $\frac { \pi ( 197 \sqrt { 197 } - 1 ) } { 3 }$
C) $\frac { \pi ( 197 \sqrt { 197 } - 1 ) } { 6 }$
D) $\frac { 5 \pi ( 197 \sqrt { 197 } - 1 ) } { 12 }$

Question 22

Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. in the first octant,
with orientation toward the origin.

Question 23

Find a parametric representation for the part of the elliptic paraboloid $x + y ^ { 2 } + 2 z ^ { 2 } = 7$ that lies in front of the plane x = 0.

A) $x = 7 - y ^ { 2 } - 2 z ^ { 2 } , y = y , z = y , y ^ { 2 } + 2 z ^ { 2 } \leq 7$
B) $x = x , y = \sqrt { 7 - x + 2 z ^ { 2 } } , z = z$
C) $x = x , y = \pm \sqrt { 7 - x + 2 z ^ { 2 } } , z = z$
D) $x = 7 - y ^ { 2 } - 2 z ^ { 2 } , y = y , z = y , 0 \leq y ^ { 2 } + 2 z ^ { 2 } \leq 3$
E) $x = 7 - y ^ { 2 } - 2 z ^ { 2 } , y = y , z = y , y ^ { 2 } + 2 z ^ { 2 } \geq 7$

Question 24

Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S.

Question 25

Find the mass of the surface S having the given mass density. S is the hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ , $z \geq 0$ ; the density at a point P on S is equal to the distance between P and the xy-plane.

A) $9 \pi$
B) $27 \pi$
C) 9
D) $3 \pi$

Question 26

Match the equation with one of the graphs below. $$\mathbf { r } ( u , v ) = u ^ { 2 } \mathbf { i } + u \cos v \mathbf { j } + u \sin v \mathbf { k }$$

A)
B)
C)
D)

Question 27

Find the mass of the surface S having the given mass density. S is part of the plane $x + y + 3 z = 3$ in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.

A) $\sqrt { 11 }$
B) $\frac { \sqrt { 11 } } { 4 }$
C) 49
D) 20

Question 28

Find the area of the part of the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ that is cut off by the cylinder $( x - 3 ) ^ { 2 } + y ^ { 2 } = 4$

A) $9 \pi \sqrt { 2 }$
B) $3 \pi \sqrt { 2 }$
C) $2 \pi \sqrt { 2 }$
D) $4 \pi \sqrt { 2 }$

Question 29

Evaluate $\iint _ { S } f ( x , y , z ) d S$ . $f ( x , y , z ) = z$ ; S is the part of the torus with vector representation $\mathbf { r } ( u , v ) = ( 5 + 3 \cos v ) \cos u \mathbf { i } + ( 5 + 3 \cos v ) \sin u \mathbf { j } + 3 \sin v \mathbf { k }$ , $0 \leq u \leq 2 \pi$ , $0 \leq v \leq \frac { \pi } { 2 }$ .

A) 0
B) $117 \pi$
C) $117$
D) $27 \pi$

Question 30

Evaluate , that is, find the flux of F across S. ; S is the part of the paraboloid between the planes z = 0 and z = 5; n points upward.

Question 31

A fluid with density flows with velocity Find the rate of flow upward through the paraboloid

Question 32

Use Gauss's Law to find the charge contained in the solid hemisphere , if the electric field is

Question 33

Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone if its density function is

Question 34

Evaluate $\iint _ { S } f ( x , y , z ) d S$ . $f ( x , y , z ) = 5 x + 6 y + z$ ; S is the part of the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ between the planes $z = 1$ and $z = 4$ .

A) $126 \pi \sqrt { 2 }$
B) $42$ $\pi \sqrt { 2 }$
C) $126 \sqrt { 2 }$
D) 0

Question 35

Find the area of the surface. The part of the plane $$\mathbf { r } ( u , v ) = ( 5 u + 3 v + 4 ) \mathbf { i } + ( 2 u + 3 v + 4 ) \mathbf { j } + ( 4 u + 2 v + 8 ) \mathbf { k }$$ ; $0 \leq u \leq 1$ , $0 \leq v \leq 5$

A) $5 \sqrt { 149 }$
B) $\sqrt { 58 }$
C) $\sqrt { 51 }$
D) $2 \sqrt { 11 }$

Question 36

Evaluate the surface integral where S is the surface with parametric equations $x = 7 u v$ , $y = 6 ( u + v ) , z = 6 ( u - v ) , u ^ { 2 } + v ^ { 2 } = 3$ . $\iint _ { S } 10 y z d S$

A) $I = 0$
B) $I = 10 + 7 \pi$
C) $I = 3$
D) $I = 10$
E) $I = 10 + 3 \pi$

Question 37

Evaluate the surface integral. S is the part of the cylinder between the planes and in the first octant.

Question 38

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , that is, find the flux of F across S. $\mathbf { F } ( x , y , z ) = - 5 z \mathbf { i } + 3 y \mathbf { j } - 5 x \mathbf { k }$ ; S is the hemisphere $z = \sqrt { 9 - x ^ { 2 } - y ^ { 2 } }$ ; n points upward.

A) $27$ $\pi$
B) 162
C) $54$ $\pi$
D) 162 $\pi$

Question 39

Let S be the cube with vertices $( \pm 1 , \pm 1 , \pm 1 )$ . Approximate $\iint _ { S } \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 7 z ^ { 2 } }$ by using a Riemann sum as in Definition 1, taking the patches $S _ { i j }$ to be the squares that are the faces of the cube and the points $P _ { i j }$ to be the centers of the squares.

A) $8 ( 3 + \sqrt { 2 } )$
B) $8 ( 3 + \sqrt { 7 } )$
C) $8 ( 1 + \sqrt { 2 } + \sqrt { 7 } )$
D) $4 ( 1 + \sqrt { 2 } + \sqrt { 7 } )$
E) none of these

Question 40

Find the area of the surface. The part of the paraboloid $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + u ^ { 2 } \mathbf { k }$ ; $0 \leq u \leq 2$ , $0 \leq v \leq 2 \pi$

A) $\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 6 }$
B) $\frac { 5 \pi ( 17 \sqrt { 17 } - 1 ) } { 12 }$
C) $\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 12 }$
D) $\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 3 }$

Question 41

Find the area of the surface S where S is the part of the sphere that lies inside the cylinder

Question 42

Find the correct identity, if f is a scalar field, F and G are vector fields.

A) $\operatorname { div } ( f \mathbf { F } ) = f \operatorname { div } ( \mathbf { F } ) + \mathbf { F } \cdot \nabla f$
B) $\operatorname { curl } ( f \mathbf { F } ) = f \operatorname { div } ( \mathbf { F } ) + \mathbf { F } \cdot \nabla f$
C) $\operatorname { div } ( f \mathbf { F } ) = f \operatorname { curl } ( \mathbf { F } ) + ( \nabla f ) \times \mathbf { F }$
D) None of these

Question 43

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ;

Question 44

Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and

Question 45

Use the Divergence Theorem to find the flux of F across S; that is, calculate . ; S is the sphere

Question 46

Find a parametric representation for the part of the sphere that lies above the cone

Question 47

Find the area of the part of paraboloid that lies inside the cylinder

Question 48

Find an equation in rectangular coordinates, and then identify the surface.

Question 49

Find an equation in rectangular coordinates, and then identify the surface.

Question 50

Set up, but do not evaluate, a double integral for the area of the surface with parametric equations

Question 51

Find a parametric representation for the part of the plane that lies inside the cylinder

Question 52

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ; u = ln 5, v = 0

Question 53

Find the area of the surface S where S is the part of the sphere that lies to the right of the xz-plane and inside the cylinder

Question 54

Below is given the plot of a vector field F in the xy-plane. (The z-component of F is 0.) By studying the plot, determine whether div F is positive, negative, or zero.

A) cannot be determined
B) positive
C) negative
D) zero

Question 55

Find the area of the surface S where S is the part of the surface that lies inside the cylinder

Question 56

Find the area of the part of the surface $y = 20 x + z ^ { 2 }$ that lies between the planes x = 0, x = 4, $z = 0$ , and z = 1.

A) $89.3563$
B) $80.2329$
C) $87.3563$
D) $90.3563$
E) $92.3563$

Question 57

Let $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k } \text { and } r = | \mathbf { r } | \text {. }$ $\text { Find } \nabla \cdot ( 9 \mathbf { r } )$

A) 9
B) 45
C) 18
D) 27
E) None of these

Question 58

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ; u = ln 9, v = 0

Question 59

Find a vector representation for the surface.
The plane that passes through the point and contains the vectors and ..

Question 60

Find the divergence of the vector field F. $$\mathbf { F } ( x , y , x ) = x z ^ { 4 } \mathbf { i } + 2 x ^ { 4 } z \mathbf { j } - 5 y ^ { 3 } z \mathbf { k }$$

A) $z ^ { 4 } \mathbf { i } + 2 x ^ { 4 } z \mathbf { j } - 5 y ^ { 3 } \mathbf { k }$
B) $z ^ { 4 } \mathbf { i } - 5 y ^ { 3 } \mathbf { k }$
C) $z ^ { 4 } - 5 y ^ { 3 }$
D) $$z ^ { 4 } + 2 x ^ { 4 } z - 5 y ^ { 3 }$$

Question 61

Determine whether or not vector field is conservative. If it is conservative, find a function f such that

Question 62

Use Green's Theorem and/or a computer algebra system to evaluate $\int _ { C } x ^ { 2 } y d x - x y ^ { 2 } d y$ where C is the circle $x ^ { 2 } + y ^ { 2 } = 64$ with counterclockwise orientation.

A) $8 \pi$
B) $64 \pi$
C) $2 \pi$
D) $- 32 \pi$
E) None of these

Question 63

Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.

Question 64

Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid $( \bar { x } , \bar { y } ) \text { of } D \text { are } \bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ where A is the area of D. Find the centroid of the triangle with vertices (0, 0), ( $4$ , 0) and (0, $8$ ).

A) $\left( \frac { 4 } { 3 } , - \frac { 8 } { 3 } \right)$
B) $\left( \frac { 1 } { 8 } , \frac { 1 } { 8 } \right)$
C) $\left( \frac { 1 } { 8 } , \frac { 1 } { 64 } \right)$
D) $\left( \frac { 1 } { 8 } , \frac { 1 } { 4 } \right)$
E) $\left( \frac { 8 } { 3 } , \frac { 8 } { 3 } \right)$

Question 65

Find the curl of .

Question 66

Determine whether or not vector field is conservative. If it is conservative, find a function f such that

Question 67

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.

Question 68

Find (a) the divergence and (b) the curl of the vector field F.

Question 69

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. $\oint _ { C } e ^ { x ^ { 2 } } d x + \left( \sin y - 4 x ^ { 2 } \right) d y$ , where C is the boundary of the region bounded by the parabolas $y = x ^ { 2 }$ and $x = y ^ { 2 }$ .

A) $- \frac { 12 } { 5 }$ + e
B) $- \frac { 6 } { 5 }$ + e
C) $- \frac { 6 } { 5 }$
D) $- \frac { 12 } { 5 }$

Question 70

Let

Question 71

Find the div F if .

Question 72

Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = 10 x y \mathbf { i } + 10 y z \mathbf { j } + 7 x z \mathbf { k }$

A) $10 y \mathbf { i } - 7 z \mathbf { j } - 10 x \mathbf { k }$
B) $- 10 x \mathbf { i } + 7 z \mathbf { j } - 10 y \mathbf { k }$
C) $- 10 x \mathbf { i } + 7 y \mathbf { j } + 10 z \mathbf { k }$
D) $- 10 y \mathbf { i } - 7 z \mathbf { j } - 10 x \mathbf { k }$
E) None of these

Question 73

Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.
curl f

Question 74

Find the curl of the vector field.

Question 75

Find the curl of the vector field.

Question 76

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.
curl (div F)

Question 77

Let

Question 78

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. $\oint _ { C } 3 x y d x + 4 x ^ { 2 } d y$ , where C is the triangle with vertices $( 0,0 )$ , $( 3,4 )$ , and $( 0,4 )$ .

A) $6$
B) $60$
C) $30$
D) $66$

Question 79

Find the divergence of the vector field.

Question 80

Find the curl of the vector field F. $\mathbf { F } ( x , y , x ) = 3 y z ^ { 3 } \mathbf { i } + 8 x ^ { 5 } y ^ { 4 } \mathbf { j } + 4 x \mathbf { k }$

A) $32 x ^ { 5 } y ^ { 3 }$
B) $32 x ^ { 5 } y ^ { 3 } \mathbf { j }$
C) $9 y z ^ { 2 } + 40 x ^ { 4 } y ^ { 4 } - 3 z ^ { 3 } - 4$
D) $\left( 9 y z ^ { 2 } - 4 \right) \mathbf { j } + \left( 40 x ^ { 4 } y ^ { 4 } - 3 z ^ { 3 } \right) \mathbf { k }$