Math 2374 Final Exam FALL 2002 PRINT NAME____________________________
12/16/02 SIGNATURE_____________________________
Time Limit: Three Hours WORKSHOP INSTRUCTOR________________
SECTION #_____________________
This exam contains 9 problems on 11 pages, 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 and one 8.5 by 11 inch crib sheet.
Do not give numerical approximations to quantities such as or or . Do simplify , e^{0} = 1, etc.
Show your work, in a reasonably neat and coherent way, in the space provided. All answers must be justified by valid mathematical reasoning, including the evaluation of definite and indefinite integrals. Using your calculator to find an indefinite integral is not "valid mathematical reasoning", even if you then check the answer by differentiation.
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.
Recall that .
If f(r ,q ,f ) = (r sinf cosq , r sinf sinq , r cosf ) is the transformation from spherical coordinates to rectangular coordinates, then the Jacobian determinant of this transformation is r ^{2} sinf . You can use this without doing the calculation.
1 
35 pts 

2 
35 pts 

3 
40 pts 

4 
40 pts 

5 
40 pts 

6 
40 pts 

7 
40 pts 

8 
40 pts 

9 
40 pts 

TOTAL 
350 pts 
(b) Let F(x,y,z) = (x+yz, sin(x^{9}z^{6}), cos (x^{7}y^{8})). Use the Divergence Theorem to find the flux of F through the boundary of S. Use the outward pointing normal.
(a) Find the Jacobian matrix for h at the point (2,3).
(b) Assuming that h(2,3) = 4, use the result of part (a) to find the equation of the tangent plane to the graph of h at the point (2,3). Write your answer in the form z =Ax+By+C.
(b) Compute the line integral of the vector field F(x,y,z) = (2x,
y/2, 2z/9) over the curve C, where C is parametrized by
x(t) = (cos^{4 }t^{2}, sin^{4 }t^{2},
t +1), 0 < t < (2p
)^{1/2}