Deck 16: Vector Calculus

Question 1

Use Stoke's theorem to evaluate C is the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.

Answer 1

Question 2

Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.

Answer 2

Question 3

Use Stoke's theorem to evaluate where and C is the boundary of the part of the plane in the first octant.

A)16
B)0
C)49
D)69
E)23

Answer 3

0

Question 4

Suppose that where g is a function of one variable such that . Evaluate where S is the sphere

A)
B)
C)
D)
E)None of these

Question 5

Use Stokes' Theorem to evaluate . ;
C is the boundary of the triangle with vertices , , and oriented in a counterclockwise direction when viewed from above

Question 6

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.

A)5
B)3
C)8
D)7
E)6

Question 7

Use a computer algebra system to compute the flux of F across S. S is the surface of the cube cut from the first octant by the planes

A)3
B)4
C)1
D)0.67
E)

Question 8

Evaluate the surface integral. Round your answer to four decimal places. S is surface

A)
B)
C)
D)
E)

Question 9

Use Stoke's theorem to calculate the surface integral where and S is the part of the cone

Question 10

Use Stokes' Theorem to evaluate . ;
S is the part of the ellipsoid lying above the xy-plane and oriented with normal pointing upward.

Question 11

The temperature at the point in a substance with conductivity is Find the rate of heat flow inward across the cylindrical

A)
B)
C)
D)
E)

Question 12

Use Stokes' Theorem to evaluate . ; C is the curve obtained by intersecting the cylinder with the hyperbolic paraboloid , oriented in a counterclockwise direction when viewed from above

A)
B)
C)0
D)

Question 13

Use Stoke's theorem to evaluate C is the boundary of the part of the paraboloid in the first octant. C is oriented counterclockwise as viewed from above.

Question 14

Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of across . S is the surface of the box bounded by the coordinate planes and the planes .

A)
B)
C)
D)
E)

Question 15

Evaluate . ; S is the part of the plane in the first octant.

A)
B)
C)0
D)

Question 16

Use Stokes' Theorem to evaluate S consists of the top and the four sides (but not the bottom) of the cube with vertices oriented outward.

Question 17

Use Stoke's theorem to evaluate C is the curve of intersection of the plane z = x + 9 and the cylinder

Question 18

Use Stokes' Theorem to evaluate . ; S is the part of the paraboloid lying below the plane and oriented with normal pointing downward.

A)
B)
C)0
D)

Question 19

Evaluate the surface integral. S is the part of the plane that lies in the first octant.

A)
B)
C)
D)
E)

Question 20

Use Stokes' Theorem to evaluate S consists of the four sides of the pyramid with vertices (0, 0, 0), (3, 0, 0), (0, 0, 3), (3, 0,3) and (0, 3, 0) that lie to the right of the xz-plane, oriented in the direction of the positive y-axis.

A)1
B)16
C)49
D)0
E)12

Question 21

Find the area of the surface. The part of the plane ; ,

A)
B)
C)
D)

Question 22

Find the area of the part of the cone that is cut off by the cylinder

A)
B)
C)
D)

Question 23

Find the mass of the surface S having the given mass density. S is the hemisphere , ; the density at a point P on S is equal to the distance between P and the xy-plane.

A)
B)
C)9
D)

Question 24

A fluid with density flows with velocity Find the rate of flow upward through the paraboloid

Question 25

Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone if its density function is

Question 26

Evaluate , that is, find the flux of F across S. ; S is the hemisphere ; n points upward.

A)162
B)162
C)
D)

Question 27

Evaluate . ; S is the part of the torus with vector representation , , .

A)
B)
C)0
D)

Question 28

Find a parametric representation for the part of the elliptic paraboloid that lies in front of the plane x = 0.

A)
B)
C)
D)
E)

Question 29

Evaluate the surface integral. S is the part of the cylinder between the planes and in the first octant.

Question 30

Use Gauss's Law to find the charge contained in the solid hemisphere , if the electric field is

Question 31

Match the equation with one of the graphs below.

A)
B)
C)
D)

Question 32

Evaluate . ; S is the part of the cone between the planes and .

A)
B)
C)
D)0

Question 33

Evaluate , that is, find the flux of F across S. ; S is the part of the paraboloid between the planes z = 0 and z = 5; n points upward.

Question 34

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. in the first octant,
with orientation toward the origin.

Question 35

Let S be the cube with vertices . Approximate by using a Riemann sum as in Definition 1, taking the patches to be the squares that are the faces of the cube and the points to be the centers of the squares.

A)
B)
C)
D)
E)none of these

Question 36

Find the area of the surface. The part of the paraboloid ; ,

A)
B)
C)
D)

Question 37

Find the mass of the surface S having the given mass density. S is part of the plane in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.

A)
B)49
C)
D)20

Question 38

Find the area of the surface. The part of the paraboloid ; ,

A)
B)
C)
D)

Question 39

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.

Question 40

Evaluate the surface integral where S is the surface with parametric equations , .

A)
B)
C)
D)
E)

Question 41

Find the area of the part of paraboloid that lies inside the cylinder

Question 42

Below is given the plot of a vector field F in the xy-plane. (The z-component of F is 0.) By studying the plot, determine whether div F is positive, negative, or zero.

A)cannot be determined
B)positive
C)negative
D)zero

Question 43

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

Question 44

Use the Divergence Theorem to find the flux of F across S; that is, calculate . ; S is the sphere

Question 45

Find the area of the surface S where S is the part of the sphere that lies to the right of the xz-plane and inside the cylinder

Question 46

Let

A)27
B)18
C)45
D)9
E)None of these

Question 47

Find an equation in rectangular coordinates, and then identify the surface.

Question 48

Find the correct identity, if f is a scalar field, F and G are vector fields.

A)
B)
C)
D)None of these

Question 49

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

Question 50

Find the area of the part of the surface that lies between the planes x = 0, x = 4, , and z = 1.

A)
B)
C)
D)
E)

Question 51

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ; u = ln 5, v = 0

Question 52

Find a parametric representation for the part of the sphere that lies above the cone

Question 53

Find the divergence of the vector field F.

A)
B)
C)
D)

Question 54

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

Question 55

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ; u = ln 9, v = 0

Question 56

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ;

Question 57

Find a vector representation for the surface.
The plane that passes through the point and contains the vectors and ..

Question 58

Find a parametric representation for the part of the plane that lies inside the cylinder

Question 59

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

Question 60

Find an equation in rectangular coordinates, and then identify the surface.

Question 61

Determine whether or not vector field is conservative. If it is conservative, find a function f such that

Question 62

Find the curl of the vector field F.

A)
B)
C)
D)

Question 63

Let

Question 64

Find the curl of the vector field.

Question 65

Find the curl of .

Question 66

Use Green's Theorem and/or a computer algebra system to evaluate where C is the circle with counterclockwise orientation.

A)
B)
C)
D)
E)None of these

Question 67

Find the divergence of the vector field.

Question 68

Find the div F if .

Question 69

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the boundary of the region bounded by the parabolas and .

A)
B) + e
C) + e
D)

Question 70

Let

Question 71

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

Question 72

Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.

Question 73

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.
curl (div F)

Question 74

Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.
curl f

Question 75

Determine whether or not vector field is conservative. If it is conservative, find a function f such that

Question 76

Find the curl of the vector field.

A)
B)
C)
D)
E)None of these

Question 77

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

A)
B)
C)
D)

Question 78

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.

Question 79

Find the curl of the vector field.

Question 80

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, ).

A)
B)
C)
D)
E)