Deck 18: Fundamental Theorems of Vector Analysis

Question 1

Calculate the circulation of the vector field around the circle , traversed in a counterclockwise direction.

Answer 1

Question 2

Use Green's Theorem to evaluate where is the closed curve shown in the following figure.

Answer 2

Question 3

Evaluate the line integral where is the circle oriented in the positive direction.

Answer 3

Question 4

Compute where c is the curve starting at the origin and ending at .

Question 5

Evaluate where c is the closed curve traversed in a counterclockwise direction.

Question 6

Let where c is the ellipse oriented in the positive direction. The value of is which of the following?

A)
B)
C) 0
D)
E) The integral cannot be evaluated analytically.

Question 7

Evaluate where c is the closed curve traversed in a counterclockwise direction.

Question 8

Compute where C is the polar curve in the positive direction and is the vector field .

Question 9

Evaluate where c is the circle oriented counterclockwise and .

Question 10

Compute where C is the curve consisting of the line segment : on the x-axis together with the curve : in the positive direction.

Question 11

Use Green's Theorem to calculate the counterclockwise circulation of the vector field around the boundary of the region that is bounded above by the curve and below by the curve

Question 12

Use Green's Theorem to evaluate the line integral along the contour of the triangle ABD with vertices

Question 13

Compute the area of the shaded region whose boundary consists of the line segment on the x axis and the curve , .

Question 14

Compute where c is the curve shown in the figure.

Question 15

Compute where C is the path shown in the following figure and

Question 16

Compute the area of the region bounded by the astroid

Question 17

Use Green's Theorem to evaluate where c is the piecewise linear path starting at (0,- 2) and then traveling to (2,4), (- 2,2), and (0,- 2), in that order.

Question 18

Find the circulation of the field around the boundary of the region that is bounded above by the curve and below by .

Question 19

Calculate the circulation of the vector field around the circle oriented counterclockwise.

Question 20

Let where C is the ellipse oriented counterclockwise. The value of I is which of the following?

A) 0
B)
C)
D)
E)

Question 21

Let S be the part of the paraboloid which is above the plane oriented upwards, and let .

A) Explain why where is the disc of radius 3 in the xy-plane oriented upward.
B) Compute the surface integral

Question 22

Compute where is the curve of intersection of the sphere and the plane .

Question 23

Let and S be the part of the sphere between the planes and , oriented outward. The integral is equal to which of the following?

A)
B)
C)
D)
E)

Question 24

Use Stokes' Theorem to compute , where and S is the part of the surface satisfying S is oriented with outward-pointing normals.

Question 25

Compute where and is the part of the cylinder which is inside the sphere , oriented with outward-pointing normal.

Question 26

Use Stokes' Theorem to evaluate where and c is the boundary of the part of the plane over the region , oriented counterclockwise.

Question 27

Compute the area of the region bounded by the curve

Question 28

Evaluate where is the curve , oriented clockwise.

Question 29

Compute where and S is the upper half of the sphere of radius 3, that is, with upward-pointing normal.

Question 30

Evaluate where is the boundary of the unit square oriented clockwise.

Question 31

Compute where c is the curve of intersection between the sphere and the plane .
The integration on c is counterclockwise when viewing from the point .

Question 32

Evaluate where is the parallelogram with vertices and , oriented counterclockwise.

Question 33

Evaluate where is the circle , traversed in a counterclockwise direction.

Question 34

Evaluate where is the triangle with vertices and , traversed in a counterclockwise direction.

Question 35

Compute where and S is the upper half of the sphere of radius 2; that is, with upward-pointing normal.

Question 36

Evaluate where is the triangle with vertices and , traversed in a counterclockwise direction.

Question 37

Use Green's Theorem to evaluate the integral of along the quarter circle in the positive direction.

Question 38

Let where and , oriented upward.
I is equal to which of the following?

A)
B)
C)
D) 0
E) None of the above.

Question 39

Compute where and S is the surface defined by ,
oriented with outward pointing normal.

Question 40

Compute where c is the curve of intersection of the surfaces and , oriented counterclockwise.

Question 41

Use Stokes' Theorem to compute the line integral of counterclockwise (as viewed from above) around the triangle with vertices and .

Question 42

Compute where S is the ellipsoid with outward-pointing normal and .

Question 43

Compute where is the intersection line of the surfaces and

Question 44

Let , and let S be the surface , together with the two vertical sides.
Compute where

Question 45

Let and S be a closed and smooth surface enclosing a region V.
If V and its boundary do not include the origin, what is the value of ?

Question 46

Compute where and S is the surface with outward pointing normal.

Question 47

Compute where S is the portion of the surface of the sphere with radius and center that is above the plane oriented upward, and F is the vector field .

Question 48

Let S be the boundary of the region V defined by oriented with outward-pointing normal.
Let .
The surface integral is equal to which of the following?

A)
B)
C)
D)
E) None of the above

Question 49

Let S be the upper half of the hemisphere including the bottom .
S is oriented with outward-pointing normal, and F is the vector field Compute .

Question 50

Use Stokes' Theorem to find the line integral of the vector field around the curve which is the intersection of the plane with the cylinder , oriented counterclockwise as viewed from above.

A) 0
B)
C)
D)
E)

Question 51

Let where C is the circle of intersection between the sphere and the plane , and .
The value of is which of the following?

A)
B)
C) 0
D)
E)

Question 52

Use Stokes' Theorem to compute the line integral where and c is the path made up of the sequence of three line segments:
0 to A, A to B, and B to C. (See the figure.)

Question 53

Use Stokes' Theorem to compute where S is the part of the surface , oriented outward, and .

Question 54

Let and c be the closed curve defined by the parametric equations .
Compute the line integral in two ways (with c oriented in the positive direction):
a) direct computation
b) using Stokes' Theorem

Question 55

Let and B be a sphere of radius R centered at the origin.
Referring to the integral , which of the following statements is correct?

A) since
B) by the Divergence Theorem.
C) The Divergence Theorem cannot be applied and by direct computation.
D) The integral is not defined since is not defined at the origin.
E) and violate the Divergence Theorem.

Question 56

Use Stokes' Theorem to evaluate the line integral where C is the boundary of the portion of the paraboloid and .

Question 57

Let S be a closed and smooth surface with outward-pointing normal which is the boundary of a solid V in Let be a vector field whose components have continuous partial derivatives.

A) Compute .
B) What is ? Explain.

Question 58

Compute where and S is the closed boundary of the cylinder , with outward-pointing normal.

Question 59

Use Stokes' Theorem to compute the flux of through the surface S which is the part of the paraboloid below the plane , oriented upward.
The vector field is given by .

Question 60

Compute where is the intersection line of the surfaces and

Question 61

Use the Divergence Theorem to compute the surface integral where and Assume is oriented so that the normal vector points away from the z-axis.

Question 62

Let where and . Compute for .

Question 63

Compute the surface integral where S is the half sphere , , oriented with outward pointing normal, and

Question 64

Evaluate where is the ellipsoid oriented outward, and

Question 65

Let where , is oriented with normal pointing to the origin, and is oriented in the opposite direction.
Let be the vector field . Compute

Question 66

Evaluate where and S is the sphere oriented with outward-pointing normal.

Question 67

Let , and w be a region in whose boundary is a closed piecewise smooth surface S. The integral is equal to which of the following?

A)
B)
C)
D)
E)

Question 68

Use the Divergence Theorem to calculate the surface integral where S is the sphere , oriented with outward-pointing normal and F is the vector field .

Question 69

Use the Divergence Theorem to compute the surface integral , where S is the surface , oriented outward, and F is the vector field .

Question 70

Compute where and , oriented outward.

Question 71

Compute where and S is the boundary of the pyramid determined by the planes .

Question 72

Evaluate where is the boundary of the region enclosed by the surfaces and and is oriented with outward-pointing normal.

Question 73

Evaluate where is the ellipsoid oriented outward, and

Question 74

Let . Write the condition for f so that for all , .

Question 75

Let be the surface area of the box in the figure with dimensions , and let W be the interior of the box. Let F be the vector field Which of the following integrals are equal?

A)
B)
C)
D)
E)

Question 76

Let . Compute the surface integral where S is the surface oriented so that the normal points toward increasing y.

Question 77

Compute where and is the part of the paraboloid above the plane , oriented outward.

Question 78

Let be the vector field , w be a region in containing the origin in its interior , and S be the boundary of w that is a closed surface with outward-pointing normal.
Compute

Question 79

Evaluate where is the boundary of the region enclosed by the surfaces and is oriented with inward-pointing normal.

Question 80

Use the Divergence Theorem to evaluate where and S is the surface , oriented outward.