Study Guide for Final Exam s2

22M:28 Spring 05 J. Simon Ch. 6 Study Guide for Final Exam page 1 of 3

22M:28 Spring 05 J. Simon

Study Guide for Final Exam Chapter 7 Portion (part 2)

(This is a continuation of the Ch 7 handout posted May 6.]

Problem 3.

[Each of the following parts could be a whole exam problem. I just want to make sure you practice using several coordinate systems.]

Here is some data about spherical coordinates that I will provide on the exam.

x sin() cos( ) y sin() sin( ) z cos()

2 d(Volxyz) =  sin() d(Vol)sph 2 T  T =  sin() [ sin  cos  , sin  sin  , cos  ]

(Suggestion: Some time during your studying for the exam, calculate the norm of the vector [ sin  cos  , sin  sin  , cos  ].)

For each of the following, set up [but do not evaluate] an iterated integral whose value is the integral (xy  z)dA X where X is the surface

(a) z = x+y , x = 0..2, y=0..3

(b) The upper hemisphere of the sphere of radius 3 centered at the origin

(c) The cylinder x2 + y2 = 25 , z=0..8 .

(Answers next page) 22M:28 Spring 05 J. Simon Ch. 6 Study Guide for Final Exam page 2 of 3

Answers: 3 2   (a)   (x yxy) 3 dx dy 0 0

2     2 (b)   9 (9 sin() cos( ) sin( )3 cos()) sin() d d 0 0

8 2    125 cos( ) sin( )5 z d dz (c)   0 0

Problem 4 (As with Problem 3, each of the three parts of this problem could be one whole problem on the exam.)

For each of the surfaces in Problem 3 , find the flux of the vector field F(x,y,z) = z i + x2 k through the surface. For each of the surfaces, there are two choices of overall direction for the flux. Include a quick sketch of each surface and indicate on your sketch which direction you are using.

Answers 3 2   (a)   (x yxy) 3 dx dy 0 0

2     2 (b)   27 cos() sin() cos( ) (13 sin() cos( )) d d 0 0

(c) [this one actually is easy to evaluate] 400  22M:28 Spring 05 J. Simon Ch. 6 Study Guide for Final Exam page 3 of 3

Problem 5

Suppose F is the vector field F(x,y,z) = (z2) k . Let W be the solid cube in 3-space given by x=0..2, y=0..3, z=0..4 .

Recall that the divergence of a vector field M(x,y,z) i + N(x,y,z) j + P(x,y,z) k M N P in R3 is the scalar function div(F) =   . x y z

(a) Calculate the integral of div(F) on the set W. Answer: 96

(b) Calculate the outward flux of F across the boundary of W . (Hint: The boundary consists of six rectangles. The flux integral for each of those six pieces is very simple.)

Answer: 96 [Remark: The divergence theorem says that those two integrals are always equal.]