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Math
20E~
Midterm
Exwn 2
August 28, 2008
Thm
off and put away your cell phone.
No calculators or any other devices are allowed on this exam.
You may
use
one page of notes, but no
books
or other assistance on this exam.
Read each question carefully, answer each question completely, and show all of your work.
Write your solutions clearly and legibly; no credit will be given for illegible solutions.
If any question is not clear, ask for clarification
.
.1.
(6
points) The vector field
F(x,y,z)
=
(2
xyz
2, X
2
Z
2
, 2x
2
yz)
is
the gradient of the
function
!(x,
y,
z)
=
x
2
yz2.
Let c be the path between (0,0,0) and (2,5, I) defined for
0:5
t:5
1
by
c(t}
=
(2
tan
(it)
I
5 sin
(~t)
I
t
cos
(~t)
+
t
2
).
Evaluate
F . ds.
1
s
\nut,
~ ~
~wA ~
t
C
~lo.\)
~ ~.~s.
'=
.\
(c.~
l
\1\1
tA
t
L
\\:::
~\o.,",
~)
S"
s;"Qf)
I
CQ'5clD
I
\)

(2,5,\)
L
lo)
::
l2.
~'C\\o)
'\
S'5~
\
o}
l
0
')
I
#
II
Points
I
Score
1
2
~
°/
°,0)
3
4
5
6
6
6
6
6
I
E
II
30
Sc
f
..
~'S
=
\l'L,S,\,)
f
lO,O,o)
2
2
'
5.
(

0
.20