∫∫S

F · dS, where F = 〈0, 0, 4x〉, and S is the surface

shown, a hemisphere of radius 3.

C

∫∫S

F · dS, where F = 〈0, 0, 4x〉, and S is the surface

shown, a circular paraboloid with radius 3 at the top.

F

∫C

F ·T ds, where F = 〈−2xy, x2, 1〉, and C is the

curve shown.

−3

3

−3

3−1

1

xy

z

G

∫∫∫E

2y dV , where E is the solid shown, half of a

ball of radius 3.

D

∫∫S

F · n dS, where F = 〈2xy, x2, 1〉, and S is the

surface shown, the boundary of a hemispherical ballof radius 3.

A

∫∫S

F · n dS, where F = 〈2zy, y2, 1〉, and S is the

surface shown, the boundary of a hemispherical ballof radius 3.

I

To evaluate this integral it is helpful to noticethat F is a conservative field

with potential function f(x, y, z) = x2y + z.

E

0

H

∫C

F ·T ds, where F = 〈2xy, x2, 1〉, and C is the

curve shown.

−3

3

−3

3−1

1

xy

z

B

1

Project 15 MATH 2400 Week 15

1. In your own words, state what Green’s Theorem, Stokes’ Theorem and Divergence Theoremallow you to do in calculus. (For example, if F is a conservative vector field (in R2 or R3) withpotential function f , then the Fundamental Theorem for Line Integrals states that ∫C F ⋅ dr canbe computed by taking the difference of f evaluated at ending and starting points.)

2. There are many different vector fields in these cards. What are some of the criteria for avector field F (in R2 or R3) to be conservative?

3. Applying the Fundamental Theorem for Line Integrals, which cards can be grouped together?

5. Applying Stokes’ Theorem, which cards can be grouped together? Evaluate the line integralon Card G.

6. By Stokes’ Theorem, if F is the curl of some vector field, then the surface integrals

SF ⋅ dS

is equal to the surface integral over any other surface that shares with S.

7. Applying Divergence Theorem, which cards can be grouped together? Evaluate the tripleintegral on Card D.