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(math204)[2007](s)final~PPSpider^_10479.pdf
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Math 204 Final Exam
May 23, 2007

Your Name

Student Number

Number Score
1
2
3
4
5
6
Total


(1) (15 points) Suppose f g h and A is a subset with volume. Suppose f and h are
Riemann integrable on A with fd = hd. Prove that g is Riemann integrable on
AA

A.
(2) (15 points) Suppose f(t) is Riemann integrable on [a, b]. Suppose A . R2 is a subset with area, such that (x, y) A =. a x + y b. Prove that f(x + y) is integrable on
A.
(3) (15 points) Suppose C is any closed curve on the sphere x2 + y2 + z2 = R2 . Prove
222
that (y + z 2)dx +(z + x 2)dy +(x + y 2)dz = 0. In general, what is the condition for
C .
f, g, h so that fdx + gdy + hdz = 0 for any closed curve C on any sphere centered at
C
the origin?
(4) (15 points) Let CR be the counterclockwise circle of radius R centered at the origin.
Prove that . ey[(x sin x + y cos x)dx +(y sin x . x cos x)]dy = cR2
CR
for a constant c..Then .nd the constant c.
y
Ie
Hint: = [(x sin x + y cos x)dx +(y sin x . x cos x)]dy.
R2 CR x2 + y2
(5) (15+10 points) If possible, .nd the potential function. (x + y)dx +(x + y)dy
1. Di.erential form =.
(ax2 + by2)p

.a
2. Vector .eld F =, b .0.
b .x
(6) (15 points) True or false (no reason needed).
1.
If A has volume, then .A has volume.

2.
If .A has volume, then A has volume.

3.
If A, B .

= . and A B has volume, then A and B have volumes.

4.
If A and B have volumes, then A B has volume.

5.
If A . Rm+n has volume, then the projections 1(A) . Rm and 2(A) . Rn have volumes.

6.
If A . Rm+n has the property that the projections 1(A) . Rm and 2(A) . Rn have volumes, then A has volume.

7.
If f is Riemann integrable on A, then |f| is Riemann integrable on A.

8.
If |f| is Riemann integrable on A, then f is Riemann integrable on A.

9.
If a parametrized curve C . Rm+n is recti.able, then the projections 1(C) . Rm and 2(C) . Rn are recti.able.

10.
If a parametrized curve C . Rm+n has the property that the projections 1(C) . Rm and 2(C) . Rn are recti.able, then C is recti.able.

11.
If . and are potential functions of a vector .eld F on an open subset U, then . = + C for some constant C.

12.
If a vector .eld F has potential functions on open subsets U and V , then F has potential function on U V .

13.
If a vector .eld F has potential functions on open subsets U and V . U, then F has potential function on V .

14.
If vector .elds F and G have potential functions on open subsets U and V , then the vector .eld (F, G) has potential function on U V .

15.
If a vector .eld F (.x, .y) has potential functions for .xed .x and for .xed .y, then F has a potential function.


Answer to Math 204 Final, Spring 2007
not absolutely guaranteed to be correct
(1) (15 points) Let I = fd = hd. For any .> 0, there is > 0, such that for any
AA