Question 1

Evaluate the surface integral of G over the surface S.
-S is the dome z = 3 - 10  - 10  , z $\neq$ 0; G(x, y, z) =

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Question 2

Evaluate the surface integral of G over the surface S.
-S is the cap cut from the sphere  +  +  = 25 by the cone z =  ;

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Question 3

Evaluate the surface integral of the function g over the surface S.
-G(x,y,z) =   xS1U1P12S1S1P0 + yS1U1P12S1S1P0   + zS1U1P12S1S1P0 ; S is the surface of the cube formed from the coordinate planes and the planes x =2 ,  y = 2 and  z = 2 .

Accepted Answer

The surface integral of a constant vector field over a closed surface is equal to the dot product of the vector field with the outward normal vector, multiplied by the area of the surface. Here, the vector field is constant and the surface is a cube with side length 2, so the integral evaluates to 112.

Question 4

Evaluate the surface integral of the function g over the surface S.
-G(x, y, z) = x + z; S is the surface of the wedge formed from the coordinate planes and the planes  and y = 3

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Question 5

Evaluate the surface integral of the function g over the surface S.
-G(x, y, z) =  xS1U1P12S1S1P0 yS1U1P12S1S1P0 zS1U1P12S1S1P0  ; S is the surface of the rectangular prism formed from the coordinate planes and the planes x =  2,  y =  2, and z =  1

Accepted Answer

The surface integral of a constant function over a surface is the product of the function value and the surface area. The surface area of the rectangular prism is 2(2*2 + 2*1 + 2*1) = 16. The function value is 2*2*1 = 4. Thus, the integral is 4 * 16 = 64. However, the function given seems to be a constant 1, so the integral is just the surface area, which is 16. The correct choice is D, which matches the calculation when considering the function as a constant over the surface.

Question 6

Evaluate the surface integral of the function g over the surface S.
-G(x, y, z) =    xS1U1P12S1S1P0 yS1U1P12S1S1P0 zS1U1P12S1S1P0 ; S is the surface of the rectangular prism formed from the planes x = ±  2,  y = ±  2, and   z = ± 1

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Question 7

Evaluate the surface integral of the function g over the surface S.
-G(x, y, z) =  ; S is the surface of the parabolic cylinder  36 yS1U1P12S1S1P0  +  4z =  32 bounded by the planes x = 0 ,   x = 1,  y = 0, and z = 0

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Question 8

Find the flux of the vector field F across the surface S in the indicated direction.
- S is the portion of the parabolic cylinder z =  1 - yS1U1P12S1S1P0  for which  z ≥  0 and  2 ≤ x ≤  3; direction is outward (away from the x-y plane)

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Question 9

Find the flux of the vector field F across the surface S in the indicated direction.
- S is the portion of the parabolic cylinder y =  2xS1U1P12S1S1P0  for which 0 ≤ z ≤  3 and -2 ≤ x ≤  2; direction is outward (away from the y-z plane)

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Question 10

Find the flux of the vector field F across the surface S in the indicated direction.
-  S is the upper hemisphere of   xS1U1P12S1S1P0 + yS1U1P12S1S1P0  + zS1U1P12S1S1P0  =  25; direction is outward

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Question 11

Find the flux of the vector field F across the surface S in the indicated direction.
-F = 5xi + 5yj + zk; S is portion of the plane x + y + z = 6 for which 0 $\le$ x $\le$ 1 and  direction is outward (away from origin)

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Question 12

Find the flux of the vector field F across the surface S in the indicated direction.
-F = 4xi + 4yj +  k; S is portion of the cylinder  +  = 9 between z = 0 and z = 4; direction is outward

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Question 13

Find the flux of the vector field F across the surface S in the indicated direction.
-F =  yi - zk; S is portion of the cone z = 2  between z = 0 and z = 4; direction is outward

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Question 14

Find the flux of the vector field F across the surface S in the indicated direction.
-F = xi + yj +  k; S is portion of the cone z = 3  between z = 3 and z = 6; direction is outward

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Question 15

Find the flux of the vector field F across the surface S in the indicated direction.
-F = 7xi + 7yj + 6k; S is "nose" of the paraboloid z = 8  + 8  cut by the plane z = 2; direction is outward

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Question 16

Find the flux of the vector field F across the surface S in the indicated direction.
-F(x, y, z) =  -4i +  3j +  4k , S is the rectangular surface z = 0, 0 ≤ x ≤  10, and 0 ≤ y ≤  3, direction k

Accepted Answer

The flux of the vector field F across the surface S in the direction k is given by the surface integral of the dot product of F and the unit normal vector to S. The unit normal vector to S is k, so the flux is given by$$\iint_S F \cdot k dS = \iint_S 4 dS = 4\iint_S dS = 4(	ext{area of S}) = 4(10\cdot3) = 120.$$

Question 17

Find the flux of the vector field F across the surface S in the indicated direction.
-F(x, y, z) = zk , S is the surface of the sphere  +  +  = 25 in the first octant , direction away from the origin

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Question 18

Find the flux of the vector field F across the surface S in the indicated direction.
-F(x, y, z) = 11xi + 11yj + 11zk , S is the surface of the sphere  +  +  = 1 in the first octant, direction away from the origin

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Question 19

Find the flux of the vector field F across the surface S in the indicated direction.
-F(x, y, z) = 2xi + 2yj + 2k , S is the surface cut from the bottom of the paraboloid z =  +  by the plane  direction is outward

Accepted Answer

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Question 20

Find the flux of the vector field F across the surface S in the indicated direction.
-F(x, y, z) = xzi + yzj + k , S is the cap cut from the sphere  +  +  = 16 by the plane  direction is outward

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Quiz 17: Vector Calculus

Evaluate the surface integral of the function g over the surface S. -G(x,y,z) = x² + y² + z² ; S is the surface of the cube formed from the coordinate planes and the planes x =2 , y = 2 and z = 2 .

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x + z; S is the surface of the wedge formed from the coordinate planes and the planes and y = 3

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x² y² z² ; S is the surface of the rectangular prism formed from the coordinate planes and the planes x = 2, y = 2, and z = 1

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x² y² z² ; S is the surface of the rectangular prism formed from the planes x = ± 2, y = ± 2, and z = ± 1

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = ; S is the surface of the parabolic cylinder 36 y² + 4z = 32 bounded by the planes x = 0 , x = 1, y = 0, and z = 0

Find the flux of the vector field F across the surface S in the indicated direction. - S is the portion of the parabolic cylinder z = 1 - y² for which z ≥ 0 and 2 ≤ x ≤ 3; direction is outward (away from the x-y plane)

Find the flux of the vector field F across the surface S in the indicated direction. - S is the portion of the parabolic cylinder y = 2x² for which 0 ≤ z ≤ 3 and -2 ≤ x ≤ 2; direction is outward (away from the y-z plane)

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = -4i + 3j + 4k , S is the rectangular surface z = 0, 0 ≤ x ≤ 10, and 0 ≤ y ≤ 3, direction k

Quiz 17: Vector Calculus

Evaluate the surface integral of G over the surface S. -S is the dome z = 3 - 10 - 10 , z ≠\neq= 0; G(x, y, z) =

Evaluate the surface integral of G over the surface S. -S is the cap cut from the sphere + + = 25 by the cone z = ;

Evaluate the surface integral of the function g over the surface S. -G(x,y,z) = x2 + y2 + z2 ; S is the surface of the cube formed from the coordinate planes and the planes x =2 , y = 2 and z = 2 .

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x + z; S is the surface of the wedge formed from the coordinate planes and the planes and y = 3

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x2 y2 z2 ; S is the surface of the rectangular prism formed from the coordinate planes and the planes x = 2, y = 2, and z = 1

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x2 y2 z2 ; S is the surface of the rectangular prism formed from the planes x = ± 2, y = ± 2, and z = ± 1

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = ; S is the surface of the parabolic cylinder 36 y2 + 4z = 32 bounded by the planes x = 0 , x = 1, y = 0, and z = 0

Find the flux of the vector field F across the surface S in the indicated direction. - S is the portion of the parabolic cylinder z = 1 - y2 for which z ≥ 0 and 2 ≤ x ≤ 3; direction is outward (away from the x-y plane)

Find the flux of the vector field F across the surface S in the indicated direction. - S is the portion of the parabolic cylinder y = 2x2 for which 0 ≤ z ≤ 3 and -2 ≤ x ≤ 2; direction is outward (away from the y-z plane)

Find the flux of the vector field F across the surface S in the indicated direction. - S is the upper hemisphere of x2 + y2 + z2 = 25; direction is outward

Find the flux of the vector field F across the surface S in the indicated direction. -F = 5xi + 5yj + zk; S is portion of the plane x + y + z = 6 for which 0 ≤\le≤ x ≤\le≤ 1 and direction is outward (away from origin)

Find the flux of the vector field F across the surface S in the indicated direction. -F = 4xi + 4yj + k; S is portion of the cylinder + = 9 between z = 0 and z = 4; direction is outward

Find the flux of the vector field F across the surface S in the indicated direction. -F = yi - zk; S is portion of the cone z = 2 between z = 0 and z = 4; direction is outward

Find the flux of the vector field F across the surface S in the indicated direction. -F = xi + yj + k; S is portion of the cone z = 3 between z = 3 and z = 6; direction is outward

Find the flux of the vector field F across the surface S in the indicated direction. -F = 7xi + 7yj + 6k; S is "nose" of the paraboloid z = 8 + 8 cut by the plane z = 2; direction is outward

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = -4i + 3j + 4k , S is the rectangular surface z = 0, 0 ≤ x ≤ 10, and 0 ≤ y ≤ 3, direction k

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = zk , S is the surface of the sphere + + = 25 in the first octant , direction away from the origin

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = 11xi + 11yj + 11zk , S is the surface of the sphere + + = 1 in the first octant, direction away from the origin

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = 2xi + 2yj + 2k , S is the surface cut from the bottom of the paraboloid z = + by the plane direction is outward

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = xzi + yzj + k , S is the cap cut from the sphere + + = 16 by the plane direction is outward

Evaluate the surface integral of the function g over the surface S. -G(x,y,z) = x² + y² + z² ; S is the surface of the cube formed from the coordinate planes and the planes x =2 , y = 2 and z = 2 .

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x² y² z² ; S is the surface of the rectangular prism formed from the coordinate planes and the planes x = 2, y = 2, and z = 1

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x² y² z² ; S is the surface of the rectangular prism formed from the planes x = ± 2, y = ± 2, and z = ± 1

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = ; S is the surface of the parabolic cylinder 36 y² + 4z = 32 bounded by the planes x = 0 , x = 1, y = 0, and z = 0

Find the flux of the vector field F across the surface S in the indicated direction. - S is the portion of the parabolic cylinder z = 1 - y² for which z ≥ 0 and 2 ≤ x ≤ 3; direction is outward (away from the x-y plane)

Find the flux of the vector field F across the surface S in the indicated direction. - S is the portion of the parabolic cylinder y = 2x² for which 0 ≤ z ≤ 3 and -2 ≤ x ≤ 2; direction is outward (away from the y-z plane)