Deck 16: Vector Calculus

Question 1

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)

Answer 1

Question 2

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

A)
B)
C)
D)

Answer 2

Question 3

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

A) -16
B) -32
C) 2
D) -80

Answer 3

-16

Question 4

Find equations for the flow lines of the vector field.

A)
B)
C)
D)

Question 5

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

Question 6

Sketch several vectors in the vector field .

A)
B)
C)
D)

Question 7

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 8

Find equations for the flow lines of the vector field.

A)
B)
C)
D)

Question 9

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 10

Find the gradient field corresponding to

A)
B)
C)
D)

Question 11

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 12

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 13

Find the gradient field corresponding to

A)
B)
C)
D)

Question 14

Sketch several vectors in the vector field .

A)
B)
C)
D)

Question 15

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 16

Evaluate , where C is the line segment from to .

A)
B)
C)
D)

Question 17

Find the gradient field corresponding to

A)
B)
C)
D)

Question 18

Sketch the vector field.

Question 19

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

A)
B)
C)
D)

Question 20

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

A)
B)
C)
D)

Question 21

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

A) 0
B)
C)
D)

Question 22

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 23

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

Question 24

Find the curl of .

A)
B)
C)
D)

Question 25

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

Question 26

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

A)
B)
C)
D)

Question 27

Evaluate where and C is the line segment from to

A)
B)
C)
D)

Question 28

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

A) -1080
B) -135
C) -45
D) 0

Question 29

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 30

Find the divergence of .

A)
B)
C)
D)

Question 31

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

A)
B)
C)
D)

Question 32

Find the curl of .

A)
B)
C)
D)

Question 33

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

A)
B) 64
C)
D) 0

Question 34

Evaluate for and C is the quarter circle from to .

A) 0
B)
C)
D)

Question 35

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

A)
B)
C) 0
D)

Question 36

Find the curl of .

A)
B)
C)
D)

Question 37

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

A)
B)
C)
D) 0

Question 38

Show that the line integral is independent of the path.

Question 39

Find the divergence of .

A) 0
B)
C)
D)

Question 40

Find the divergence of .

A)
B)
C)
D)

Question 41

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

A)
B) 0
C)
D)

Question 42

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

A)
B)
C)
D) 0

Question 43

Find the surface area of the cone below the plane .

A)
B)
C)
D)

Question 44

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

A)
B)
C)
D) 0

Question 45

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

A) 0
B)
C)
D)

Question 46

Find the flux of across the portion of below

A)
B)
C)
D)

Question 47

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 48

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

A)
B)
C)
D)

Question 49

Evaluate the flux integral S is the sphere

A) 0
B)
C)
D)

Question 50

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

A) 0
B)
C)
D)

Question 51

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

Question 52

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 53

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 54

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 55

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

A)
B) 0
C)
D) 4

Question 56

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

A) 0
B)
C)
D)

Question 57

Find a parametric representation of the surface from to .

A)
B)
C)
D)

Question 58

Find a parametric representation of the surface .

A)
B)
C)
D)

Question 59

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 60

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 61

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

A)
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

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

A) 0
B)
C)
D)

Question 64

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

A) 0
B)
C)
D)

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

Question 67

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)