Question 1

Let   . Find divF.

Accepted Answer

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

Question 2

Use Green's Theorem to evaluate   , where C is 4x2 + 9y2 = 36.

Accepted Answer

A)  6 $\pi$ 
B)  6 
C)  12 $\pi$ 
D)  12 
E)  0 
A)  6 $\pi$ 
B)  6 
C)  12 $\pi$ 
D)  12 
E)  0

Question 3

Use Green's Theorem to evaluate   , where C is the square bounded by x = y = 0, and x = y = 1.

Accepted Answer

A)  1/2 
B)  1 
C)  2 
D)  0 
E)  3 
A)  1/2 
B)  1 
C)  2 
D)  0 
E)  3

Question 4

Use Stokes' Theorem to evaluate $\int$C 8z dx - 4x dy + 4x dz where C is the intersection of the cylinder x2 + y2 = 1 and the plane z = y + 1.

Accepted Answer

The answer of Use Stokes' Theorem to evaluate $\int$C 8z...

Question 5

Use Stokes' Theorem to evaluate   over the circle x2 + y2 = 16, z = 4.

Accepted Answer

A)  0 
B)  64 
C)  -64 
D)    
E)    
A)  0 
B)  64 
C)  -64 
D)    
E)

Question 6

The work done by   along the curve   from (0, 0) to (1, 1) is

Accepted Answer

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

Question 7

Use Green's Theorem to evaluate   , where C is x2 + y2 = 16.

Accepted Answer

A)  6 $\pi$ 
B)  9 $\pi$ 
C)  18 $\pi$ 
D)  0 
E)  12 $\pi$ 
A)  6 $\pi$ 
B)  9 $\pi$ 
C)  18 $\pi$ 
D)  0 
E)  12 $\pi$

Question 8

Determine whether F(x, y) = 6(y2 - 2 sin y)i + 6(2xy - 2x cos y)j is conservative. If it is, find a potential function for it.

Accepted Answer

Question 9

F(x, y, z) = 12x3 i + y2 j + z3 k. Find curl F.

Accepted Answer

Question 10

Evaluate $\int$C 8xy dx + 4(e x + x2)dy where C is the line segment from (0, 0) to (1, 1).

Accepted Answer

The answer of Evaluate $\int$C 8xy dx + 4(e x...

Question 11

Let F(x, y, z) = 8x2 i + 6y j + 3z k . Find the outward flux of the vector field F across the unit cube in the first octant and including the origin as a vertex.

Accepted Answer

A)  17 
B)  18 
C)  16 
D)  27 
E)  37 
A)  17 
B)  18 
C)  16 
D)  27 
E)  37

Question 12

Use Green's Theorem to evaluate   where C is   .

Accepted Answer

A)  2 $\pi$ 
B)  2 
C)  12 $\pi$ 
D)  12 
E)  0 
A)  2 $\pi$ 
B)  2 
C)  12 $\pi$ 
D)  12 
E)  0

Question 13

Find the flux of the vector field F(x, y, z) = 3z k across the sphere x2 + y2 + z2 = 9 oriented outward.

Accepted Answer

A)   $\pi$ 
B)  3 $\pi$ 
C)  0 
D)  108 $\pi$ 
E)     $\pi$ 
A)   $\pi$ 
B)  3 $\pi$ 
C)  0 
D)  108 $\pi$ 
E)     $\pi$

Question 14

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

Accepted Answer

The answer of Evaluate the surface integral   where...

Question 15

Use Green's Theorem to evaluate   , where C is the square bounded by x = y = 0, and x = y = 1.

Accepted Answer

A)  1/2 
B)  -5 
C)  2 
D)  0 
E)  3 
A)  1/2 
B)  -5 
C)  2 
D)  0 
E)  3

Question 16

Evaluate   where F(x, y) = 5x2y i + 20j and C is the curve   for 0 $\le$ t $\le$ 1.

Accepted Answer

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

Question 17

Find the outward flux of the vector field   across the sphere   .

Accepted Answer

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

Question 18

For   the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(1, 2) is

Accepted Answer

A)    Nm 
B)    Nm 
C)    Nm 
D)    Nm 
E)    Nm 
A)    Nm 
B)    Nm 
C)    Nm 
D)    Nm 
E)    Nm

Question 19

Use Green's Theorem to evaluate $\int$C -2x2y dx + 2xy2dy) where C is the boundary of the circle x2 + y2 = 16 traversed in a counterclockwise manner.

Accepted Answer

The answer of Use Green's Theorem to evaluate $\int$C -2x2y...

Question 20

Use Stokes' Theorem to evaluate $\int$C 3(z - y)dx + 3(x - z)dy + 3(y - x)dz where C is the boundary, in the xy-plane, of the surface $\sigma$ given by z = 4 - (x2 + y2), z$\ge$0.

Accepted Answer

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

Let . Find divF.

Use Green's Theorem to evaluate , where C is 4x² + 9y² = 36.

Use Green's Theorem to evaluate , where C is the square bounded by x = y = 0, and x = y = 1.

Use Stokes' Theorem to evaluate $\int$ _C 8z dx - 4x dy + 4x dz where C is the intersection of the cylinder x² + y² = 1 and the plane z = y + 1.

Use Stokes' Theorem to evaluate over the circle x² + y² = 16, z = 4.

The work done by along the curve from (0, 0) to (1, 1) is

Use Green's Theorem to evaluate , where C is x² + y² = 16.

Determine whether F(x, y) = 6(y² - 2 sin y)i + 6(2xy - 2x cos y)j is conservative. If it is, find a potential function for it.

F(x, y, z) = 12x³ i + y² j + z³ k. Find curl F.

Evaluate $\int$ _C 8xy dx + 4(e ^x + x²)dy where C is the line segment from (0, 0) to (1, 1).

Let F(x, y, z) = 8x² i + 6y j + 3z k . Find the outward flux of the vector field F across the unit cube in the first octant and including the origin as a vertex.

Use Green's Theorem to evaluate where C is .

Find the flux of the vector field F(x, y, z) = 3z k across the sphere x² + y² + z² = 9 oriented outward.

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

Use Green's Theorem to evaluate , where C is the square bounded by x = y = 0, and x = y = 1.

Evaluate $Evaluate where F(x, y) = 5x<sup>2</sup>y i + 20j and C is the curve for 0 \le t \le 1.$ where F(x, y) = 5x²y i + 20j and C is the curve $Evaluate where F(x, y) = 5x<sup>2</sup>y i + 20j and C is the curve for 0 \le t \le 1.$ for 0 $\le$ t $\le$ 1.

Find the outward flux of the vector field across the sphere .

For the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(1, 2) is

Use Green's Theorem to evaluate $\int$ _C -2x²y dx + 2xy²dy) where C is the boundary of the circle x² + y² = 16 traversed in a counterclockwise manner.

Use Stokes' Theorem to evaluate $\int$ _C 3(z - y)dx + 3(x - z)dy + 3(y - x)dz where C is the boundary, in the xy-plane, of the surface $\sigma$ given by z = 4 - (x² + y²), z $\ge$ 0.

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

Let . Find divF.

Use Green's Theorem to evaluate , where C is 4x2 + 9y2 = 36.

Use Green's Theorem to evaluate , where C is the square bounded by x = y = 0, and x = y = 1.

Use Stokes' Theorem to evaluate ∫\int∫ C 8z dx - 4x dy + 4x dz where C is the intersection of the cylinder x2 + y2 = 1 and the plane z = y + 1.

Use Stokes' Theorem to evaluate over the circle x2 + y2 = 16, z = 4.

The work done by along the curve from (0, 0) to (1, 1) is

Use Green's Theorem to evaluate , where C is x2 + y2 = 16.

Determine whether F(x, y) = 6(y2 - 2 sin y)i + 6(2xy - 2x cos y)j is conservative. If it is, find a potential function for it.

F(x, y, z) = 12x3 i + y2 j + z3 k. Find curl F.

Evaluate ∫\int∫ C 8xy dx + 4(e x + x2)dy where C is the line segment from (0, 0) to (1, 1).

Let F(x, y, z) = 8x2 i + 6y j + 3z k . Find the outward flux of the vector field F across the unit cube in the first octant and including the origin as a vertex.

Use Green's Theorem to evaluate where C is .

Find the flux of the vector field F(x, y, z) = 3z k across the sphere x2 + y2 + z2 = 9 oriented outward.

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

Use Green's Theorem to evaluate , where C is the square bounded by x = y = 0, and x = y = 1.

Evaluate where F(x, y) = 5x2y i + 20j and C is the curve for 0 ≤\le≤ t ≤\le≤ 1.

Find the outward flux of the vector field across the sphere .

For the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(1, 2) is

Use Green's Theorem to evaluate ∫\int∫ C -2x2y dx + 2xy2dy) where C is the boundary of the circle x2 + y2 = 16 traversed in a counterclockwise manner.

Use Stokes' Theorem to evaluate ∫\int∫ C 3(z - y)dx + 3(x - z)dy + 3(y - x)dz where C is the boundary, in the xy-plane, of the surface σ\sigmaσ given by z = 4 - (x2 + y2), z ≥\ge≥ 0.

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 Green's Theorem to evaluate , where C is 4x² + 9y² = 36.

Use Stokes' Theorem to evaluate $\int$ _C 8z dx - 4x dy + 4x dz where C is the intersection of the cylinder x² + y² = 1 and the plane z = y + 1.

Use Stokes' Theorem to evaluate over the circle x² + y² = 16, z = 4.

Use Green's Theorem to evaluate , where C is x² + y² = 16.

Determine whether F(x, y) = 6(y² - 2 sin y)i + 6(2xy - 2x cos y)j is conservative. If it is, find a potential function for it.

F(x, y, z) = 12x³ i + y² j + z³ k. Find curl F.

Evaluate $\int$ _C 8xy dx + 4(e ^x + x²)dy where C is the line segment from (0, 0) to (1, 1).

Let F(x, y, z) = 8x² i + 6y j + 3z k . Find the outward flux of the vector field F across the unit cube in the first octant and including the origin as a vertex.

Find the flux of the vector field F(x, y, z) = 3z k across the sphere x² + y² + z² = 9 oriented outward.

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

Evaluate $Evaluate where F(x, y) = 5x<sup>2</sup>y i + 20j and C is the curve for 0 \le t \le 1.$ where F(x, y) = 5x²y i + 20j and C is the curve $Evaluate where F(x, y) = 5x<sup>2</sup>y i + 20j and C is the curve for 0 \le t \le 1.$ for 0 $\le$ t $\le$ 1.

Use Green's Theorem to evaluate $\int$ _C -2x²y dx + 2xy²dy) where C is the boundary of the circle x² + y² = 16 traversed in a counterclockwise manner.

Use Stokes' Theorem to evaluate $\int$ _C 3(z - y)dx + 3(x - z)dy + 3(y - x)dz where C is the boundary, in the xy-plane, of the surface $\sigma$ given by z = 4 - (x² + y²), z $\ge$ 0.