Question 1

Is F(x, y) = 7x i + 8y j is a conservative vector field?

Accepted Answer

The answer of Is F(x, y) = 7x i +...

Question 2

Use Stokes' Theorem to evaluate $\int$C 2(x + y)dx + 2(2x - 3)dy + 2(y + z)dz over the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) traversed in a counterclockwise manner.

Accepted Answer

A)  2 
B)  12 
C)  0 
D)  24 
E)  18 
A)  2 
B)  12 
C)  0 
D)  24 
E)  18

Question 3

Evaluate   where F(x, y, z) = 12z i + 12x j + 12y k and C is the helix r(t) = sin t i + 3 sin t j + sin2t k for   .

Accepted Answer

Question 4

Use Green's Theorem to evaluate $\int$4(3x2 + y)dx + 8xy3dy where C is the rectangle bounded by x = -1, x = 3, y = 0, and y = 2.

Accepted Answer

The answer of Use Green's Theorem to evaluate $\int$4(3x2 +...

Question 5

Use the divergence theorem to evaluate   where F(x, y, z) = 11yz i + 11xy j + 11xz k, n is the outer unit normal to $\sigma$, and $\sigma$ is the surface enclosed by the cylinder x2 + z2 = 1 and the planes y = -1 and y = 1.

Accepted Answer

The answer of Use the divergence theorem to evaluate ...

Question 6

Evaluate the surface integral   where   is the part of the plane   in the first octant.

Accepted Answer

A)  3 
B)  192 
C)    
D)  400 
E)  800 $\pi$ 
A)  3 
B)  192 
C)    
D)  400 
E)  800 $\pi$

Question 7

If F(x, y, z) = 7y j + 7z k, the magnitude of the flux through the portion of the surface $\sigma$ that lies in front of the xz-plane, where $\sigma$ is defined by y = 1 - x2 - z2, is

Accepted Answer

A)    
B)    
C)  7 $\pi$ 
D)  0 
E)   $\pi$ 
A)    
B)    
C)  7 $\pi$ 
D)  0 
E)   $\pi$

Question 8

Use Green's Theorem to evaluate   , where C is the circle   .

Accepted Answer

A)   $\pi$ 
B)  7 
C)  49 $\pi$ 
D)  7 $\pi$ 
E)  0 
A)   $\pi$ 
B)  7 
C)  49 $\pi$ 
D)  7 $\pi$ 
E)  0

Question 9

Sketch the vector field, F(x, y) = -5x j, by drawing some typical non-intersecting vectors. The vectors need not be drawn to the same scale as the coordinate axes, but they should be in the correct proportions relative to each other.

Accepted Answer

The answer of Sketch the vector field, F(x, y) =...

Question 10

Evaluate the surface integral   where $\sigma$ is that portion of the plane x + 2y + 3z = 6 which lies in the first octant.

Accepted Answer

The answer of Evaluate the surface integral   where...

Question 11

Find the surface area of (x - 7)2 + (y + 1)2 + (z - 4)2 = 4 that lies below z = 6.

Accepted Answer

A)  8 $\pi$ 
B)  32 $\pi$ 
C)  16 $\pi$ 
D)  64 $\pi$ 
E)  128 $\pi$ 
A)  8 $\pi$ 
B)  32 $\pi$ 
C)  16 $\pi$ 
D)  64 $\pi$ 
E)  128 $\pi$

Question 12

Evaluate the surface integral   where $\sigma$ is that portion of the cylinder x2 + z2 = 1 that lies above the xy-plane enclosed by 0 $\le$ y $\le$ 5.

Accepted Answer

The answer of Evaluate the surface integral   where...

Question 13

Find the outward flux of F(x, y, z) = 4x i + 4y j + 4(z - 2)k across the rectangular box with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (3, 0, 0), (3, 1, 0), (3, 0, 1), and (3, 1, 1).

Accepted Answer

A)  12 
B)  36 
C)  0 
D)  1 
E)  4 
A)  12 
B)  36 
C)  0 
D)  1 
E)  4

Question 14

Use Stokes' Theorem to evaluate $\int$C -21 dx + 21x dy + 7z dz over the circle x2 + y2 = 1, z = 1 traversed counterclockwise.

Accepted Answer

The answer of Use Stokes' Theorem to evaluate $\int$C -21...

Question 15

Find the surface area of (x - 1)2 + (y + 1)2 + (z - 4)2 = 4 that lies below z = 4.

Accepted Answer

A)  8 $\pi$ 
B)  16 $\pi$ 
C)  32 $\pi$ 
D)  64 $\pi$ 
E)  128 $\pi$ 
A)  8 $\pi$ 
B)  16 $\pi$ 
C)  32 $\pi$ 
D)  64 $\pi$ 
E)  128 $\pi$

Question 16

Evaluate the surface integral   where $\sigma$ is the portion of the cone   for 0 $\le$ x $\le$ 3.

Accepted Answer

The answer of Evaluate the surface integral   where...

Question 17

F(x, y, z) = 9x2 i + (2y - 4) j + 3k. Find div F.

Accepted Answer

A)  18x + 2 
B)  36x 
C)  18x 
D)  18x + 5 
E)  36x + 5 
A)  18x + 2 
B)  36x 
C)  18x 
D)  18x + 5 
E)  36x + 5

Question 18

Evaluate   where F(x, y, z) = 2x i + 2y j + 4z k and $\sigma$ is that portion of the surface z = 4 - x2 - y2 above the xy-plane oriented by upward unit normals.

Accepted Answer

The answer of Evaluate   where F(x, y, z)...

Question 19

Evaluate the surface integral   where $\sigma$ is that portion of the cylinder y2 + z2 = 1 that lies above the xy-plane enclosed by 0 $\le$  x $\le$ 5 and -1 $\le$  y $\le$ 1.

Accepted Answer

A)    
B)    
C)    
D)    
E)  0 
A)    
B)    
C)    
D)    
E)  0

Question 20

Is F(x, y) = 2y i + 2x j is a conservative vector field?

Accepted Answer

The answer of Is F(x, y) = 2y i +...

Is F(x, y) = 7x i + 8y j is a conservative vector field?

Use Stokes' Theorem to evaluate $\int$ _C 2(x + y)dx + 2(2x - 3)dy + 2(y + z)dz over the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) traversed in a counterclockwise manner.

Evaluate where F(x, y, z) = 12z i + 12x j + 12y k and C is the helix r(t) = sin t i + 3 sin t j + sin²t k for .

Use Green's Theorem to evaluate $\int$ 4(3x² + y)dx + 8xy³dy where C is the rectangle bounded by x = -1, x = 3, y = 0, and y = 2.

Evaluate the surface integral where is the part of the plane in the first octant.

If F(x, y, z) = 7y j + 7z k, the magnitude of the flux through the portion of the surface $\sigma$ that lies in front of the xz-plane, where $\sigma$ is defined by y = 1 - x² - z², is

Use Green's Theorem to evaluate , where C is the circle .

Sketch the vector field, F(x, y) = -5x j, by drawing some typical non-intersecting vectors. The vectors need not be drawn to the same scale as the coordinate axes, but they should be in the correct proportions relative to each other.

Evaluate the surface integral $Evaluate the surface integral where \sigma is that portion of the plane x + 2y + 3z = 6 which lies in the first octant.$ where $\sigma$ is that portion of the plane x + 2y + 3z = 6 which lies in the first octant.

Find the surface area of (x - 7)² + (y + 1)² + (z - 4)² = 4 that lies below z = 6.

Find the outward flux of F(x, y, z) = 4x i + 4y j + 4(z - 2)k across the rectangular box with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (3, 0, 0), (3, 1, 0), (3, 0, 1), and (3, 1, 1).

Use Stokes' Theorem to evaluate $\int$ _C -21 dx + 21x dy + 7z dz over the circle x² + y² = 1, z = 1 traversed counterclockwise.

Find the surface area of (x - 1)² + (y + 1)² + (z - 4)² = 4 that lies below z = 4.

Evaluate the surface integral $Evaluate the surface integral where \sigma is the portion of the cone for 0 \le x \le 3.$ where $\sigma$ is the portion of the cone $Evaluate the surface integral where \sigma is the portion of the cone for 0 \le x \le 3.$ for 0 $\le$ x $\le$ 3.

F(x, y, z) = 9x² i + (2y - 4) j + 3k. Find div F.

Is F(x, y) = 2y i + 2x j is a conservative vector field?

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Vector-Valued Functions

Partial Derivatives

Multiple Integrals

Filters

Exam 15: Topics in Vector Calculus

Is F(x, y) = 7x i + 8y j is a conservative vector field?

Use Stokes' Theorem to evaluate ∫\int∫ C 2(x + y)dx + 2(2x - 3)dy + 2(y + z)dz over the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) traversed in a counterclockwise manner.

Evaluate where F(x, y, z) = 12z i + 12x j + 12y k and C is the helix r(t) = sin t i + 3 sin t j + sin2t k for .

Use Green's Theorem to evaluate ∫\int∫ 4(3x2 + y)dx + 8xy3dy where C is the rectangle bounded by x = -1, x = 3, y = 0, and y = 2.

Use the divergence theorem to evaluate where F(x, y, z) = 11yz i + 11xy j + 11xz k, n is the outer unit normal to σ\sigmaσ , and σ\sigmaσ is the surface enclosed by the cylinder x2 + z2 = 1 and the planes y = -1 and y = 1.

Evaluate the surface integral where is the part of the plane in the first octant.

If F(x, y, z) = 7y j + 7z k, the magnitude of the flux through the portion of the surface σ\sigmaσ that lies in front of the xz-plane, where σ\sigmaσ is defined by y = 1 - x2 - z2, is

Use Green's Theorem to evaluate , where C is the circle .

Sketch the vector field, F(x, y) = -5x j, by drawing some typical non-intersecting vectors. The vectors need not be drawn to the same scale as the coordinate axes, but they should be in the correct proportions relative to each other.

Evaluate the surface integral where σ\sigmaσ is that portion of the plane x + 2y + 3z = 6 which lies in the first octant.

Find the surface area of (x - 7)2 + (y + 1)2 + (z - 4)2 = 4 that lies below z = 6.

Evaluate the surface integral where σ\sigmaσ is that portion of the cylinder x2 + z2 = 1 that lies above the xy-plane enclosed by 0 ≤\le≤ y ≤\le≤ 5.

Find the outward flux of F(x, y, z) = 4x i + 4y j + 4(z - 2)k across the rectangular box with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (3, 0, 0), (3, 1, 0), (3, 0, 1), and (3, 1, 1).

Use Stokes' Theorem to evaluate ∫\int∫ C -21 dx + 21x dy + 7z dz over the circle x2 + y2 = 1, z = 1 traversed counterclockwise.

Find the surface area of (x - 1)2 + (y + 1)2 + (z - 4)2 = 4 that lies below z = 4.

Evaluate the surface integral where σ\sigmaσ is the portion of the cone for 0 ≤\le≤ x ≤\le≤ 3.

F(x, y, z) = 9x2 i + (2y - 4) j + 3k. Find div F.

Evaluate where F(x, y, z) = 2x i + 2y j + 4z k and σ\sigmaσ is that portion of the surface z = 4 - x2 - y2 above the xy-plane oriented by upward unit normals.

Evaluate the surface integral where σ\sigmaσ is that portion of the cylinder y2 + z2 = 1 that lies above the xy-plane enclosed by 0 ≤\le≤ x ≤\le≤ 5 and -1 ≤\le≤ y ≤\le≤ 1.

Is F(x, y) = 2y i + 2x j is a conservative vector field?

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Vector-Valued Functions

Partial Derivatives

Multiple Integrals

Filters

Use Stokes' Theorem to evaluate $\int$ _C 2(x + y)dx + 2(2x - 3)dy + 2(y + z)dz over the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) traversed in a counterclockwise manner.

Evaluate where F(x, y, z) = 12z i + 12x j + 12y k and C is the helix r(t) = sin t i + 3 sin t j + sin²t k for .

Use Green's Theorem to evaluate $\int$ 4(3x² + y)dx + 8xy³dy where C is the rectangle bounded by x = -1, x = 3, y = 0, and y = 2.

If F(x, y, z) = 7y j + 7z k, the magnitude of the flux through the portion of the surface $\sigma$ that lies in front of the xz-plane, where $\sigma$ is defined by y = 1 - x² - z², is

Evaluate the surface integral $Evaluate the surface integral where \sigma is that portion of the plane x + 2y + 3z = 6 which lies in the first octant.$ where $\sigma$ is that portion of the plane x + 2y + 3z = 6 which lies in the first octant.

Find the surface area of (x - 7)² + (y + 1)² + (z - 4)² = 4 that lies below z = 6.

Use Stokes' Theorem to evaluate $\int$ _C -21 dx + 21x dy + 7z dz over the circle x² + y² = 1, z = 1 traversed counterclockwise.

Find the surface area of (x - 1)² + (y + 1)² + (z - 4)² = 4 that lies below z = 4.

Evaluate the surface integral $Evaluate the surface integral where \sigma is the portion of the cone for 0 \le x \le 3.$ where $\sigma$ is the portion of the cone $Evaluate the surface integral where \sigma is the portion of the cone for 0 \le x \le 3.$ for 0 $\le$ x $\le$ 3.

F(x, y, z) = 9x² i + (2y - 4) j + 3k. Find div F.