Evaluate where F(x, y, z) = 18 i - 2z j + 2y k and$\sigma$is that portion of the paraboloid x = 4 - y2 - z2 to the right of x = 0 oriented by forward unit normals.

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

Verify Stokes' Theorem if $\sigma$ is the portion of the sphere x2 + y2 + z2 = 1 for which z$\ge$0 and F(x, y, z) = (2x - y)i - yz2 j - y2z k.

If $\sigma$ is oriented by upward normals, then C is the intersection of the sphere x 2 + y 2 + z 2 = 1 with z = 0, thus, C is the circle x 2 + y 2 = 1 which can be parameterized as r(t) = cos t i + sin t j for 0 $\le$ t $\le$ 2 $\pi$, so, ;Curl F = k, , and R is the circular region in the xy-plane enclosed by C, so area of the circle of radius 1 = $\pi$ (1) 2 = $\pi$.

Find the flux of the vector field F(x, y, z) = 5z k across the sphere 5x2 + 5y2 + 5z2 = 20 oriented outward.

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

Evaluate $\int$C 2x2y dx + 2xy dy, where C is the triangle with vertices (0, 0), (1, 0), and (1, 3).

A) 2.333333 B) 3.25 C) 6.5 D) 0 E) 1.5 A) 2.333333 B) 3.25 C) 6.5 D) 0 E) 1.5

Evaluate where F(x, y, z) = 4y i + 8x j + 4xy k and $\sigma$ is that portion of the cylinder x2 + y2 = 9 in the first octant between z = 1 and z = 4. The surface is oriented by right unit normals.

A) 0 B) 4 C) 162 D) 8 E) -32 A) 0 B) 4 C) 162 D) 8 E) -32

Evaluate where F(x, y, z) = 12x i + 24j + 24x2 k and $\sigma$ is that portion of the paraboloid z = x2 + y2 which lies above the xy-plane enclosed by the parabolas y = 1 - x2 and y = x2 - 1. The surface is oriented by downward unit normals.

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

F(x, y, z) = 4xy i + 4xyz j + (z2 - 4zx)k. Find div F.

A) 0 B) 4xy + 4xyz + 4x C) 4y + 4xz + 2z - 4x D) 2z E) 4xy + z2 A) 0 B) 4xy + 4xyz + 4x C) 4y + 4xz + 2z - 4x D) 2z E) 4xy + z2

Exam 15: Topics in Vector Calculus

Evaluate $Evaluate where F(x, y, z) = 18 i - 2z j + 2y k and \sigma is that portion of the paraboloid x = 4 - y2 - z2 to the right of x = 0 oriented by forward unit normals.$ where F(x, y, z) = 18 i - 2z j + 2y k and $\sigma$ is that portion of the paraboloid x = 4 - y² - z² to the right of x = 0 oriented by forward unit normals.

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Question 1

Correct Answer:

Verified

Verify Stokes' Theorem if $\sigma$ is the portion of the sphere x² + y² + z² = 1 for which z $\ge$ 0 and F(x, y, z) = (2x - y)i - yz² j - y²z k.

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Question 2

Correct Answer:

Verified

If $\sigma$ is oriented by upward normals, then C is the intersection of the sphere x² + y² + z² = 1 with z = 0, thus, C is the circle x² + y² = 1 which can be parameterized as r(t) = cos t i + sin t j for 0 $\le$ t $\le$ 2 $\pi$ , so, $If \sigma is oriented by upward normals, then C is the intersection of the sphere x2 + y2 + z2 = 1 with z = 0, thus, C is the circle x2 + y2 = 1 which can be parameterized as r(t) = cos t i + sin t j for 0 \le t \le 2 \pi , so, ;Curl F = k, , and R is the circular region in the xy-plane enclosed by C, so area of the circle of radius 1 = \pi (1)2 = \pi .$ ;Curl F = k, $If \sigma is oriented by upward normals, then C is the intersection of the sphere x2 + y2 + z2 = 1 with z = 0, thus, C is the circle x2 + y2 = 1 which can be parameterized as r(t) = cos t i + sin t j for 0 \le t \le 2 \pi , so, ;Curl F = k, , and R is the circular region in the xy-plane enclosed by C, so area of the circle of radius 1 = \pi (1)2 = \pi .$ , and R is the circular region in the xy-plane enclosed by C, so $If \sigma is oriented by upward normals, then C is the intersection of the sphere x2 + y2 + z2 = 1 with z = 0, thus, C is the circle x2 + y2 = 1 which can be parameterized as r(t) = cos t i + sin t j for 0 \le t \le 2 \pi , so, ;Curl F = k, , and R is the circular region in the xy-plane enclosed by C, so area of the circle of radius 1 = \pi (1)2 = \pi .$ area of the circle of radius 1 = $\pi$ (1)² = $\pi$ .

Evaluate the surface integral $Evaluate the surface integral where \sigma is that portion of the plane x + y + z = 1 that lies in the first octant.$ where $\sigma$ is that portion of the plane x + y + z = 1 that lies in the first octant.

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Question 3

Correct Answer:

Verified

Find the flux of the vector field F(x, y, z) = 5z k across the sphere 5x² + 5y² + 5z² = 20 oriented outward.

(Multiple Choice)

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Question 4

Evaluate $\int$ _C 2x²y dx + 2xy dy, where C is the triangle with vertices (0, 0), (1, 0), and (1, 3).

(Multiple Choice)

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Question 5

Let . Is the field conservative?

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Question 6

The work done by the force F(x, y) = 6x² i + 12y² j acting on a particle that moves along the circle x = cos t, y = sin t, from (1, 0) to (0, 1).

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Question 7

For F(x, y) = 2i + 2j the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(8, 5) is

(Multiple Choice)

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Question 8

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

(Short Answer)

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Question 9

Evaluate . Note: This integral is independent of path.

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Question 10

Use Green's Theorem to evaluate $\int$ _C 10(2xy - y²)dx + 10x²dy where C is the boundary of the region enclosed by y = x + 1 and y = x² + 1, traversed in a counterclockwise manner.

(Short Answer)

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Question 11

Let . Find curlF.

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Question 12

Use Green's Theorem to evaluate $\int$ _C 4(y - sin x)dx + 4cos x dy where C is the boundary of the region with vertices (0, 0), $Use Green's Theorem to evaluate \int C 4(y - sin x)dx + 4cos x dy where C is the boundary of the region with vertices (0, 0), , and traversed in a counterclockwise manner.$ , and $Use Green's Theorem to evaluate \int C 4(y - sin x)dx + 4cos x dy where C is the boundary of the region with vertices (0, 0), , and traversed in a counterclockwise manner.$ traversed in a counterclockwise manner.

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Question 13

Evaluate $Evaluate where F(x, y, z) = 4y i + 8x j + 4xy k and \sigma is that portion of the cylinder x2 + y2 = 9 in the first octant between z = 1 and z = 4. The surface is oriented by right unit normals.$ where F(x, y, z) = 4y i + 8x j + 4xy k and $\sigma$ is that portion of the cylinder x² + y² = 9 in the first octant between z = 1 and z = 4. The surface is oriented by right unit normals.

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Question 14

Evaluate $Evaluate where F(x, y, z) = 12x i + 24j + 24x2 k and \sigma is that portion of the paraboloid z = x2 + y2 which lies above the xy-plane enclosed by the parabolas y = 1 - x2 and y = x2 - 1. The surface is oriented by downward unit normals.$ where F(x, y, z) = 12x i + 24j + 24x² k and $\sigma$ is that portion of the paraboloid z = x² + y² which lies above the xy-plane enclosed by the parabolas y = 1 - x² and y = x² - 1. The surface is oriented by downward unit normals.

(Short Answer)

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Question 15

Is F(x, y) = 12cos y i + 12cos x j is a conservative vector field?

(Short Answer)

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Question 16

F (x, y, z) = 5xyz i + 5xyz j + 5xyz 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.

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Question 17

F(x, y, z) = 4xy i + 4xyz j + (z² - 4zx)k. Find div F.

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Question 18

Evaluate the line integral. along y = ln x from the point (1, 0) to the point (3, ln 3).

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Question 19

Determine whether F(x, y) = 4(3 cos y + 2 sin x)i + 4(3y² - 3x sin y)j is conservative. If it is, find a potential function for it.

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Question 20

Showing 1 - 20 of 149

Verify Stokes' Theorem if $\sigma$ is the portion of the sphere x² + y² + z² = 1 for which z $\ge$ 0 and F(x, y, z) = (2x - y)i - yz² j - y²z k.

Evaluate the surface integral $Evaluate the surface integral where \sigma is that portion of the plane x + y + z = 1 that lies in the first octant.$ where $\sigma$ is that portion of the plane x + y + z = 1 that lies in the first octant.

Find the flux of the vector field F(x, y, z) = 5z k across the sphere 5x² + 5y² + 5z² = 20 oriented outward.

Evaluate $\int$ _C 2x²y dx + 2xy dy, where C is the triangle with vertices (0, 0), (1, 0), and (1, 3).

Let . Is the field conservative?

The work done by the force F(x, y) = 6x² i + 12y² j acting on a particle that moves along the circle x = cos t, y = sin t, from (1, 0) to (0, 1).

For F(x, y) = 2i + 2j the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(8, 5) is

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

Evaluate . Note: This integral is independent of path.

Use Green's Theorem to evaluate $\int$ _C 10(2xy - y²)dx + 10x²dy where C is the boundary of the region enclosed by y = x + 1 and y = x² + 1, traversed in a counterclockwise manner.

Let . Find curlF.

Is F(x, y) = 12cos y i + 12cos x j is a conservative vector field?

F (x, y, z) = 5xyz i + 5xyz j + 5xyz 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.

F(x, y, z) = 4xy i + 4xyz j + (z² - 4zx)k. Find div F.

Evaluate the line integral. along y = ln x from the point (1, 0) to the point (3, ln 3).

Determine whether F(x, y) = 4(3 cos y + 2 sin x)i + 4(3y² - 3x sin y)j is conservative. If it is, find a potential function for it.

Limits and Continuity

The Derivative

Topics in Deifferentiation

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Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

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Exam 15: Topics in Vector Calculus

Evaluate where F(x, y, z) = 18 i - 2z j + 2y k and σ\sigmaσ is that portion of the paraboloid x = 4 - y2 - z2 to the right of x = 0 oriented by forward unit normals.

Verify Stokes' Theorem if σ\sigmaσ is the portion of the sphere x2 + y2 + z2 = 1 for which z ≥\ge≥ 0 and F(x, y, z) = (2x - y)i - yz2 j - y2z k.

Evaluate the surface integral where σ\sigmaσ is that portion of the plane x + y + z = 1 that lies in the first octant.

Find the flux of the vector field F(x, y, z) = 5z k across the sphere 5x2 + 5y2 + 5z2 = 20 oriented outward.

Evaluate ∫\int∫ C 2x2y dx + 2xy dy, where C is the triangle with vertices (0, 0), (1, 0), and (1, 3).

Let . Is the field conservative?

The work done by the force F(x, y) = 6x2 i + 12y2 j acting on a particle that moves along the circle x = cos t, y = sin t, from (1, 0) to (0, 1).

For F(x, y) = 2i + 2j the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(8, 5) is

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

Evaluate . Note: This integral is independent of path.

Use Green's Theorem to evaluate ∫\int∫ C 10(2xy - y2)dx + 10x2dy where C is the boundary of the region enclosed by y = x + 1 and y = x2 + 1, traversed in a counterclockwise manner.

Let . Find curlF.

Use Green's Theorem to evaluate ∫\int∫ C 4(y - sin x)dx + 4cos x dy where C is the boundary of the region with vertices (0, 0), , and traversed in a counterclockwise manner.

Evaluate where F(x, y, z) = 4y i + 8x j + 4xy k and σ\sigmaσ is that portion of the cylinder x2 + y2 = 9 in the first octant between z = 1 and z = 4. The surface is oriented by right unit normals.

Evaluate where F(x, y, z) = 12x i + 24j + 24x2 k and σ\sigmaσ is that portion of the paraboloid z = x2 + y2 which lies above the xy-plane enclosed by the parabolas y = 1 - x2 and y = x2 - 1. The surface is oriented by downward unit normals.

Is F(x, y) = 12cos y i + 12cos x j is a conservative vector field?

F (x, y, z) = 5xyz i + 5xyz j + 5xyz 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.

F(x, y, z) = 4xy i + 4xyz j + (z2 - 4zx)k. Find div F.

Evaluate the line integral. along y = ln x from the point (1, 0) to the point (3, ln 3).

Determine whether F(x, y) = 4(3 cos y + 2 sin x)i + 4(3y2 - 3x sin y)j is conservative. If it is, find a potential function for it.

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

Verify Stokes' Theorem if $\sigma$ is the portion of the sphere x² + y² + z² = 1 for which z $\ge$ 0 and F(x, y, z) = (2x - y)i - yz² j - y²z k.

Evaluate the surface integral $Evaluate the surface integral where \sigma is that portion of the plane x + y + z = 1 that lies in the first octant.$ where $\sigma$ is that portion of the plane x + y + z = 1 that lies in the first octant.

Find the flux of the vector field F(x, y, z) = 5z k across the sphere 5x² + 5y² + 5z² = 20 oriented outward.

Evaluate $\int$ _C 2x²y dx + 2xy dy, where C is the triangle with vertices (0, 0), (1, 0), and (1, 3).

The work done by the force F(x, y) = 6x² i + 12y² j acting on a particle that moves along the circle x = cos t, y = sin t, from (1, 0) to (0, 1).

Use Green's Theorem to evaluate $\int$ _C 10(2xy - y²)dx + 10x²dy where C is the boundary of the region enclosed by y = x + 1 and y = x² + 1, traversed in a counterclockwise manner.

F(x, y, z) = 4xy i + 4xyz j + (z² - 4zx)k. Find div F.

Determine whether F(x, y) = 4(3 cos y + 2 sin x)i + 4(3y² - 3x sin y)j is conservative. If it is, find a potential function for it.