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Exam 13: Vector Calculus

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Exam 1: Functions and Limits54 Questionsadd course
Exam 2: Derivatives50 Questionsadd course
Exam 3: Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions43 Questionsadd course
Exam 4: Applications of Differentiation68 Questionsadd course
Exam 5: Integrals33 Questionsadd course
Exam 6: Techniques of Integration46 Questionsadd course
Exam 7: Applications of Integration69 Questionsadd course
Exam 8: Series51 Questionsadd course
Exam 9: Parametric Equations and Polar Coordinates30 Questionsadd course
Exam 10: Vectors and the Geometry of Space68 Questionsadd course
Exam 11: Partial Derivatives73 Questionsadd course
Exam 12: Multiple Integrals59 Questionsadd course
Exam 13: Vector Calculus54 Questionsadd course
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Determine whether or not vector field is conservative. If it is conservative, find a function f such that Determine whether or not vector field is conservative. If it is conservative, find a function f such that Determine whether or not vector field is conservative. If it is conservative, find a function f such that

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Question 21

Use Stokes' Theorem to evaluate Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above. where Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above. Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above. is the curve of intersection of the plane Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above. and the cylinder Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above. Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above. is oriented counterclockwise as viewed from above.

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Question 22

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the triangle with vertices , , and . , where C is the triangle with vertices Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the triangle with vertices , , and . , Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the triangle with vertices , , and . , and Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the triangle with vertices , , and . .

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Question 23

Find (a) the divergence and (b) the curl of the vector field F. Find (a) the divergence and (b) the curl of the vector field F.

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Question 24

Evaluate the line integral over the given curve C. Evaluate the line integral over the given curve C. ; , ; Evaluate the line integral over the given curve C. ; , , Evaluate the line integral over the given curve C. ; ,

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Question 25

Find the curl of the vector field F. Find the curl of the vector field F.

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Question 26

Find the area of the surface S where S is the part of the surface Find the area of the surface S where S is the part of the surface that lies inside the cylinder that lies inside the cylinder Find the area of the surface S where S is the part of the surface that lies inside the cylinder

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Question 27

A plane lamina with constant density A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C. Its moments of inertia about the axes are Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and . occupies a region in the xy-plane bounded by a simple closed path C. Its moments of inertia about the axes are A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C. Its moments of inertia about the axes are Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and . Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C. Its moments of inertia about the axes are Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and . .

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Question 28

Set up, but do not evaluate, a double integral for the area of the surface with parametric equations Set up, but do not evaluate, a double integral for the area of the surface with parametric equations

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Question 29

Evaluate the surface integral Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. for the given vector field F and the oriented surface S. In other words, find the flux of F across S. Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S.

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Question 30

Suppose that F is an inverse square force field, that is, Suppose that F is an inverse square force field, that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin. where Suppose that F is an inverse square force field, that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin. Find the work done by F in moving an object from a point Suppose that F is an inverse square force field, that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin. along a path to a point Suppose that F is an inverse square force field, that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin. in terms of the distances Suppose that F is an inverse square force field, that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin. and Suppose that F is an inverse square force field, that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin. from these points to the origin.

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Question 31

Find the area of the surface S where S is the part of the plane Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and that lies above the triangular region with vertices Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and , and Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and

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Question 32

Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid where A is the area of D.Find the centroid of the triangle with vertices (0, 0), ( , 0) and (0, ). where A is the area of D.Find the centroid of the triangle with vertices (0, 0), ( Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid where A is the area of D.Find the centroid of the triangle with vertices (0, 0), ( , 0) and (0, ). , 0) and (0, Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid where A is the area of D.Find the centroid of the triangle with vertices (0, 0), ( , 0) and (0, ). ).

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Question 33

Find a function f such that Find a function f such that , and use it to evaluate along the given curve C. , and use it to evaluate Find a function f such that , and use it to evaluate along the given curve C. along the given curve C. Find a function f such that , and use it to evaluate along the given curve C. Find a function f such that , and use it to evaluate along the given curve C.

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Question 34

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find , where a is the constant vector. , where a is the constant vector.

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Question 35

Determine whether F is conservative. If so, find a function f such that Determine whether F is conservative. If so, find a function f such that . . Determine whether F is conservative. If so, find a function f such that .

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Question 36

Find the curl of the vector field. Find the curl of the vector field.

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Question 37

A thin wire is bent into the shape of a semicircle A thin wire is bent into the shape of a semicircle If the linear density is , find the exact mass of the wire. If the linear density is A thin wire is bent into the shape of a semicircle If the linear density is , find the exact mass of the wire. , find the exact mass of the wire.

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Question 38

Let Let Let

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Question 39

Use Green's Theorem to find the work done by the force Use Green's Theorem to find the work done by the force in moving a particle from the origin along the x-axis to (1, 0) then along the line segment to (0, 1) and then back to the origin along the y-axis. in moving a particle from the origin along the x-axis to (1, 0) then along the line segment to (0, 1) and then back to the origin along the y-axis.

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Question 40
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