MATH 2263 Midterm 3 Review
Cora Brown
1. Sketch the vector field
(a)
F
x, y
1
2
i
y
x
j
(b)
F
x, y
y, y
2
2. Evaluate the line integral (since none of these curves are closed, we cannot use Green’s
Theorem).
(a)
C
x ds
,
C
is the arc of the parabola
y
x
2
from
0
,
0
to
1
,
1
(b)
C
yz
cos
x
ds
,
C
:
x
t, y
3 cos
t , z
3 sin
t
, 0
t
π
(c)
C
xy dx
y
2
dy
yz dz
where
C
is the line segment from
1
,
0
,
1
to
3
,
4
,
2
(d)
C
F
d
r
where
C
is given by
r
t
sin
t ,
1
t
and 0
t
π
.
3. Show that
F
is conservative (without finding the potential) and then find the potential
function
f
such that
F
∇
f
.
(a)
F
x, y
1
xy e
xy
, e
y
x
2
e
xy
(b)
F
x, y, z
sin
y , x
cos
y ,
sin
z
4. Show that
F
x, y
4
x
3
y
2
2
xy
3
,
2
xy
4
3
x
2
y
2
,
4
y
3
is conservative and use that to
evaluate
C
F
d
r
where
C
is the curve given by
r
t
t
sin
πt ,
2
t
cos
πt
for
0
t
1.
5. Use Green’s theorem to evaluate
C
xy
2
dx
x
2
y dy
where
C
consists of the parabola
y
x
2
from
1
,
1
to
1
,
1
and the line segment from
1
,
1
to
1
,
1 . Check to make
sure
C
has positive orientation first.
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 Spring '08
 Staff
 Math, Multivariable Calculus

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