F(x, y, z) = (9xy - 9) i + 6xyz j + (z2 - 9)k. Find div F.

A) 9y + 6xz + 2z B) 9xy + 6xyz + z2 C) 0 D) 2z E) 9xy + z2 A) 9y + 6xz + 2z B) 9xy + 6xyz + z2 C) 0 D) 2z E) 9xy + z2

Exam 15: Topics in Vector Calculus

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

(Essay)

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Question 121

F(x, y, z) = xyz i + 11y j + x k. Find divF.

(Multiple Choice)

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Question 122

Use a line integral to find the area of the region enclosed by y = 2 - 2x⁴ and y = 0.

(Short Answer)

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Question 123

Use Stokes' Theorem to evaluate $Use Stokes' Theorem to evaluate where F(x, y, z) = 3y i and \sigma is that portion of the ellipsoid 4x2 + 4y2 + z2 = 4 for which z \ge 0.$ where F(x, y, z) = 3y i and $\sigma$ is that portion of the ellipsoid 4x² + 4y² + z² = 4 for which z $\ge$ 0.

(Essay)

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Question 124

Use Stokes' Theorem to evaluate $\int$ _C 3sin z dx - 3cos x dy + 3sin y dz over the rectangle 0 $\le$ x $\le$ $\pi$ , 0 $\le$ y $\le$ 1, and z = 2 traversed in a counterclockwise manner.

(Short Answer)

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Question 125

Determine whether is conservative. If it is, find a potential function for it. ( K is an arbitrary constant.)

(Multiple Choice)

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Question 126

F(x, y, z) = 4xyz i + 4xyz j + 4xyz k. Find the outward flux of the vector field F across the sphere x² + y² + z² = 25.

(Short Answer)

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Question 127

Find where F(x, y) = 3(2x + 3y)i + 3(3x - 2y)j and C is the curve r(t) = sin t i + cos t sin²t j; .

(Short Answer)

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Question 128

Evaluate the line integral $Evaluate the line integral , where C is the curve x = t, , 0 \le t \le 2.$ , where C is the curve x = t, $Evaluate the line integral , where C is the curve x = t, , 0 \le t \le 2.$ , 0 $\le$ t $\le$ 2.

(Essay)

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Question 129

Determine whether F(x, y) = 5(2x + y³)i + 5(3xy² - e^-²^y )j is conservative. If it is, find a potential function for it.

(Essay)

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Question 130

If F(x, y, z) = 3y j + 3z k, the magnitude of the flux through the portion of the surface $\sigma$ that lies right of the yz-plane, where $\sigma$ is defined by x = 1 - y² - z², is

(Multiple Choice)

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Question 131

F(x, y, z) = (9xy - 9) i + 6xyz j + (z² - 9)k. Find div F.

(Multiple Choice)

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Question 132

Evaluate where C is the boundary of the region in the first quadrant, enclosed by the circle and the coordinate axes.

(Multiple Choice)

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Question 133

Use the divergence theorem to evaluate $Use the divergence theorem to evaluate where F(x, y, z) = 2e x i - 2ye x j + 6z k, n is the outer unit normal to \sigma , and \sigma is the surface of the sphere by x2 + y2 + z2 = 36.$ where F(x, y, z) = 2e ^x i - 2ye ^x j + 6z k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere by x² + y² + z² = 36.

(Multiple Choice)

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Question 134

Determine whether the flow field F(x, y, z) = 10x² i + 10y² j + 10x² k is free of all sources and sinks. If it is not, find the location of all sources and sinks.

(Short Answer)

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Question 135

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

(Multiple Choice)

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Question 136

Use a line integral to find the area of the region enclosed by y = 5sin x, y = 5cos x, and x = 0.

(Essay)

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Question 137

Use Stokes' Theorem to evaluate $\int$ _C 28z dx - 14x dy + 14x dz where C is the intersection of the cylinder x² + y² = 1 and the plane z = y + 1.

(Multiple Choice)

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Question 138

F(x, y, z) = 3e ^x i + e ^y j + e ^z k. Find div F.

(Essay)

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Question 139

Let Find the outward flux of the vector field F across the surface of the region bounded above by the sphere by x² + y² + z² = 4 and below by the plane z = 0.

(Multiple Choice)

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Question 140

Showing 121 - 140 of 149

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

F(x, y, z) = xyz i + 11y j + x k. Find divF.

Use a line integral to find the area of the region enclosed by y = 2 - 2x⁴ and y = 0.

Use Stokes' Theorem to evaluate $\int$ _C 3sin z dx - 3cos x dy + 3sin y dz over the rectangle 0 $\le$ x $\le$ $\pi$ , 0 $\le$ y $\le$ 1, and z = 2 traversed in a counterclockwise manner.

Determine whether is conservative. If it is, find a potential function for it. ( K is an arbitrary constant.)

F(x, y, z) = 4xyz i + 4xyz j + 4xyz k. Find the outward flux of the vector field F across the sphere x² + y² + z² = 25.

Find where F(x, y) = 3(2x + 3y)i + 3(3x - 2y)j and C is the curve r(t) = sin t i + cos t sin²t j; .

Evaluate the line integral $Evaluate the line integral , where C is the curve x = t, , 0 \le t \le 2.$ , where C is the curve x = t, $Evaluate the line integral , where C is the curve x = t, , 0 \le t \le 2.$ , 0 $\le$ t $\le$ 2.

Determine whether F(x, y) = 5(2x + y³)i + 5(3xy² - e^-²^y )j is conservative. If it is, find a potential function for it.

If F(x, y, z) = 3y j + 3z k, the magnitude of the flux through the portion of the surface $\sigma$ that lies right of the yz-plane, where $\sigma$ is defined by x = 1 - y² - z², is

F(x, y, z) = (9xy - 9) i + 6xyz j + (z² - 9)k. Find div F.

Evaluate where C is the boundary of the region in the first quadrant, enclosed by the circle and the coordinate axes.

Determine whether the flow field F(x, y, z) = 10x² i + 10y² j + 10x² k is free of all sources and sinks. If it is not, find the location of all sources and sinks.

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

Use a line integral to find the area of the region enclosed by y = 5sin x, y = 5cos x, and x = 0.

Use Stokes' Theorem to evaluate $\int$ _C 28z dx - 14x dy + 14x dz where C is the intersection of the cylinder x² + y² = 1 and the plane z = y + 1.

F(x, y, z) = 3e ^x i + e ^y j + e ^z k. Find div F.

Let Find the outward flux of the vector field F across the surface of the region bounded above by the sphere by x² + y² + z² = 4 and below by the plane z = 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

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Exam 15: Topics in Vector Calculus

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

F(x, y, z) = xyz i + 11y j + x k. Find divF.

Use a line integral to find the area of the region enclosed by y = 2 - 2x4 and y = 0.

Use Stokes' Theorem to evaluate where F(x, y, z) = 3y i and σ\sigmaσ is that portion of the ellipsoid 4x2 + 4y2 + z2 = 4 for which z ≥\ge≥ 0.

Use Stokes' Theorem to evaluate ∫\int∫ C 3sin z dx - 3cos x dy + 3sin y dz over the rectangle 0 ≤\le≤ x ≤\le≤ π\piπ , 0 ≤\le≤ y ≤\le≤ 1, and z = 2 traversed in a counterclockwise manner.

Determine whether is conservative. If it is, find a potential function for it. ( K is an arbitrary constant.)

F(x, y, z) = 4xyz i + 4xyz j + 4xyz k. Find the outward flux of the vector field F across the sphere x2 + y2 + z2 = 25.

Find where F(x, y) = 3(2x + 3y)i + 3(3x - 2y)j and C is the curve r(t) = sin t i + cos t sin2t j; .

Evaluate the line integral , where C is the curve x = t, , 0 ≤\le≤ t ≤\le≤ 2.

Determine whether F(x, y) = 5(2x + y3)i + 5(3xy2 - e-2y )j is conservative. If it is, find a potential function for it.

If F(x, y, z) = 3y j + 3z k, the magnitude of the flux through the portion of the surface σ\sigmaσ that lies right of the yz-plane, where σ\sigmaσ is defined by x = 1 - y2 - z2, is

F(x, y, z) = (9xy - 9) i + 6xyz j + (z2 - 9)k. Find div F.

Evaluate where C is the boundary of the region in the first quadrant, enclosed by the circle and the coordinate axes.

Use the divergence theorem to evaluate where F(x, y, z) = 2e x i - 2ye x j + 6z k, n is the outer unit normal to σ\sigmaσ , and σ\sigmaσ is the surface of the sphere by x2 + y2 + z2 = 36.

Determine whether the flow field F(x, y, z) = 10x2 i + 10y2 j + 10x2 k is free of all sources and sinks. If it is not, find the location of all sources and sinks.

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

Use a line integral to find the area of the region enclosed by y = 5sin x, y = 5cos x, and x = 0.

Use Stokes' Theorem to evaluate ∫\int∫ C 28z dx - 14x dy + 14x dz where C is the intersection of the cylinder x2 + y2 = 1 and the plane z = y + 1.

F(x, y, z) = 3e x i + e y j + e z k. Find div F.

Let Find the outward flux of the vector field F across the surface of the region bounded above by the sphere by x2 + y2 + z2 = 4 and below by the plane z = 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

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

Use a line integral to find the area of the region enclosed by y = 2 - 2x⁴ and y = 0.

Use Stokes' Theorem to evaluate $\int$ _C 3sin z dx - 3cos x dy + 3sin y dz over the rectangle 0 $\le$ x $\le$ $\pi$ , 0 $\le$ y $\le$ 1, and z = 2 traversed in a counterclockwise manner.

F(x, y, z) = 4xyz i + 4xyz j + 4xyz k. Find the outward flux of the vector field F across the sphere x² + y² + z² = 25.

Find where F(x, y) = 3(2x + 3y)i + 3(3x - 2y)j and C is the curve r(t) = sin t i + cos t sin²t j; .

Evaluate the line integral $Evaluate the line integral , where C is the curve x = t, , 0 \le t \le 2.$ , where C is the curve x = t, $Evaluate the line integral , where C is the curve x = t, , 0 \le t \le 2.$ , 0 $\le$ t $\le$ 2.

Determine whether F(x, y) = 5(2x + y³)i + 5(3xy² - e^-²^y )j is conservative. If it is, find a potential function for it.

If F(x, y, z) = 3y j + 3z k, the magnitude of the flux through the portion of the surface $\sigma$ that lies right of the yz-plane, where $\sigma$ is defined by x = 1 - y² - z², is

F(x, y, z) = (9xy - 9) i + 6xyz j + (z² - 9)k. Find div F.

Determine whether the flow field F(x, y, z) = 10x² i + 10y² j + 10x² k is free of all sources and sinks. If it is not, find the location of all sources and sinks.

Use Stokes' Theorem to evaluate $\int$ _C 28z dx - 14x dy + 14x dz where C is the intersection of the cylinder x² + y² = 1 and the plane z = y + 1.

F(x, y, z) = 3e ^x i + e ^y j + e ^z k. Find div F.

Let Find the outward flux of the vector field F across the surface of the region bounded above by the sphere by x² + y² + z² = 4 and below by the plane z = 0.