Math 2374 Final Exam PRINT NAME____________________________
SIGNATURE_____________________________
Time Limit: 3 Hours WORKSHOP INSTRUCTOR________________
SECTION #_____________________
This exam contains 7 problems on 7 pages, not including this cover page. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated.
You may not use your books and notes on this exam, but you may use your graphing calculator. However, all answers must be justified by valid mathematical reasoning. This includes the evaluation of definite and indefinite integrals.
Show your work, in a reasonably neat and coherent way, in the space provided on the following pages. If you need more space, use the back of the preceding page.
Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written.
A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit.
Full credit will be given only for work that is presented neatly and logically; work scattered all over the page without a clear ordering will receive very little credit.
Do not give numerical approximations to quantities such as or p or .
Do simplify expressions
such as
e^{0
}= 1, cos (p /2) = 0, etc.
The transformation from spherical coordinates to rectangular coordinates is given by:
x = r sinf cosq , y = r sinf sinq , z = r cosf
The Jacobian determinant is
Do not write in the table below.
1 
30 pts 

2 
40 pts 

3 
30 pts 

4 
40 pts 

5 
40 pts 

6 
40 pts 

7 
30 pts 

TOTAL 
250 pts 
Let S be the
hemisphere given by
, z > 0. Using
spherical coordinates, set up, but do not evaluate, an
integral which represents the flux of curl F across S, using the outward
pointing normal. Your answer should be expressed as the integral of some function, with
appropriate endpoints of integration. Your answer should not contain any dot products,
cross products, etc. These should all be worked out and simplified. Then
use Stokes’s Theorem to evaluate the integral.