TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 251-507/509
Final Exam version A, 11 Dec 2009
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.
Name (print):
In all questions, no analytical work — no points.
1.
Let P be the plane 2x − y + z = 3 and L1 be the line h2 − t, 6 + t, 4 + 2ti. Further, let L2 be
the line that passes through the point A(1, 0, 2), is parallel to the plane P and intersects
the line L1 . Find the point of intersection of the lines L1 and L2 .
2.
Find the absolute minimum and maximum values of f (x, y) = x3 + 2y 2 + x on the set
D = {(x, y)|x2 + y 2 ≤ 1}.
3.
Evaluate the integral
ZZZ
zdV,
E
where E is bounded by the surfaces x2 + y 2 + z 2 = 1 and z =
z ≥ 0).
p
x2 /3 + y 2 /3 (in particular,
4.
Is the field F = hx2 y, xy 2 i conservative? Evaluate
parabola y = x2 going from (0, 0) to (1, 1).
R
C
F · dr, where C is the part of the
5.
Evaluate the integral
2 oriented clockwise.
H
C
F · dr, where F = hcos(x) + y, x2 + ey i and C is the circle of radius
6.
Find the flux of the field F = hy + z, x + z, 1i down through the part of the paraboloid
z = x2 + y 2 with z ≤ 4.
RR
In this question you must evaluate the integral S F · dS directly (without using “the
theorems”).
7.
Bonus question +2 points (no partial credit): Cheburashka wonders if the Divergence
Theorem can be used to solve question 6. Krokodil Gena believes that he read something
like this in the textbook. Shapoklyak, on the other hand, mocks them becuase she knows
that the Theorem can only be applied to closed surfaces. Can you help Cheburashka and
Gena devise a solution to question 6 that utilizes the Divergence Theorem? (An answer
“yes” is not sufficient here, please provide the solution as well).
Points:
/30