Question 1

The line integral $\int $C $y d s$ where C is the curve $x = 3 t ^ { 2 } , y = t , 0 \leq t \leq 1$ is

Accepted Answer

A)  $\frac { 37 \sqrt { 37 } - 1 } { 108 }$ 
B)  $\frac { 37 \sqrt { 37 } + 1 } { 108 }$ 
C)  $\frac { 37 \sqrt { 37 } - 1 } { 105 }$ 
D)  $\frac { 37 \sqrt { 37 } + 1 } { 105 }$ 
E)  $- \frac { 37 \sqrt { 37 } - 1 } { 108 }$ 
A)  $\frac { 37 \sqrt { 37 } - 1 } { 108 }$ 
B)  $\frac { 37 \sqrt { 37 } + 1 } { 108 }$ 
C)  $\frac { 37 \sqrt { 37 } - 1 } { 105 }$ 
D)  $\frac { 37 \sqrt { 37 } + 1 } { 105 }$ 
E)  $- \frac { 37 \sqrt { 37 } - 1 } { 108 }$

Question 2

Using Green's Theorem, the line integral $\int $C $x ^ { 2 } y d x - y ^ { 2 } x d y$ where C is the circle $x ^ { 2 } + y ^ { 2 } = 1 \text {, }$ is

Accepted Answer

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

Question 3

If I = $\int $C $( \ln x + 2 ) d x + \left( e ^ { y } + 2 x \right) d y$ is independent of the path where C is a curve from (3, 1) to (1, 3), then I is

Accepted Answer

A)  $e ^ { 3 } + e + 3 \ln 3 + 6$ 
B)  $e ^ { 3 } - e - 3 \ln 3 - 6$ 
C)  $e ^ { 3 } - e + 3 \ln 3 + 6$ 
D)  $e ^ { 3 } - e - 3 \ln 3 + 6$ 
E)  $e ^ { 3 } + e - 3 \ln 3 + 6$ 
A)  $e ^ { 3 } + e + 3 \ln 3 + 6$ 
B)  $e ^ { 3 } - e - 3 \ln 3 - 6$ 
C)  $e ^ { 3 } - e + 3 \ln 3 + 6$ 
D)  $e ^ { 3 } - e - 3 \ln 3 + 6$ 
E)  $e ^ { 3 } + e - 3 \ln 3 + 6$

Question 4

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ in the first octant. Then the flux of $\text { F }$ through S is

Accepted Answer

A) 78? 
B) 84? 
C) 99? 
D) 108? 
E) 121? 
A) 78? 
B) 84? 
C) 99? 
D) 108? 
E) 121?

Question 5

Using Green's Theorem, the line integral $\int $C $( x + y ) d x + x y d y$ where C is the closed curve consisting of the arc of $4 y = x ^ { 3 }$ from (0, 0) to (2, 2), the line segment from (2, 2) to (2, 0), and the line segment from (2, 0) to (0, 0), is

Accepted Answer

A)  $- \frac { 10 } { 7 }$ 
B)  $- \frac { 9 } { 7 }$ 
C)  $- 1$ 
D)  $\frac { 3 } { 7 }$ 
E)  $\frac { 2 } { 7 }$ 
A)  $- \frac { 10 } { 7 }$ 
B)  $- \frac { 9 } { 7 }$ 
C)  $- 1$ 
D)  $\frac { 3 } { 7 }$ 
E)  $\frac { 2 } { 7 }$

Question 6

If I = $\int $C $\left( \frac { y } { x - 1 } \right) d x + \left( \ln ( 2 x - 2 ) + \frac { 1 } { y } \right) d y$ is independent of the path where C is a curve from (3, 1) to (2, 2), then I is

Accepted Answer

A) 0 
B)  $\ln 2$ 
C)  $\ln 3$ 
D)  $\ln 5$ 
E)  $\ln 6$ 
A) 0 
B)  $\ln 2$ 
C)  $\ln 3$ 
D)  $\ln 5$ 
E)  $\ln 6$

Question 7

Let $f ( x , y , z ) = \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { - \frac { 1 } { 2 } }$ .Its gradient vector field is

Accepted Answer

A)  $\nabla f ( x , y , z ) = - \frac { x \mathbf { i } - y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
B)  $\nabla f ( x , y , z ) = \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
C)  $\nabla f ( x , y , z ) = - \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
D)  $\nabla f ( x , y , z ) = \frac { x \mathbf { i } - y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
E)  $\nabla f ( x , y , z ) = - \frac { x \mathbf { i } - y \mathbf { j } - z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
A)  $\nabla f ( x , y , z ) = - \frac { x \mathbf { i } - y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
B)  $\nabla f ( x , y , z ) = \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
C)  $\nabla f ( x , y , z ) = - \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
D)  $\nabla f ( x , y , z ) = \frac { x \mathbf { i } - y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$ 
E)  $\nabla f ( x , y , z ) = - \frac { x \mathbf { i } - y \mathbf { j } - z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$

Question 8

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k }$ and S is $z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ Then the flux of $\text { F }$ through S is

Accepted Answer

A) 8? 
B) 6? 
C) 4? 
D) 2? 
E) 0 
A) 8? 
B) 6? 
C) 4? 
D) 2? 
E) 0

Question 9

The domain of the vector field $\mathbf { F } ( x , y ) = \sin ( x + y ) \mathbf { i } - \frac { 5 } { x ^ { 2 } - 2 x y + y ^ { 2 } } \mathbf { j }$ is

Accepted Answer

A)  $\{ ( x , y ) : x \neq y \}$ 
B)  $\{ ( x , y ) : x y > 0 \}$ 
C)  $\{ ( x , y ) : x \neq | y | \}$ 
D)  $\{ ( x , y ) : x > 1 , y > 0 \}$ 
E)  $\{ ( x , y ) : x \geq | y | \}$ 
A)  $\{ ( x , y ) : x \neq y \}$ 
B)  $\{ ( x , y ) : x y > 0 \}$ 
C)  $\{ ( x , y ) : x \neq | y | \}$ 
D)  $\{ ( x , y ) : x > 1 , y > 0 \}$ 
E)  $\{ ( x , y ) : x \geq | y | \}$

Question 10

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + y \mathbf { k }$ and S is $2 x + 3 y + z = 6$ in the first octant. Then the flux of $\text { F }$ through S is

Accepted Answer

A) 8 
B) 9 
C) 10 
D) 14 
E) 15 
A) 8 
B) 9 
C) 10 
D) 14 
E) 15

Question 11

Let $\mathbf { F } ( x , y , z ) = x y \mathbf { i } + z \mathbf { j } + ( x + y ) \mathbf { k }$ and S is the region bounded by $x = 4 - z , y = 4$ and the coordinate planes. Using the Divergence Theorem, $\iint S$ $\mathbf { F } \cdot \mathbf { n } d S$ is

Accepted Answer

A) 48 
B) 54 
C) 60 
D) 64 
E) 72 
A) 48 
B) 54 
C) 60 
D) 64 
E) 72

Question 12

If $\mathbf { F } ( x , y ) = \left( 6 x ^ { 2 } y ^ { 2 } - 14 x y + 3 \right) \mathbf { i } + \left( 4 x ^ { 3 } y - 7 x ^ { 2 } - 8 \right) \mathbf { j }$ is a conservative field, then its potential function $f ( x , y )$ is

Accepted Answer

A)  $2 x ^ { 3 } y ^ { 2 } + 7 x ^ { 2 } y + 3 x - 8 y + C$ 
B)  $2 x ^ { 3 } y ^ { 2 } - 7 x ^ { 2 } y + 3 x - 8 y + C$ 
C)  $2 x ^ { 3 } y ^ { 2 } - 7 x ^ { 2 } y - 3 x - 8 y + C$ 
D)  $2 x ^ { 3 } y ^ { 2 } - 7 x ^ { 2 } y + 3 x + 8 y + C$ 
E)  $2 x ^ { 3 } y ^ { 2 } + 7 x ^ { 2 } y - 3 x - 8 y + C$ 
A)  $2 x ^ { 3 } y ^ { 2 } + 7 x ^ { 2 } y + 3 x - 8 y + C$ 
B)  $2 x ^ { 3 } y ^ { 2 } - 7 x ^ { 2 } y + 3 x - 8 y + C$ 
C)  $2 x ^ { 3 } y ^ { 2 } - 7 x ^ { 2 } y - 3 x - 8 y + C$ 
D)  $2 x ^ { 3 } y ^ { 2 } - 7 x ^ { 2 } y + 3 x + 8 y + C$ 
E)  $2 x ^ { 3 } y ^ { 2 } + 7 x ^ { 2 } y - 3 x - 8 y + C$

Question 13

The surface integral $\int \int$S $z ^ { 2 } d S$ where S is $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ between $z = 1$ and $z = 2$ is

Accepted Answer

A)  $- \frac { 15 \pi } { \sqrt { 2 } }$ 
B)  $\frac { 15 \pi } { \sqrt { 2 } }$ 
C)  $\frac { 15 \pi } { \sqrt { 3 } }$ 
D)  $- \frac { 15 \pi } { \sqrt { 3 } }$ 
E)  $\frac { 15 \pi } { 2 }$ 
A)  $- \frac { 15 \pi } { \sqrt { 2 } }$ 
B)  $\frac { 15 \pi } { \sqrt { 2 } }$ 
C)  $\frac { 15 \pi } { \sqrt { 3 } }$ 
D)  $- \frac { 15 \pi } { \sqrt { 3 } }$ 
E)  $\frac { 15 \pi } { 2 }$

Question 14

Let $\mathbf { F } ( x , y , z ) = 3 z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Using the Divergence Theorem, $\iint S$ $\mathbf { F } \bullet \mathbf { n } d S$ is

Accepted Answer

A) 64 $\pi$ 
B) 72 $\pi$ 
C) 84 $\pi$ 
D) 96 $\pi$ 
E)  $108 \pi$ 
A) 64 $\pi$ 
B) 72 $\pi$ 
C) 84 $\pi$ 
D) 96 $\pi$ 
E)  $108 \pi$

Question 15

The line integral $\int $C $\left( x ^ { 2 } + x y \right) d x + \left( y ^ { 2 } - x y \right) d y$ , where C is the line $y = x$ from (0,0) to (2,2), is

Accepted Answer

A)  $\frac { 32 } { 3 }$ 
B)  $\frac { 28 } { 3 }$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 14 } { 3 }$ 
E)  $\frac { 13 } { 3 }$ 
A)  $\frac { 32 } { 3 }$ 
B)  $\frac { 28 } { 3 }$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 14 } { 3 }$ 
E)  $\frac { 13 } { 3 }$

Question 16

The work done by the force $\mathbf { F } ( x , y ) = - x ^ { 2 } y \mathbf { i } + 2 y \mathbf { j }$ moving along $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j }$ with $0 \leq t \leq \frac { \pi } { 2 }$ is

Accepted Answer

A)  $4 ( \pi + 1 )$ 
B)  $8 ( \pi + 1 )$ 
C)  $16 ( \pi + 1 )$ 
D)  $22 ( \pi + 1 )$ 
E)  $24 ( \pi + 1 )$ 
A)  $4 ( \pi + 1 )$ 
B)  $8 ( \pi + 1 )$ 
C)  $16 ( \pi + 1 )$ 
D)  $22 ( \pi + 1 )$ 
E)  $24 ( \pi + 1 )$

Question 17

The work done by the force $\mathbf { F } ( x , y ) = ( y - x ) \mathbf { i } + x ^ { 2 } y \mathbf { j }$ moving along $y = 3 x - 2$ from (1, 1) to (2, 4) is

Accepted Answer

A)  $\frac { 85 } { 4 }$ 
B)  $\frac { 83 } { 4 }$ 
C)  $\frac { 81 } { 4 }$ 
D)  $\frac { 77 } { 4 }$ 
E)  $\frac { 71 } { 5 }$ 
A)  $\frac { 85 } { 4 }$ 
B)  $\frac { 83 } { 4 }$ 
C)  $\frac { 81 } { 4 }$ 
D)  $\frac { 77 } { 4 }$ 
E)  $\frac { 71 } { 5 }$

Question 18

The line integral $\int $C $4 x y d x + \left( 2 x ^ { 2 } - 3 x y \right) d y$ , where C is the line $y = x$ from (0,0) to (2,2), is

Accepted Answer

A) 12 
B) 8 
C) 6 
D) 4 
E) 2 
A) 12 
B) 8 
C) 6 
D) 4 
E) 2

Question 19

The work done by the force $\mathbf { F } ( x , y ) = - x ^ { 2 } y \mathbf { i } + 2 y \mathbf { j }$ moving along the line segment from (3, 0) to (0, 3) is

Accepted Answer

A)  $\frac { 71 } { 4 }$ 
B)  $\frac { 69 } { 4 }$ 
C)  $\frac { 67 } { 4 }$ 
D)  $\frac { 65 } { 4 }$ 
E)  $\frac { 63 } { 4 }$ 
A)  $\frac { 71 } { 4 }$ 
B)  $\frac { 69 } { 4 }$ 
C)  $\frac { 67 } { 4 }$ 
D)  $\frac { 65 } { 4 }$ 
E)  $\frac { 63 } { 4 }$

Question 20

If $\mathbf { F } ( x , y ) = \left( 4 y ^ { 2 } + 6 x y - 2 \right) \mathbf { i } + \left( 3 x ^ { 2 } + 8 x y + 1 \right) \mathbf { j }$ is a conservative field, then its potential function $f ( x , y )$ is

Accepted Answer

A)  $4 x y ^ { 2 } - 3 x ^ { 2 } y - 2 x - y + C$ 
B)  $4 x y ^ { 2 } - 3 x ^ { 2 } y + 2 x + y + C$ 
C)  $4 x y ^ { 2 } + 3 x ^ { 2 } y + 2 x + y + C$ 
D)  $4 x y ^ { 2 } + 3 x ^ { 2 } y - 2 x + y + C$ 
E)  $4 x y ^ { 2 } - 3 x ^ { 2 } y - 2 x + y + C$ 
A)  $4 x y ^ { 2 } - 3 x ^ { 2 } y - 2 x - y + C$ 
B)  $4 x y ^ { 2 } - 3 x ^ { 2 } y + 2 x + y + C$ 
C)  $4 x y ^ { 2 } + 3 x ^ { 2 } y + 2 x + y + C$ 
D)  $4 x y ^ { 2 } + 3 x ^ { 2 } y - 2 x + y + C$ 
E)  $4 x y ^ { 2 } - 3 x ^ { 2 } y - 2 x + y + C$

The line integral $\int$ C $y d s$ where C is the curve $x = 3 t ^ { 2 } , y = t , 0 \leq t \leq 1$ is

Using Green's Theorem, the line integral $\int$ C $x ^ { 2 } y d x - y ^ { 2 } x d y$ where C is the circle $x ^ { 2 } + y ^ { 2 } = 1 \text {, }$ is

If I = $\int$ C $( \ln x + 2 ) d x + \left( e ^ { y } + 2 x \right) d y$ is independent of the path where C is a curve from (3, 1) to (1, 3), then I is

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ in the first octant. Then the flux of $\text { F }$ through S is

Using Green's Theorem, the line integral $\int$ C $( x + y ) d x + x y d y$ where C is the closed curve consisting of the arc of $4 y = x ^ { 3 }$ from (0, 0) to (2, 2), the line segment from (2, 2) to (2, 0), and the line segment from (2, 0) to (0, 0), is

If I = $\int$ C $\left( \frac { y } { x - 1 } \right) d x + \left( \ln ( 2 x - 2 ) + \frac { 1 } { y } \right) d y$ is independent of the path where C is a curve from (3, 1) to (2, 2), then I is

Let $f ( x , y , z ) = \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { - \frac { 1 } { 2 } }$ .Its gradient vector field is

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k }$ and S is $z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ Then the flux of $\text { F }$ through S is

The domain of the vector field $\mathbf { F } ( x , y ) = \sin ( x + y ) \mathbf { i } - \frac { 5 } { x ^ { 2 } - 2 x y + y ^ { 2 } } \mathbf { j }$ is

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + y \mathbf { k }$ and S is $2 x + 3 y + z = 6$ in the first octant. Then the flux of $\text { F }$ through S is

Let $\mathbf { F } ( x , y , z ) = x y \mathbf { i } + z \mathbf { j } + ( x + y ) \mathbf { k }$ and S is the region bounded by $x = 4 - z , y = 4$ and the coordinate planes. Using the Divergence Theorem, $\iint S$ $\mathbf { F } \cdot \mathbf { n } d S$ is

If $\mathbf { F } ( x , y ) = \left( 6 x ^ { 2 } y ^ { 2 } - 14 x y + 3 \right) \mathbf { i } + \left( 4 x ^ { 3 } y - 7 x ^ { 2 } - 8 \right) \mathbf { j }$ is a conservative field, then its potential function $f ( x , y )$ is

The surface integral $\int \int$ S $z ^ { 2 } d S$ where S is $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ between $z = 1$ and $z = 2$ is

Let $\mathbf { F } ( x , y , z ) = 3 z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Using the Divergence Theorem, $\iint S$ $\mathbf { F } \bullet \mathbf { n } d S$ is

The line integral $\int$ C $\left( x ^ { 2 } + x y \right) d x + \left( y ^ { 2 } - x y \right) d y$ , where C is the line $y = x$ from (0,0) to (2,2), is

The work done by the force $\mathbf { F } ( x , y ) = - x ^ { 2 } y \mathbf { i } + 2 y \mathbf { j }$ moving along $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j }$ with $0 \leq t \leq \frac { \pi } { 2 }$ is

The work done by the force $\mathbf { F } ( x , y ) = ( y - x ) \mathbf { i } + x ^ { 2 } y \mathbf { j }$ moving along $y = 3 x - 2$ from (1, 1) to (2, 4) is

The line integral $\int$ C $4 x y d x + \left( 2 x ^ { 2 } - 3 x y \right) d y$ , where C is the line $y = x$ from (0,0) to (2,2), is

The work done by the force $\mathbf { F } ( x , y ) = - x ^ { 2 } y \mathbf { i } + 2 y \mathbf { j }$ moving along the line segment from (3, 0) to (0, 3) is

If $\mathbf { F } ( x , y ) = \left( 4 y ^ { 2 } + 6 x y - 2 \right) \mathbf { i } + \left( 3 x ^ { 2 } + 8 x y + 1 \right) \mathbf { j }$ is a conservative field, then its potential function $f ( x , y )$ is

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Exam 16: Vector Calculus

The line integral ∫\int ∫ C ydsy d syds where C is the curve x=3t2,y=t,0≤t≤1x = 3 t ^ { 2 } , y = t , 0 \leq t \leq 1x=3t2,y=t,0≤t≤1 is

Using Green's Theorem, the line integral ∫\int ∫ C x2ydx−y2xdyx ^ { 2 } y d x - y ^ { 2 } x d yx2ydx−y2xdy where C is the circle x2+y2=1, x ^ { 2 } + y ^ { 2 } = 1 \text {, }x2+y2=1, is

If I = ∫\int ∫ C (ln⁡x+2)dx+(ey+2x)dy( \ln x + 2 ) d x + \left( e ^ { y } + 2 x \right) d y(lnx+2)dx+(ey+2x)dy is independent of the path where C is a curve from (3, 1) to (1, 3), then I is

Let F(x,y,z)=xi+yj+zk\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }F(x,y,z)=xi+yj+zk and S is x2+y2+z2=36x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36x2+y2+z2=36 in the first octant. Then the flux of F \text { F } F through S is

Using Green's Theorem, the line integral ∫\int ∫ C (x+y)dx+xydy( x + y ) d x + x y d y(x+y)dx+xydy where C is the closed curve consisting of the arc of 4y=x34 y = x ^ { 3 }4y=x3 from (0, 0) to (2, 2), the line segment from (2, 2) to (2, 0), and the line segment from (2, 0) to (0, 0), is

If I = ∫\int ∫ C (yx−1)dx+(ln⁡(2x−2)+1y)dy\left( \frac { y } { x - 1 } \right) d x + \left( \ln ( 2 x - 2 ) + \frac { 1 } { y } \right) d y(x−1y​)dx+(ln(2x−2)+y1​)dy is independent of the path where C is a curve from (3, 1) to (2, 2), then I is

Let f(x,y,z)=(x2+y2+z2)−12f ( x , y , z ) = \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { - \frac { 1 } { 2 } }f(x,y,z)=(x2+y2+z2)−21​ .Its gradient vector field is

Let F(x,y,z)=xi+yj−2zk\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k }F(x,y,z)=xi+yj−2zk and S is z=1−x2−y2z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }z=1−x2−y2​ Then the flux of F \text { F } F through S is

The domain of the vector field F(x,y)=sin⁡(x+y)i−5x2−2xy+y2j\mathbf { F } ( x , y ) = \sin ( x + y ) \mathbf { i } - \frac { 5 } { x ^ { 2 } - 2 x y + y ^ { 2 } } \mathbf { j }F(x,y)=sin(x+y)i−x2−2xy+y25​j is

Let F(x,y,z)=xi+yj+yk\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + y \mathbf { k }F(x,y,z)=xi+yj+yk and S is 2x+3y+z=62 x + 3 y + z = 62x+3y+z=6 in the first octant. Then the flux of F \text { F } F through S is

The surface integral ∫∫\int \int∫∫ S z2dSz ^ { 2 } d Sz2dS where S is z2=x2+y2z ^ { 2 } = x ^ { 2 } + y ^ { 2 }z2=x2+y2 between z=1z = 1z=1 and z=2z = 2z=2 is

Let F(x,y,z)=3zk\mathbf { F } ( x , y , z ) = 3 z \mathbf { k }F(x,y,z)=3zk and S is x2+y2+z2=9x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9x2+y2+z2=9 . Using the Divergence Theorem, ∬S\iint S∬S F∙ndS\mathbf { F } \bullet \mathbf { n } d SF∙ndS is

The line integral ∫\int ∫ C (x2+xy)dx+(y2−xy)dy\left( x ^ { 2 } + x y \right) d x + \left( y ^ { 2 } - x y \right) d y(x2+xy)dx+(y2−xy)dy , where C is the line y=xy = xy=x from (0,0) to (2,2), is

The work done by the force F(x,y)=(y−x)i+x2yj\mathbf { F } ( x , y ) = ( y - x ) \mathbf { i } + x ^ { 2 } y \mathbf { j }F(x,y)=(y−x)i+x2yj moving along y=3x−2y = 3 x - 2y=3x−2 from (1, 1) to (2, 4) is

The line integral ∫\int ∫ C 4xydx+(2x2−3xy)dy4 x y d x + \left( 2 x ^ { 2 } - 3 x y \right) d y4xydx+(2x2−3xy)dy , where C is the line y=xy = xy=x from (0,0) to (2,2), is

The work done by the force F(x,y)=−x2yi+2yj\mathbf { F } ( x , y ) = - x ^ { 2 } y \mathbf { i } + 2 y \mathbf { j }F(x,y)=−x2yi+2yj moving along the line segment from (3, 0) to (0, 3) is

If F(x,y)=(4y2+6xy−2)i+(3x2+8xy+1)j\mathbf { F } ( x , y ) = \left( 4 y ^ { 2 } + 6 x y - 2 \right) \mathbf { i } + \left( 3 x ^ { 2 } + 8 x y + 1 \right) \mathbf { j }F(x,y)=(4y2+6xy−2)i+(3x2+8xy+1)j is a conservative field, then its potential function f(x,y)f ( x , y )f(x,y) is

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The line integral $\int$ C $y d s$ where C is the curve $x = 3 t ^ { 2 } , y = t , 0 \leq t \leq 1$ is

Using Green's Theorem, the line integral $\int$ C $x ^ { 2 } y d x - y ^ { 2 } x d y$ where C is the circle $x ^ { 2 } + y ^ { 2 } = 1 \text {, }$ is

If I = $\int$ C $( \ln x + 2 ) d x + \left( e ^ { y } + 2 x \right) d y$ is independent of the path where C is a curve from (3, 1) to (1, 3), then I is

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ in the first octant. Then the flux of $\text { F }$ through S is

Using Green's Theorem, the line integral $\int$ C $( x + y ) d x + x y d y$ where C is the closed curve consisting of the arc of $4 y = x ^ { 3 }$ from (0, 0) to (2, 2), the line segment from (2, 2) to (2, 0), and the line segment from (2, 0) to (0, 0), is

If I = $\int$ C $\left( \frac { y } { x - 1 } \right) d x + \left( \ln ( 2 x - 2 ) + \frac { 1 } { y } \right) d y$ is independent of the path where C is a curve from (3, 1) to (2, 2), then I is

Let $f ( x , y , z ) = \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { - \frac { 1 } { 2 } }$ .Its gradient vector field is

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k }$ and S is $z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ Then the flux of $\text { F }$ through S is

The domain of the vector field $\mathbf { F } ( x , y ) = \sin ( x + y ) \mathbf { i } - \frac { 5 } { x ^ { 2 } - 2 x y + y ^ { 2 } } \mathbf { j }$ is

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + y \mathbf { k }$ and S is $2 x + 3 y + z = 6$ in the first octant. Then the flux of $\text { F }$ through S is

The surface integral $\int \int$ S $z ^ { 2 } d S$ where S is $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ between $z = 1$ and $z = 2$ is

Let $\mathbf { F } ( x , y , z ) = 3 z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Using the Divergence Theorem, $\iint S$ $\mathbf { F } \bullet \mathbf { n } d S$ is

The line integral $\int$ C $\left( x ^ { 2 } + x y \right) d x + \left( y ^ { 2 } - x y \right) d y$ , where C is the line $y = x$ from (0,0) to (2,2), is

The work done by the force $\mathbf { F } ( x , y ) = ( y - x ) \mathbf { i } + x ^ { 2 } y \mathbf { j }$ moving along $y = 3 x - 2$ from (1, 1) to (2, 4) is

The line integral $\int$ C $4 x y d x + \left( 2 x ^ { 2 } - 3 x y \right) d y$ , where C is the line $y = x$ from (0,0) to (2,2), is

The work done by the force $\mathbf { F } ( x , y ) = - x ^ { 2 } y \mathbf { i } + 2 y \mathbf { j }$ moving along the line segment from (3, 0) to (0, 3) is

If $\mathbf { F } ( x , y ) = \left( 4 y ^ { 2 } + 6 x y - 2 \right) \mathbf { i } + \left( 3 x ^ { 2 } + 8 x y + 1 \right) \mathbf { j }$ is a conservative field, then its potential function $f ( x , y )$ is