Question 1

Find the value of the gradient vector field of the function $f ( x , y , z ) = x ^ { 2 } y ^ { 3 } z$ at the point $( 1,1 , - 1 )$ .

Accepted Answer

A)  $- 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }$ 
B) $2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ 
C) $3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ 
D) $3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ 
E) $6 \mathbf { i } - 6 \mathbf { j } + \mathbf { k }$ 
F)$6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$
G)$\mathbf { i } - \mathbf { j } + \mathbf { k }$
H)$\mathbf { i } + \mathbf { j } - \mathbf { k }$ 
A)  $- 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }$ 
B) $2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ 
C) $3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ 
D) $3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$ 
E) $6 \mathbf { i } - 6 \mathbf { j } + \mathbf { k }$ 
F)$6 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }$
G)$\mathbf { i } - \mathbf { j } + \mathbf { k }$
H)$\mathbf { i } + \mathbf { j } - \mathbf { k }$

Question 2

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( e ^ { x ^ { 2 } } - y ^ { 3 } , \cos y ^ { 2 } + x ^ { 3 } \right)$ and the curve C is given by C : $x = 2 \sin t , y = 2 \cos t , 0 \leq t \leq 2 \pi$ .

Accepted Answer

A)  $24 \pi$ 
B) $12 \pi$ 
C) $48 \pi$ 
D)  $96 \pi$ 
E)  $- 24 \pi$ 
F)$- 12 \pi$ 
G) $- 48 \pi$

H)$0$ 
A)  $24 \pi$ 
B) $12 \pi$ 
C) $48 \pi$ 
D)  $96 \pi$ 
E)  $- 24 \pi$ 
F)$- 12 \pi$ 
G) $- 48 \pi$

H)$0$

Question 3

Use Green's Theorem to find the area of the region bounded by one arch of the cycloid $x = t - \sin t , y = 1 - \cos t$ , and the x-axis.

Accepted Answer

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

Question 4

According to Green's Theorem, the line integral $\int _ { C } x ^ { 2 } d x + y ^ { 2 } d y$ over a positively oriented, piecewise-smooth, simple closed curve C is equal to the double integral $\iint _ { D } f ( x , y ) d A$ over the region D bounded by C. Find the function $f ( x , y )$ .

Accepted Answer

A)  $2 x - y$ 
B) $x - 2 y$ 
C) $2 x - 2 y$ 
D)  $0$ 
E) $1$
F)$2 x + y$ 
G)$x + 2 y$ 
H) $2 x + 2 y$ 
A)  $2 x - y$ 
B) $x - 2 y$ 
C) $2 x - 2 y$ 
D)  $0$ 
E) $1$
F)$2 x + y$ 
G)$x + 2 y$ 
H) $2 x + 2 y$

Question 5

Find a formula for the vector field graphed below. (There are many possible answers.)

Accepted Answer

@#IMG-DLM& , where @#IMG-DLM&

Question 6

Find the gradient vector field of the function $f ( x ; y ) = x + y ^ { 2 }$

Accepted Answer

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

Question 7

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ , find $\nabla \cdot ( \nabla f )$ at the point $( 1,0,1 )$ .

Accepted Answer

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

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

G)
$- 6$ 
H)
$0$

Question 8

Let $\mathbf { F } ( x , y , z ) = 3 x y \mathbf { i } + y ^ { 2 } \mathbf { j } - x ^ { 2 } y ^ { 4 } \mathbf { k }$ and let S be the surface of the tetrahedron with vertices $( 0,0,0 ) , ( 1,0,0 ) , ( 0,1,0 )$ , and $( 0,0,1 )$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Accepted Answer

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

Question 9

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangle$ and C is the circle $( x + 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 1$ in the clockwise direction.

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 10

Find the flux of $\mathbf { F } ( x , y , z ) = \left( x ^ { 3 } \right) \mathbf { i } + \left( y ^ { 3 } + \frac { 1 } { x + 3 } \right) \mathbf { j } + \left( z ^ { 2 } + \frac { x } { ( y + 3 ) ^ { 2 } } \right) \mathbf { k }$ across the surface of the solid bounded by $x ^ { 2 } + z ^ { 2 } = 9$ , and the planes $y = 0 , y = 2$ .

Accepted Answer

The answer of Find the flux of \(\mathbf { F...

Question 11

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

Accepted Answer

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

G) $\pi$

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

G) $\pi$

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

Question 12

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - z } , \frac { z } { y - z } , \ln ( x - y ) \right\rangle$ and C is any curve from $( 2,1 , - 5 )$ to $\left( e ^ { 2 } , 0,2 \right)$ .

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 13

Evaluate the line integral $\int _ { c } x d y$ , where the curve $C$ is given in the figure below.

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

Evaluate the surface integral $\iint _ { S } x d S$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ .

Accepted Answer

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

Question 16

Find the flux of $\mathbf { F } ( x , y , z ) = x ^ { 3 } \mathbf { i } + 2 x z ^ { 2 } \mathbf { j } + 3 y ^ { 2 } z \mathbf { k }$ across the surface of the solid bounded by the paraboloid $z = 4 - x ^ { 2 } - y ^ { 2 }$ and the $x y -$ plane.

Accepted Answer

The answer of Find the flux of \(\mathbf { F...

Question 17

Find the flux of the vector field $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + \mathbf { k }$ across the paraboloid given by $x = u \cos v , y = u \sin v , z = 1 - u ^ { 2 }$ with $1 \leq u \leq 2,0 \leq v \leq 2 \pi$ and upward orientation:

Accepted Answer

The answer of Find the flux of the vector field...

Question 18

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: $\int _ { C } 2 x y d x + x ^ { 2 } d y$ , where C is the cardioid $r = 1 + \cos $ .

Accepted Answer

The answer of Use Green's Theorem to evaluate the line...

Question 19

Evaluate the line integral $\int _ { C } \frac { x y } { 1 + x } d x - \ln ( 1 + x ) d y$ , where C is the triangular path consisting of the line segment from $( 0,0 )$ to $( 4,0 )$ , followed by the line segment from $( 4,0 )$ to $( 0,2 )$ , followed by the line segment from $( 0,2 )$ to $( 0,0 )$ .

Accepted Answer

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

Question 20

Evaluate the surface integral $\iint _ { S } ( x + y + z ) d S$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ .

Accepted Answer

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

Find the value of the gradient vector field of the function $f ( x , y , z ) = x ^ { 2 } y ^ { 3 } z$ at the point $( 1,1 , - 1 )$ .

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( e ^ { x ^ { 2 } } - y ^ { 3 } , \cos y ^ { 2 } + x ^ { 3 } \right)$ and the curve C is given by C : $x = 2 \sin t , y = 2 \cos t , 0 \leq t \leq 2 \pi$ .

Use Green's Theorem to find the area of the region bounded by one arch of the cycloid $x = t - \sin t , y = 1 - \cos t$ , and the x-axis.

According to Green's Theorem, the line integral $\int _ { C } x ^ { 2 } d x + y ^ { 2 } d y$ over a positively oriented, piecewise-smooth, simple closed curve C is equal to the double integral $\iint _ { D } f ( x , y ) d A$ over the region D bounded by C. Find the function $f ( x , y )$ .

Find a formula for the vector field graphed below. (There are many possible answers.)

Find the gradient vector field of the function $f ( x ; y ) = x + y ^ { 2 }$

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ , find $\nabla \cdot ( \nabla f )$ at the point $( 1,0,1 )$ .

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangle$ and C is the circle $( x + 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 1$ in the clockwise direction.

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - z } , \frac { z } { y - z } , \ln ( x - y ) \right\rangle$ and C is any curve from $( 2,1 , - 5 )$ to $\left( e ^ { 2 } , 0,2 \right)$ .

Evaluate the line integral $\int _ { c } x d y$ , where the curve $C$ is given in the figure below. $Evaluate the line integral \int _ { c } x d y , where the curve C is given in the figure below.$

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

Evaluate the surface integral $\iint _ { S } x d S$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ .

Find the flux of $\mathbf { F } ( x , y , z ) = x ^ { 3 } \mathbf { i } + 2 x z ^ { 2 } \mathbf { j } + 3 y ^ { 2 } z \mathbf { k }$ across the surface of the solid bounded by the paraboloid $z = 4 - x ^ { 2 } - y ^ { 2 }$ and the $x y -$ plane.

Find the flux of the vector field $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + \mathbf { k }$ across the paraboloid given by $x = u \cos v , y = u \sin v , z = 1 - u ^ { 2 }$ with $1 \leq u \leq 2,0 \leq v \leq 2 \pi$ and upward orientation:

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: $\int _ { C } 2 x y d x + x ^ { 2 } d y$ , where C is the cardioid $r = 1 + \cos$ .

Evaluate the surface integral $\iint _ { S } ( x + y + z ) d S$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ .

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Exam 13: Vector Calculus

Find the value of the gradient vector field of the function f(x,y,z)=x2y3zf ( x , y , z ) = x ^ { 2 } y ^ { 3 } zf(x,y,z)=x2y3z at the point (1,1,−1)( 1,1 , - 1 )(1,1,−1) .

Use Green's Theorem to find the area of the region bounded by one arch of the cycloid x=t−sin⁡t,y=1−cos⁡tx = t - \sin t , y = 1 - \cos tx=t−sint,y=1−cost , and the x-axis.

Find a formula for the vector field graphed below. (There are many possible answers.)

Find the gradient vector field of the function f(x;y)=x+y2f ( x ; y ) = x + y ^ { 2 }f(x;y)=x+y2

Let f(x,y,z)=x2+y2+z2f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }f(x,y,z)=x2+y2+z2 , find ∇⋅(∇f)\nabla \cdot ( \nabla f )∇⋅(∇f) at the point (1,0,1)( 1,0,1 )(1,0,1) .

Evaluate the line integral ∫cxdy\int _ { c } x d y∫c​xdy , where the curve CCC is given in the figure below.

Find (a) the curl and (b) the divergence of the vector field F(x,y,z)=sin⁡xi+cos⁡xj+z2k\mathbf { F } ( x , y , z ) = \sin x \mathbf { i } + \cos x \mathbf { j } + z ^ { 2 } \mathbf { k }F(x,y,z)=sinxi+cosxj+z2k .

Evaluate the surface integral ∬SxdS\iint _ { S } x d S∬S​xdS , where S is that part of the plane z=xz = xz=x that lies above the square with vertices (0,0),(1,0),(0,1)( 0,0 ) , ( 1,0 ) , ( 0,1 )(0,0),(1,0),(0,1) , and (1,1)( 1,1 )(1,1) .

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: ∫C2xydx+x2dy\int _ { C } 2 x y d x + x ^ { 2 } d y∫C​2xydx+x2dy , where C is the cardioid r=1+cos⁡r = 1 + \cos r=1+cos .

Evaluate the surface integral ∬S(x+y+z)dS\iint _ { S } ( x + y + z ) d S∬S​(x+y+z)dS , where S is that part of the plane z=xz = xz=x that lies above the square with vertices (0,0),(1,0),(0,1)( 0,0 ) , ( 1,0 ) , ( 0,1 )(0,0),(1,0),(0,1) , and (1,1)( 1,1 )(1,1) .

Functions and Models

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Find the value of the gradient vector field of the function $f ( x , y , z ) = x ^ { 2 } y ^ { 3 } z$ at the point $( 1,1 , - 1 )$ .

Use Green's Theorem to find the area of the region bounded by one arch of the cycloid $x = t - \sin t , y = 1 - \cos t$ , and the x-axis.

Find the gradient vector field of the function $f ( x ; y ) = x + y ^ { 2 }$

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ , find $\nabla \cdot ( \nabla f )$ at the point $( 1,0,1 )$ .

Evaluate the line integral $\int _ { c } x d y$ , where the curve $C$ is given in the figure below. $Evaluate the line integral \int _ { c } x d y , where the curve C is given in the figure below.$

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

Evaluate the surface integral $\iint _ { S } x d S$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ .

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: $\int _ { C } 2 x y d x + x ^ { 2 } d y$ , where C is the cardioid $r = 1 + \cos$ .

Evaluate the surface integral $\iint _ { S } ( x + y + z ) d S$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ .