Deck 14: Vector Analysis

Question 1

Find a potential for the given vector field: $$\vec { F } ( x , y ) = \left\langle 2 x y , x ^ { 2 } + 1 \right\rangle$$

Answer 1

$f ( x , y ) = x ^ { 2 } y + y$

Question 2

Find a potential for the given vector field: $$\vec { F } ( x , y ) = \left\langle - e ^ { y } \sin \left( x e ^ { y } \right) , - x e ^ { y } \sin \left( x e ^ { y } \right) \right\rangle$$

Answer 2

$f ( x , y ) = \cos \left( x e ^ { y } \right)$

Question 3

Find a potential for the given vector field: $$\vec { F } ( x , y ) = \left\langle 3 x ^ { 2 } \cos ( x y ) - x ^ { 3 } y \sin ( x y ) , - x ^ { 4 } \sin ( x y ) \right\rangle$$

A) $f ( x , y ) = x ^ { 4 } \cos ( x y )$
B) $f ( x , y ) = x ^ { 3 } \sin ( x y )$
C) $f ( x , y ) = x ^ { 3 } \cos ( x y )$
D) $f ( x , y ) = x ^ { 4 } \sin ( x y )$

Answer 3

$f ( x , y ) = x ^ { 3 } \cos ( x y )$

Question 4

Find a potential for the given vector field: $$\vec { F } ( x , y ) = \left\langle y \cos ( x y ) e ^ { \sin ( x y ) } , x \cos ( x y ) e ^ { \sin ( x y ) } \right\rangle$$

Question 5

Find a potential for the given vector field: $$\vec { F } ( x , y ) = \left\langle \frac { 2 } { x } , \frac { 2 } { y } \right\rangle$$

Question 6

Find a potential for the given vector field: $$\vec { F } ( x , y ) = \left\langle 2 x y ^ { - 1 } , - x ^ { 2 } y ^ { - 2 } \right\rangle$$

A) $f ( x , y ) = - x ^ { 2 } y ^ { - 1 }$
B) $f ( x , y ) = x y ^ { - 2 }$
C) $f ( x , y ) = x y ^ { - 1 }$
D) $f ( x , y ) = x ^ { 2 } y ^ { - 1 }$

Question 7

Find a potential for the given vector field: $$\vec { F } ( x , y , z ) = \left\langle y z ^ { - 1 } , x z ^ { - 1 } , - x y z ^ { - 2 } \right\rangle$$

Question 8

Find a potential for the given vector field: $$\vec { F } ( x , y , z ) = \left\langle y z e ^ { x y z } , x z e ^ { x y z } , x y e ^ { x y z } \right\rangle$$

Question 9

Find a potential for the given vector field: $$\vec { F } ( x , y , z ) = \left\langle \frac { 1 } { x } , \frac { 1 } { y } , \cot ( z ) \right\rangle$$

A) $f ( x , y , z ) = \ln ( x y \tan ( z ) )$
B) $f ( x , y , z ) = \ln ( x y \cos ( z ) )$
C) $f ( x , y , z ) = \ln ( x y \sin ( z ) )$
D) $f ( x , y , z ) = \ln ( x y \sec ( z ) )$

Question 10

Is the given vector field conservative? $$\vec { F } ( x , y ) = \left\langle x ^ { y - 1 } , x ^ { y } \ln ( x ) \right\rangle$$

Question 11

Is the given vector field conservative? $$\vec { F } ( x , y ) = \left\langle 2 x y ^ { 3 } , 3 x ^ { 2 } y ^ { 2 } \right\rangle$$

Question 12

Is the given vector field conservative? $$\vec { F } ( x , y ) = \left\langle 3 x y ^ { 3 } , 3 x ^ { 2 } y ^ { 3 } \right\rangle$$

Question 13

Evaluate the line integral of the function over the given curve: $f ( x , y ) = x ^ { 3 }$ and the curve C is given by $y = x ^ { 3 } , 0 \leq x \leq 2$

Question 14

Evaluate the line integral of the function over the given curve: $f ( x , y ) = y$ and the curve C is given by $y = x ^ { 3 } , 0 \leq x \leq 2$

Question 15

Evaluate the line integral of the function over the given curve: $f ( x , y ) = x$ and the curve C is given by $y = x ^ { 2 } , 0 \leq x \leq 2$

A) $\frac { 17 \sqrt { 17 } - 1 } { 12 }$
B) $\frac { 17 \sqrt { 17 } - 1 } { 14 }$
C) $\frac { 17 \sqrt { 17 } - 1 } { 17 }$
D) $\frac { 17 \sqrt { 17 } - 2 } { 12 }$

Question 16

Evaluate the line integral of the function over the given curve: $f ( x , y ) = x$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 17

Evaluate the line integral of the function over the given curve: $f ( x , y ) = x y$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 18

Evaluate the line integral of the function over the given curve: $f ( x , y ) = x e ^ { y }$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

A) $e$
B) $e - 1$
C) $- 1$
D) $e ^ { \frac { \pi } { 2 } } - 1$

Question 19

Evaluate the line integral of the function over the given curve: $f ( x , y , z ) = y e ^ { x }$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) , t \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 20

Evaluate the line integral of the function over the given curve: $f ( x , y , z ) = e ^ { z }$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) , t \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 21

Evaluate the line integral of the function over the given curve: $f ( x , y , z ) = x e ^ { y }$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) , t \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 22

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle x , x \rangle$ and the curve C is given by $y = x ^ { 3 } , 0 \leq x \leq 2$

Question 23

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle y , y \rangle$ and the curve C is given by $y = x ^ { 3 } , 0 \leq x \leq 2$

Question 24

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle x , y \rangle$ and the curve C is given by $y = x ^ { 2 } , 0 \leq x \leq 2$

A) 4
B) $8$

C) $10$
D) $12$

Question 25

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle x , x \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 26

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle y , y \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 27

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle x , y \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

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

Question 28

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle x , y , z \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) , t \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 29

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle x , x , x \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) , t \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Question 30

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle 1 , x , 1 \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) , t \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

A) $\frac { 3 \pi } { 4 } - 1$
B) $\frac { 3 \pi } { 4 }$
C) $\frac { \pi } { 4 } - 1$
D) $\frac { 3 \pi } { 4 } - \frac { 3 } { 2 }$

Question 31

Find the area of the surface $S$ Let $S$ be the portion of the plane $z = 2 x + y + 3$ above the rectangle $0 \leq x \leq 2$ and $1 \leq y \leq 6$

Question 32

Find the area of the surface $S$ Let $S$ be the portion of the plane $z = 3 x + 4 y + 7$ above the rectangle $0 \leq x \leq 3$ and $0 \leq y \leq 4$

Question 33

Find the area of the surface $S$ Let $S$ be the portion of the plane $z = x + 9 y - 2$ above the rectangle $1 \leq x \leq 3$ and $2 \leq y \leq 4$

A) $4 \sqrt { 83 }$
B) $2 \sqrt { 83 }$
C) $\sqrt { 83 }$
D) $\frac { 1 } { 2 } \sqrt { 83 }$

Question 34

Find the area of the surface $S$ Let $S = \{ ( 2 u + v , u - v , u + 3 v ) : 0 \leq u \leq 2 \text { and } 0 \leq v \leq 1 \}$

Question 35

Find the area of the surface $S$ Let $S = \{ ( v , u , u ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$

Question 36

Find the area of the surface $S$ Let $S = \{ ( u + v , 3 u + 2 v , 6 v ) : 0 \leq u \leq 1 \text { and } 1 \leq v \leq 3 \}$

A) $\sqrt { 219 }$
B) $2 \sqrt { 83 }$
C) $\sqrt { 83 }$
D) $38$

Question 37

Integrate the function $f$ over the surface $S$ Let $S$ be the portion of the plane $z = 2 x + y + 3$ above the rectangle $0 \leq x \leq 2$ and $1 \leq x \leq 6$ and $f ( x , y ) = x$

Question 38

Integrate the function $f$ over the surface $S$ Let $S$ be the portion of the plane $z = 3 x + 4 y + 7$ above the rectangle $0 \leq x \leq 3$ and $0 \leq x \leq 4$ and $f ( x , y ) = x y$

Question 39

Integrate the function $f$ over the surface $S$ Let $S$ be the portion of the plane $z = x + 9 y - 2$ above the rectangle $1 \leq x \leq 3$ and $2 \leq x \leq 4$ and $f ( x , y ) = x ^ { 2 }$

A) $52 \sqrt { 83 }$
B) $26 \sqrt { 83 }$
C) $\frac { 52 } { 3 } \sqrt { 83 }$
D) $\frac { 26 } { 3 } \sqrt { 83 }$

Question 40

Integrate the function $f$ over the surface $S$ Let $S = \{ ( 2 u + v , u - v , u + 3 v ) : 0 \leq u \leq 2 \text { and } 0 \leq v \leq 1 \}$ and $f ( x , y ) = e ^ { x }$

Question 41

Integrate the function $f$ over the surface $S$ Let $S = \{ ( v , u , u ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$ and $f ( x , y ) = y$

Question 42

Integrate the function $f$ over the surface $S$ Let $S = \{ ( u + v , 3 u + 2 v , 6 v ) : 0 \leq u \leq 1 \text { and } 1 \leq v \leq 3 \}$ and $f ( x , y ) = \sin ( x )$

A) $\sin ( 4 ) - \sin ( 3 ) - \sin ( 2 ) + \sin ( 1 )$
B) $19 ( - \sin ( 4 ) + \sin ( 3 ) + \sin ( 2 ) - \sin ( 1 ) )$
C) $- \sin ( 4 ) + \sin ( 3 )$
D) $2 \sqrt { 219 }$

Question 43

Find the flux of $\vec { F }$ through the surface $S$ in the positive z direction. Let $S = \{ ( 2 u + v , u - v , u + 3 v ) : 0 \leq u \leq 2 \text { and } 0 \leq v \leq 1 \}$ and $\vec { F } ( x , y , z ) = \langle x , y , z - y \rangle$

Question 44

Find the flux of $\vec { F }$ through the surface $S$ in the positive z direction. Let $S = \{ ( v , u , u ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$ and $\vec { F } ( x , y , z ) = \langle x , y , z - y \rangle$

Question 45

Find the flux of $\vec { F }$ through the surface $S$ in the positive z direction. Let $S = \{ ( u + v , 3 u + 2 v , 6 v ) : 0 \leq u \leq 1 \text { and } 1 \leq v \leq 3 \}$ and $\vec { F } ( x , y , z ) = \langle z , x , y \rangle$

A) $378$
B) $- 791$
C) $- 409$
D) $0$

Question 46

Find the divergence of $\vec { F } ( x , y ) = \langle x , x y \rangle$

Question 47

Find the divergence of $\vec { F } ( x , y ) = \left\langle x ^ { 2 } y , e ^ { x y } \right\rangle$

Question 48

Find the divergence of $\vec { F } ( x , y ) = \left\langle \sin ( x y ) , \cos \left( y ^ { 2 } \right) \right\rangle$

A) $y \cos ( x y ) - 2 y \sin \left( y ^ { 2 } \right)$
B) $\cos ( x y ) - 2 \sin \left( y ^ { 2 } \right)$
C) $- y \cos ( x y ) + 2 y \sin \left( y ^ { 2 } \right)$
D) $y \cos ( x y ) - y \sin \left( y ^ { 2 } \right)$

Question 49

Find the divergence of $\vec { F } ( x , y , z ) = \left\langle x , y ^ { 2 } , z ^ { 3 } \right\rangle$

Question 50

Find the divergence of $\vec { F } ( x , y , z ) = \langle x y , y z , x z \rangle$

Question 51

Find the divergence of $\vec { F } ( x , y , z ) = \left\langle e ^ { y z } , e ^ { x z } , e ^ { x y } \right\rangle$

A) $e ^ { y z } + e ^ { x z } + e ^ { x y }$
B) $y e ^ { y z } + x e ^ { x z } + x e ^ { x y }$
C) $0$
D) $3 e ^ { x y }$

Question 52

Find the curl of $\vec { F } ( x , y , z ) = \left\langle x , y ^ { 2 } , z ^ { 3 } \right\rangle$

Question 53

Find the curl of $\vec { F } ( x , y , z ) = \langle x y , y z , x z \rangle$

Question 54

Find the curl of $\vec { F } ( x , y , z ) = \left\langle e ^ { y z } , e ^ { x z } , e ^ { x y } \right\rangle$

A) $\left\langle x e ^ { x y } - z e ^ { x z } , y e ^ { y z } - x e ^ { x y } , z e ^ { x z } - y e ^ { y z } > \right.$
B) $< x e ^ { x y } + x e ^ { x z } , y e ^ { y z } + y e ^ { x y } , z e ^ { x z } + z e ^ { y z } >$
C) $\left\langle x e ^ { x y } - x e ^ { x z } , y e ^ { y z } - y e ^ { x y } , z e ^ { x z } - z e ^ { y z } > \right.$
D) $\left\langle x e ^ { x y } , y e ^ { y z } , z e ^ { x z } \right\rangle$

Question 55

Compute the integral using Green's theorem. $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is the square with vertices (1, 1), (0, 1), (0, 0), and (1, 0) traversed counterclockwise and $\vec { F } ( x , y ) = \left\langle x ^ { 2 } y , x ^ { 2 } y ^ { 2 } \right\rangle$

Question 56

Compute the integral using Green's theorem. $$\int _ { C } \vec { F } \cdot d \vec { r }$$ , where C is the unit circle traversed counterclockwise and $\vec { F } ( x , y ) = \left\langle y ^ { 3 } , - x ^ { 3 } \right\rangle$

Question 57

Compute the integral using Green's theorem. $$\int _ { C } \vec { F } \cdot d \vec { r }$$ , where C is the triangle with vertices (1, 1), (0, 0), and (1, 0) traversed counterclockwise and $\vec { F } ( x , y ) = \left\langle x ^ { 2 } y , x ^ { 3 } y ^ { 3 } \right\rangle$

A) $- \frac { 1 } { 7 }$
B) $\frac { 1 } { 7 }$
C) $- \frac { 1 } { 14 }$
D) $\frac { 1 } { 14 }$

Question 58

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 6 - x - y$ lying above the square with vertices (0, 0), (1, 0), (1, 1), and (0, 1) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle x z , y x , x y z \rangle$

Question 59

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 6 - x - y$ lying above the square with vertices (0, 0), (1, 0), (1, 1), and (0, 1) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle z , y , 2 x \rangle$

Question 60

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 6 - x - y$ lying above the square with vertices (0, 0), (1, 0), (1, 1), and (0, 1) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \left\langle z , y ^ { 2 } , 3 x \right\rangle$

A) $- 4$
B) $- 2$
C) $- 1$
D) $0$

Question 61

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 7 - 2 x - y$ lying above the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle x z , y x , x y z \rangle$

Question 62

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 7 - 2 x - y$ lying above the unit disk with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle z , y , 2 x \rangle$

Question 63

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 7 - 2 x - y$ lying above the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \left\langle z , y ^ { 2 } , 3 x \right\rangle$

A) $- 16$
B) $- 8$
C) $- 4$
D) $- 2$

Question 64

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is the boundary curve of the surface $S = \{ ( u v , u , u - v ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$ with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle x , x , y \rangle$

Question 65

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is the boundary curve of the surface $S = \{ ( u v , u , u - v ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$ with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle z , y , 2 x \rangle$

Question 66

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is the boundary curve of the surface $S = \{ ( u v , u , u - v ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$ with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \left\langle z , y ^ { 2 } , 3 x \right\rangle$

A) 1
B) $- 1$
C) $- 2$
D) 2

Question 67

Find $\operatorname { Div } \vec { F }$ for $\vec { F } ( x , y , z ) = \left\langle 2 x , y z , z ^ { 2 } \right\rangle$

Question 68

Find $\operatorname { Div } \vec { F }$ for $\vec { F } ( x , y , z ) = \left\langle y x ^ { 2 } , e ^ { x y } , z \right\rangle$

Question 69

Find $\operatorname { Div } \vec { F }$ for $\vec { F } ( x , y , z ) = \left\langle \cos ( x y ) , \sin ( y ) , e ^ { z } \right\rangle$

A) $y \sin ( x y ) - \cos ( y ) + e ^ { z }$
B) $- \sin ( x y ) + \cos ( y ) + e ^ { z }$
C) $- y \sin ( x y ) + \cos ( y ) + e ^ { z }$
D) $- x \sin ( x y ) + \cos ( y ) + e ^ { z }$

Question 70

Find $\operatorname { Div } \vec { F }$ for $\vec { F } ( x , y , z ) = \left\langle \ln ( x y ) , \cos ( y ) , z ^ { 2 } \right\rangle$

Question 71

Find $\operatorname { Div } \vec { F }$ for $\vec { F } ( x , y , z ) = \langle x y z , x y z , x y z \rangle$

Question 72

Find $\operatorname { Div } \vec { F }$ for $\vec { F } ( x , y , z ) = \left\langle x ^ { 2 } e ^ { y } , y ^ { 2 } e ^ { x } , y \right\rangle$

A) $2 x e ^ { y } + 2 y e ^ { x }$
B) $x e ^ { y } + y e ^ { x }$
C) $2 y e ^ { y } + 2 x e ^ { x }$
D) $2 x e ^ { y } + 2 y e ^ { x } + 1$

Question 73

Compute $\iint _ { S } \vec { F } \cdot \vec { n } d S$ where S is the unit sphere, $\vec { n }$ is the unit outward normal, and $\vec { F } ( x , y , z ) = \left\langle x ^ { 3 } , y ^ { 3 } , z ^ { 3 } \right\rangle$

Question 74

Compute $\iint _ { S } \vec { F } \cdot \vec { n } d S$ where S is the unit sphere, $\vec { n }$ is the unit outward normal, and $\vec { F } ( x , y , z ) = \left\langle x ^ { 2 } , y , 4 \right\rangle$

Question 75

Compute $\iint _ { S } \vec { F } \cdot \vec { n } d S$ where S is the unit sphere, $\vec { n }$ is the unit outward normal, and $\vec { F } ( x , y , z ) = \left\langle x , y , z ^ { 2 } \right\rangle$

A) $\frac { 12 } { 3 } \pi$

B) $\frac { 8 } { 3 } \pi$

C) $\frac { 4 } { 3 } \pi$

D) $\frac { 1 } { 3 } \pi$