Question 1

Evaluate the line integral $\int _ { c } x ^ { 2 } z d s$ where C is the curve $x = \sin 2 t , y = 3 t , z = \cos 2 t , 0 \leq t \leq \frac { \pi } { 4 }$ .

Accepted Answer

The answer of Evaluate the line integral \(\int _ {...

Question 2

Find a function of $f ( x , y )$ such that $\nabla f = \mathbf { F } ( x , y ) = \left( \frac { x } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } , \frac { y } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ ..

Accepted Answer

The answer of Find a function of \(f ( x...

Question 3

Use Green's Theorem to find the area of the region interior to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$ .

Accepted Answer

The answer of Use Green's Theorem to find the area...

Question 4

Evaluate $\int _ { c } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = \left\langle y + 2 x z , x , x ^ { 2 } \right\rangle$ and $C$ is any curve from $( 1,0,2 )$ to $( 2,3,0 )$ .

Accepted Answer

A)  $6$ 
B) $- 2$ 
C)  $- 4$ 
D)  $1$ 
E) $- 6$ 
F)$2$
G) $4$
H)$0$ 
A)  $6$ 
B) $- 2$ 
C)  $- 4$ 
D)  $1$ 
E) $- 6$ 
F)$2$
G) $4$
H)$0$

Question 5

Let $\mathbf { F } ( x , y , z ) = \frac { z \mathbf { i } + x \mathbf { j } + y \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ and let S be the boundary surface of the solid $E = \left\{ ( x , y , z ) \mid 1 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 4 \right\}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Accepted Answer

The answer of Let \(\mathbf { F } ( x...

Question 6

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where F $( x , y , z ) = y z \mathbf { i } + x z \mathbf { j } + x y \mathbf { k }$ , and the curve $C$ is given by the vector function $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t \mathbf { j } + t ^ { 3 } \mathbf { k } , 0 \leq t \leq 1$ .

Accepted Answer

A)  $- 3$ 
B) $- 2$ 
C) $- 1$ 
D) $0$ 
E) $1$
F)$2$ 
G)$3$
H) $4$ 
A)  $- 3$ 
B) $- 2$ 
C) $- 1$ 
D) $0$ 
E) $1$
F)$2$ 
G)$3$
H) $4$

Question 7

Find the gradient vector field of the function $f ( x , y ) = \arctan \left( \frac { y } { x } \right)$ .

Accepted Answer

A)  $\left( \frac { y } { x ^ { 2 } + y ^ { 2 } } , \frac { x } { x ^ { 2 } + y ^ { 2 } } \right)$ 
B) $\left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } , \frac { - x } { x ^ { 2 } + y ^ { 2 } } \right)$ 
C) $\left( \frac { y } { x ^ { 2 } + y ^ { 2 } } , \frac { - x } { x ^ { 2 } + y ^ { 2 } } \right)$ 
D) $\left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } , \frac { x } { x ^ { 2 } + y ^ { 2 } } \right)$ 
E) $\left( \frac { x } { x ^ { 2 } + y ^ { 2 } } , \frac { y } { x ^ { 2 } + y ^ { 2 } } \right)$ 
F)$\left( \frac { - x } { x ^ { 2 } + y ^ { 2 } } , \frac { - y } { x ^ { 2 } + y ^ { 2 } } \right)$
G)$\left( \frac { x } { x ^ { 2 } + y ^ { 2 } } , \frac { - y } { x ^ { 2 } + y ^ { 2 } } \right)$ 
H)$\left( \frac { - x } { x ^ { 2 } + y ^ { 2 } } , \frac { y } { x ^ { 2 } + y ^ { 2 } } \right)$ 
A)  $\left( \frac { y } { x ^ { 2 } + y ^ { 2 } } , \frac { x } { x ^ { 2 } + y ^ { 2 } } \right)$ 
B) $\left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } , \frac { - x } { x ^ { 2 } + y ^ { 2 } } \right)$ 
C) $\left( \frac { y } { x ^ { 2 } + y ^ { 2 } } , \frac { - x } { x ^ { 2 } + y ^ { 2 } } \right)$ 
D) $\left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } , \frac { x } { x ^ { 2 } + y ^ { 2 } } \right)$ 
E) $\left( \frac { x } { x ^ { 2 } + y ^ { 2 } } , \frac { y } { x ^ { 2 } + y ^ { 2 } } \right)$ 
F)$\left( \frac { - x } { x ^ { 2 } + y ^ { 2 } } , \frac { - y } { x ^ { 2 } + y ^ { 2 } } \right)$
G)$\left( \frac { x } { x ^ { 2 } + y ^ { 2 } } , \frac { - y } { x ^ { 2 } + y ^ { 2 } } \right)$ 
H)$\left( \frac { - x } { x ^ { 2 } + y ^ { 2 } } , \frac { y } { x ^ { 2 } + y ^ { 2 } } \right)$

Question 8

Evaluate the flux integral $\iint _ { S } ( 2 x \mathbf { i } - y \mathbf { j } + 3 z \mathbf { k } ) \cdot \mathbf { n } d S$ over the boundary of the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 9$ .

Accepted Answer

The answer of Evaluate the flux integral \(\iint _ {...

Question 9

Find the work done by the force field $\mathbf { F } ( x , y ) = - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$ on a particle that moves along the circle C: $x = 1 + \cos t , y = 1 + \sin t , 0 \leq t \leq 2 \pi$ .

Accepted Answer

The answer of Find the work done by the force...

Question 10

Evaluate the line integral $\int _ { C } y d s$ , where $C$ is the curve $x = \cos ^ { 3 } t , y = \sin ^ { 3 } t , 0 \leq t \leq \frac { \pi } { 2 }$ .

Accepted Answer

The answer of Evaluate the line integral \(\int _ {...

Question 11

Evaluate $\int _ { C } \left[ 3 x ^ { 2 } y ^ { 2 } + 2 \cos ( 2 x + y ) \right] d x + \left[ 2 x ^ { 3 } y + \cos ( 2 x + y ) \right] d y$ if C is the closed path starting at $( 0,0 )$ and moving clockwise around the square with vertices $( 0,0 )$ , $( 1,0 )$ , $( 1,1 )$ , and $( 0,1 )$ .

Accepted Answer

The answer of Evaluate \(\int _ { C } \left[...

Question 12

Evaluate the surface integral $\iint _ { S } x y z d S$ , where S is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ that lies above the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Accepted Answer

The answer of Evaluate the surface integral \(\iint _ {...

Question 13

Determine whether $\mathbf { F } ( x , y , z ) = \left( 2 x y + \ln z , x ^ { 2 } + z \cos ( y z ) , \frac { x } { z } + y \cos ( y z ) \right)$ is conservative and if so, find a potential function.

Accepted Answer

The answer of Determine whether \(\mathbf { F } (...

Question 14

Determine whether or not $\mathbf { F } ( x , y ) = \left( x ^ { 2 } + y \right) \mathbf { i } + \left( y ^ { 2 } + x \right) \mathbf { j }$ is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f$

Accepted Answer

The answer of Determine whether or not \(\mathbf { F...

Question 15

Let $\mathbf { F } ( x , y , z ) = \mathbf { k }$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ and has upward orientation.

Accepted Answer

A)  $1$ 
B)  $\sqrt { 2 }$ 
C)  $\sqrt { 3 }$ 
D) $2$ 
E) $\frac { \sqrt { 2 } } { 2 }$ 
F) $\frac { \sqrt { 3 } } { 2 }$ 
G)$\frac { 1 } { 2 }$
H)$\frac { 1 } { 4 }$ 
A)  $1$ 
B)  $\sqrt { 2 }$ 
C)  $\sqrt { 3 }$ 
D) $2$ 
E) $\frac { \sqrt { 2 } } { 2 }$ 
F) $\frac { \sqrt { 3 } } { 2 }$ 
G)$\frac { 1 } { 2 }$
H)$\frac { 1 } { 4 }$

Question 16

Find the gradient vector field of the function $f ( x , y , z ) = x \sin y + z ^ { 2 }$ .

Accepted Answer

A)  $\sin y \mathbf { i } + \cos y \mathbf { j } + 2 \mathbf { k }$ 
B)  $\sin y \mathbf { i } + x \cos y \mathbf { j } + 2 z \mathbf { k }$ 
C) $x \sin y \mathbf { i } + \cos y \mathbf { j } + \mathbf { k }$ 
D)  $x \sin y \mathbf { i } + x \cos y \mathbf { j } + 2 z \mathbf { k }$ 
E) $\sin y \mathbf { i } - \cos y \mathbf { j } - 2 \mathbf { k }$
F)$\sin y \mathbf { i } - x \cos y \mathbf { j } + z \mathbf { k }$
G)$x \sin y \mathbf { i } - \cos y \mathbf { j } + z \mathbf { k }$
H)$x \sin y \mathbf { i } - x \cos y \mathbf { j } + z \mathbf { k }$ 
A)  $\sin y \mathbf { i } + \cos y \mathbf { j } + 2 \mathbf { k }$ 
B)  $\sin y \mathbf { i } + x \cos y \mathbf { j } + 2 z \mathbf { k }$ 
C) $x \sin y \mathbf { i } + \cos y \mathbf { j } + \mathbf { k }$ 
D)  $x \sin y \mathbf { i } + x \cos y \mathbf { j } + 2 z \mathbf { k }$ 
E) $\sin y \mathbf { i } - \cos y \mathbf { j } - 2 \mathbf { k }$
F)$\sin y \mathbf { i } - x \cos y \mathbf { j } + z \mathbf { k }$
G)$x \sin y \mathbf { i } - \cos y \mathbf { j } + z \mathbf { k }$
H)$x \sin y \mathbf { i } - x \cos y \mathbf { j } + z \mathbf { k }$

Question 17

Evaluate the line integral $\int _ { C } \left( 2 x y ^ { 2 } + \tan ^ { - 1 } x ^ { 3 } \right) d x + \left( x ^ { 2 } y + \sqrt { y ^ { 3 } + 1 } \right) d y$ , where C consists of the arc of the parabola $y = x ^ { 2 }$ from $( 1,1 )$ to $( 2,4 )$ , followed by the line segments from $( 2,4 )$ to $( 0,4 )$ , from $( 0,4 )$ to $( 0,1 )$ and from $( 0,1 )$ to $( 1,1 )$ .

Accepted Answer

The answer of Evaluate the line integral \(\int _ {...

Question 18

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $\mathbf { F } ( x , y , z ) = ( y + z ) \mathbf { i } - x ^ { 2 } \mathbf { j } - 4 y ^ { 2 } \mathbf { k }$ , and the curve $C$ is given by the vector function $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + t ^ { 4 } \mathbf { k } , \quad 0 \leq t \leq 1$

Accepted Answer

The answer of Evaluate the line integral \(\int _ {...

Question 19

Find (a) the curl and (b) the divergence of the vector field $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { y } \mathbf { i } + y z ^ { 2 } \mathbf { j } + z x ^ { 2 } \mathbf { k }$ .

Accepted Answer

The answer of Find (a) the curl and (b) the...

Question 20

Let $\mathbf { F } ( x , y , z ) = \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Accepted Answer

A)  $0$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $1$

H)
$\frac { 3 } { 2 }$ 
A)  $0$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $1$

H)
$\frac { 3 } { 2 }$

Evaluate the line integral $\int _ { c } x ^ { 2 } z d s$ where C is the curve $x = \sin 2 t , y = 3 t , z = \cos 2 t , 0 \leq t \leq \frac { \pi } { 4 }$ .

Find a function of $f ( x , y )$ such that $\nabla f = \mathbf { F } ( x , y ) = \left( \frac { x } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } , \frac { y } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ ..

Use Green's Theorem to find the area of the region interior to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$ .

Evaluate $\int _ { c } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = \left\langle y + 2 x z , x , x ^ { 2 } \right\rangle$ and $C$ is any curve from $( 1,0,2 )$ to $( 2,3,0 )$ .

Find the gradient vector field of the function $f ( x , y ) = \arctan \left( \frac { y } { x } \right)$ .

Evaluate the flux integral $\iint _ { S } ( 2 x \mathbf { i } - y \mathbf { j } + 3 z \mathbf { k } ) \cdot \mathbf { n } d S$ over the boundary of the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 9$ .

Find the work done by the force field $\mathbf { F } ( x , y ) = - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$ on a particle that moves along the circle C: $x = 1 + \cos t , y = 1 + \sin t , 0 \leq t \leq 2 \pi$ .

Evaluate the line integral $\int _ { C } y d s$ , where $C$ is the curve $x = \cos ^ { 3 } t , y = \sin ^ { 3 } t , 0 \leq t \leq \frac { \pi } { 2 }$ .

Evaluate $\int _ { C } \left[ 3 x ^ { 2 } y ^ { 2 } + 2 \cos ( 2 x + y ) \right] d x + \left[ 2 x ^ { 3 } y + \cos ( 2 x + y ) \right] d y$ if C is the closed path starting at $( 0,0 )$ and moving clockwise around the square with vertices $( 0,0 )$ , $( 1,0 )$ , $( 1,1 )$ , and $( 0,1 )$ .

Evaluate the surface integral $\iint _ { S } x y z d S$ , where S is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ that lies above the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Determine whether $\mathbf { F } ( x , y , z ) = \left( 2 x y + \ln z , x ^ { 2 } + z \cos ( y z ) , \frac { x } { z } + y \cos ( y z ) \right)$ is conservative and if so, find a potential function.

Determine whether or not $\mathbf { F } ( x , y ) = \left( x ^ { 2 } + y \right) \mathbf { i } + \left( y ^ { 2 } + x \right) \mathbf { j }$ is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f$

Let $\mathbf { F } ( x , y , z ) = \mathbf { k }$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ and has upward orientation.

Find the gradient vector field of the function $f ( x , y , z ) = x \sin y + z ^ { 2 }$ .

Find (a) the curl and (b) the divergence of the vector field $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { y } \mathbf { i } + y z ^ { 2 } \mathbf { j } + z x ^ { 2 } \mathbf { k }$ .

Let $\mathbf { F } ( x , y , z ) = \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

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Exam 13: Vector Calculus

Evaluate the line integral ∫cx2zds\int _ { c } x ^ { 2 } z d s∫c​x2zds where C is the curve x=sin⁡2t,y=3t,z=cos⁡2t,0≤t≤π4x = \sin 2 t , y = 3 t , z = \cos 2 t , 0 \leq t \leq \frac { \pi } { 4 }x=sin2t,y=3t,z=cos2t,0≤t≤4π​ .

Use Green's Theorem to find the area of the region interior to the ellipse x29+y216=1\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 19x2​+16y2​=1 .

Evaluate ∫cFdr\int _ { c } \mathbf { F } d \mathbf { r }∫c​Fdr where F(x,y,z)=⟨y+2xz,x,x2⟩\mathbf { F } ( x , y , z ) = \left\langle y + 2 x z , x , x ^ { 2 } \right\rangleF(x,y,z)=⟨y+2xz,x,x2⟩ and CCC is any curve from (1,0,2)( 1,0,2 )(1,0,2) to (2,3,0)( 2,3,0 )(2,3,0) .

Find the gradient vector field of the function f(x,y)=arctan⁡(yx)f ( x , y ) = \arctan \left( \frac { y } { x } \right)f(x,y)=arctan(xy​) .

Evaluate the flux integral ∬S(2xi−yj+3zk)⋅ndS\iint _ { S } ( 2 x \mathbf { i } - y \mathbf { j } + 3 z \mathbf { k } ) \cdot \mathbf { n } d S∬S​(2xi−yj+3zk)⋅ndS over the boundary of the ball x2+y2+z2≤9x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 9x2+y2+z2≤9 .

Evaluate the line integral ∫Cyds\int _ { C } y d s∫C​yds , where CCC is the curve x=cos⁡3t,y=sin⁡3t,0≤t≤π2x = \cos ^ { 3 } t , y = \sin ^ { 3 } t , 0 \leq t \leq \frac { \pi } { 2 }x=cos3t,y=sin3t,0≤t≤2π​ .

Evaluate the surface integral ∬SxyzdS\iint _ { S } x y z d S∬S​xyzdS , where S is the part of the sphere x2+y2+z2=1x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1x2+y2+z2=1 that lies above the cone z=x2+y2z = \sqrt { x ^ { 2 } + y ^ { 2 } }z=x2+y2​ .

Determine whether F(x,y,z)=(2xy+ln⁡z,x2+zcos⁡(yz),xz+ycos⁡(yz))\mathbf { F } ( x , y , z ) = \left( 2 x y + \ln z , x ^ { 2 } + z \cos ( y z ) , \frac { x } { z } + y \cos ( y z ) \right)F(x,y,z)=(2xy+lnz,x2+zcos(yz),zx​+ycos(yz)) is conservative and if so, find a potential function.

Determine whether or not F(x,y)=(x2+y)i+(y2+x)j\mathbf { F } ( x , y ) = \left( x ^ { 2 } + y \right) \mathbf { i } + \left( y ^ { 2 } + x \right) \mathbf { j }F(x,y)=(x2+y)i+(y2+x)j is a conservative vector field. If it is, find a function f such that F=∇f\mathbf { F } = \nabla fF=∇f

Find the gradient vector field of the function f(x,y,z)=xsin⁡y+z2f ( x , y , z ) = x \sin y + z ^ { 2 }f(x,y,z)=xsiny+z2 .

Find (a) the curl and (b) the divergence of the vector field F(x,y,z)=x2yi+yz2j+zx2k\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { y } \mathbf { i } + y z ^ { 2 } \mathbf { j } + z x ^ { 2 } \mathbf { k }F(x,y,z)=x2yi+yz2j+zx2k .

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Evaluate the line integral $\int _ { c } x ^ { 2 } z d s$ where C is the curve $x = \sin 2 t , y = 3 t , z = \cos 2 t , 0 \leq t \leq \frac { \pi } { 4 }$ .

Use Green's Theorem to find the area of the region interior to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$ .

Evaluate $\int _ { c } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = \left\langle y + 2 x z , x , x ^ { 2 } \right\rangle$ and $C$ is any curve from $( 1,0,2 )$ to $( 2,3,0 )$ .

Find the gradient vector field of the function $f ( x , y ) = \arctan \left( \frac { y } { x } \right)$ .

Evaluate the flux integral $\iint _ { S } ( 2 x \mathbf { i } - y \mathbf { j } + 3 z \mathbf { k } ) \cdot \mathbf { n } d S$ over the boundary of the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 9$ .

Evaluate the line integral $\int _ { C } y d s$ , where $C$ is the curve $x = \cos ^ { 3 } t , y = \sin ^ { 3 } t , 0 \leq t \leq \frac { \pi } { 2 }$ .

Evaluate the surface integral $\iint _ { S } x y z d S$ , where S is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ that lies above the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Determine whether $\mathbf { F } ( x , y , z ) = \left( 2 x y + \ln z , x ^ { 2 } + z \cos ( y z ) , \frac { x } { z } + y \cos ( y z ) \right)$ is conservative and if so, find a potential function.

Determine whether or not $\mathbf { F } ( x , y ) = \left( x ^ { 2 } + y \right) \mathbf { i } + \left( y ^ { 2 } + x \right) \mathbf { j }$ is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f$

Find the gradient vector field of the function $f ( x , y , z ) = x \sin y + z ^ { 2 }$ .

Find (a) the curl and (b) the divergence of the vector field $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { y } \mathbf { i } + y z ^ { 2 } \mathbf { j } + z x ^ { 2 } \mathbf { k }$ .