Deck 15: Vector Calculus

Question 1

Evaluate , where C is the quarter circle from to . [Hint: ]

A)
B)
C)
D)

Answer 1

D

Question 2

Evaluate , where C is the portion of in the plane from to .

A) 32
B) 0
C) 2
D) 64

Answer 2

A

Question 3

Determine if is conservative, and if so, find a potential function of the vector field.

A) is conservative with
B) is conservative with
C) is conservative with
D) is not conservative

Answer 3

B

Question 4

Compute the work done by the force field along the curve C, where C is the line segment from (2, 1) to (3, 5).

A)
B)
C)
D)

Question 5

Determine whether is conservative, and if so, find a potential function of the vector field.

A) is conservative with
B) is conservative with
C) is conservative with
D) is not conservative

Question 6

Find equations for the flow lines of the vector field.

A)
B)
C)
D)

Question 7

Sketch several vectors in the vector field .

A)
B)
C)
D)

Question 8

Express the line integral where C is the portion of from to in parametric form. Estimate the integral.

Question 9

Find the gradient field corresponding to

A)
B)
C)
D)

Question 10

Compute the work done by the force field along the curve C, where C is the helix from to .

A)
B)
C)
D)

Question 11

Find equations for the flow lines of the velocity vector field .

A)
B)
C)
D)

Question 12

Find the gradient field corresponding to

A)
B)
C)
D)

Question 13

Determine if is conservative, and if so, find a potential function of the vector field.

A) is conservative with
B) is conservative with
C) is conservative with
D) is not conservative

Question 14

Find the gradient field corresponding to

A)
B)
C)
D)

Question 15

Compute the work done by the force field along the curve C, where C is the portion of from to .

A)
B)
C)
D)

Question 16

Determine if is conservative, and if so, find a potential function of the vector field.

A) is conservative with
B) is conservative with
C) is conservative with
D) is not conservative

Question 17

Sketch several vectors in the vector field .

A)
B)
C)
D)

Question 18

Sketch the vector field.

Question 19

Find equations for the flow lines of the vector field.

A)
B)
C)
D)

Question 20

Evaluate , where C is the line segment from to .

A)
B)
C)
D)

Question 21

Find the curl of .

A)
B)
C)
D)

Question 22

Evaluate where and C is the line segment from to

A)
B)
C)
D)

Question 23

Find the divergence of .

A) 0
B)
C)
D)

Question 24

Use Green's Theorem to evaluate where C is curve bounded by and

A)
B) 32
C)
D) 0

Question 25

Use Green's Theorem to evaluate where and C is formed by and Assume the curve is oriented positively.

A)
B)
C)
D)

Question 26

Find the divergence of .

A)
B)
C)
D)

Question 27

Find the curl of .

A)
B)
C)
D)

Question 28

Set up a line integral to compute the area of the region bounded by and

Question 29

Evaluate for and C is the quarter circle from to .

A) 0
B)
C)
D)

Question 30

Find the curl of .

A)
B)
C)
D)

Question 31

Use Green's Theorem to evaluate where C is the rectangle from to to to .

A) -1000
B) -80
C) -40
D) 0

Question 32

Find the divergence of .

A)
B)
C)
D)

Question 33

Evaluate for and C is the top half-circle from to .

A)
B)
C) 0
D)

Question 34

Evaluate where and C is any positively oriented simple closed curve enclosing the origin.

A) 0
B)
C)
D)

Question 35

Determine whether is conservative, and if so, find a potential function of the vector field.

A) is conservative with
B) is conservative with
C) is conservative with
D) is not conservative

Question 36

Show that the line integral is independent of the path.

Question 37

Determine whether is conservative, and if so, find a potential function of the vector field.

A) is conservative with
B) is conservative with
C) is conservative with
D) is not conservative

Question 38

Find a potential function and evaluate the integral where C runs from to

Question 39

Use Green's Theorem to evaluate where C is curve bounded by and oriented clockwise.

A)
B)
C)
D) 0

Question 40

Use a line integral to compute the area of the region bounded by the ellipse .

A)
B)
C)
D)

Question 41

Find the outward flux of F over where Q is bounded by and , and .

A) 0
B)
C)
D)

Question 42

Evaluate the flux integral S is the portion of above the xy-plane (n upward)

A) 0
B)
C)
D)

Question 43

Set up a double integral that equals where S is the portion of the sphere that lies inside the cylinder and above the xy-plane.

Question 44

Use the Divergence Theorem to compute for and Q bounded by and .

A) 0
B)
C)
D)

Question 45

Set up a double integral that equals where S is the portion of the plane above the rectangle

Question 46

Evaluate the flux integral S is the sphere

A) 0
B)
C)
D)

Question 47

Compute using the easiest method available. Q is bounded by and and

A)
B)
C)
D) 0

Question 48

Use the Divergence Theorem to set up and evaluate a triple integral for computing the outward flux of F over where Q is bounded by and and .

Question 49

Find the surface area of the portion of the paraboloid inside the cylinder .

A)
B)
C)
D)

Question 50

Find the surface area of the cone below the plane .

A)
B)
C)
D)

Question 51

Use the Divergence Theorem to compute for and Q bounded by and the coordinate planes.

A)
B) 0
C)
D) 4

Question 52

Determine whether is conservative and/or incompressible.

A) conservative and incompressible
B) conservative and compressible
C) not conservative and incompressible
D) not conservative and compressible

Question 53

Find a parametric representation of the surface from to .

A)
B)
C)
D)

Question 54

Find the outward flux of F over where Q is bounded by and and .

A)
B)
C)
D) 0

Question 55

Find a parametric representation of the surface .

A)
B)
C)
D)

Question 56

Determine whether the vector field is conservative and/or incompressible on R³.

A) conservative and not incompressible
B) conservative and incompressible
C) incompressible and not conservative
D) neither incompressible nor conservative

Question 57

Use the Divergence Theorem to compute for and Q bounded by the cube , and .

A)
B) 0
C)
D)

Question 58

Determine whether the vector field is conservative and/or incompressible on R³.

A) conservative and not incompressible
B) conservative and incompressible
C) incompressible and not conservative
D) neither incompressible nor conservative

Question 59

Use Stokes' Theorem to compute for and S is the portion of a tetrahedron bounded by and the coordinate planes with and upward.

A)
B)
C)
D) 0

Question 60

Find the flux of across the portion of below

A)
B)
C)
D)

Question 61

Use Gauss' Law for electricity and the relationship to find the total charge in the hemisphere where .

A) 0
B)
C)
D)

Question 62

Use Stokes' Theorem to compute for and C is the square to to to .

A)
B)
C)
D) 0

Question 63

Compute or whichever is easier. S is the part of the cone that lies between and

A)
B)
C)
D)

Question 64

Use Guass' Law for electricity and the relationship to find the total charge in the hemisphere for the electric field .

Question 65

Use Stoke's Theorem to evaluate where C is the intersection of and oriented counterclockwise as viewed from above.

A)
B)
C)
D)

Question 66

Use Stokes' Theorem to compute for and C is the boundary of the portion of the paraboloid with and with to the right.

A)
B)
C)
D)

Question 67

Use Stokes' Theorem to compute for and S is the portion of above the xy-plane with upward.

A) 0
B)
C)
D)