Math 232
Midterm II
Name_______________________________
(Please Print)
Last,
First
•
Answer each question as accurately as possible.
Please
explain
everything clearly.
Show all
your work for full credit
.
In particular, you must
show the steps for integration.
If you need
to use any of the theorems, make sure that you check
all the hypotheses.
•
No graphing calculator.
If I suspect that you used a graphing calculator for part of the
computation of a problem, I will ask you to do the problem again in my office without a
graphing calculator.
•
Points will be
taken off
for improper writing.
No scratch work will be accepted.
•
Please start each problem on a separate sheet of paper
.
•
You may use your notes, textbook, handouts, and assignments as reference.
However, you
may
NOT
get help from the MathLab or from other teachers, students or friends.
In other
words, you
MUST
work alone.
Any
SUSPICION
of collaborating, discussing, and/or
copying amongst classmates or getting help from someone will result in receiving zero, 0, for
the exam.
•
If you have any questions, please see me.
•
Staple this sheet together with your work to turn in.
Please make sure that your solutions are
still readable after stapling.
1.
(10 pts)
Consider the transformation
( )
2
:
,
1
T x
v y
u
v
=
=
+
.
Analyze and describe
clearly
the image
D
in the
xy
plane
of
R
in the
uv
plane under
T
, where
( )
{ }
,
0
1, 0
1
R
u v
u
v
=
≤
≤
≤
≤
.
2.
(10 pts)
Consider the rectangle with vertices
( ) ( ) ( ) ( )
1, 1 , 1, 1 , 1,2 , and
1,2
 


.
Compute
the work done by the force field
( )
2
2
,
sin
,
y
x y
x
xy e
x
=
+
+
F
in moving an object along
C
,
where
C
is the path that goes around the boundary of the rectangle clockwise twice starting at
the vertex
( )
1, 1
 
.
3.
(10 pts)
Consider the vector field
( )
, ,
2 ,2 ,2
x y z
x
y
z
=
F
.
If
C
is any path from
( )
0,0,0 to
( )
1
2
3
,
,
a a a
and
1
2
3
,
,
a a a
=
a
, show that
C
d
⋅
= ⋅
∫
F
r
a a
.