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.
•
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
.