Radhika Gupta
Math 2210-005, Fall 2014
December 5
Midterm 3
INSTRUCTIONS:
1. Calculators are NOT allowed.
2. REMEMBER : Don’t spend too much time on problems which are worth very few points.
3. Dont get stuck on a problem, KEEP MOVING ON and in the end come back to the problems you
could not figure out on first try.
4. Show proper work to get full points. If your answer is wrong you still have a chance of getting
partial credit for the work.
Maximum Points
Number of Pages
60 points
7
Question No.
1
2
3
4
5
Total
Marks
BEST OF LUCK!!
1
Radhika Gupta
1
Math 2210-005, Fall 2014
December 5
Triple Integral [10 points]
Consider the region in the first octant between the spheres of radius 1 and radius 3. (That is outside the
sphere of radius 1 and inside the sphere of radius 3 and in the first octant.) If this region is filled with
goo of constant density 2 kg/m
, find the mass of goo in this region. Also draw a cartoon of the solid.
3
Hint: Mass of a solid
fff
(density) dV
3
iii
/
°ot
_
2
t
j
Hc1tct&
-
J
2
t
do
c2..
-c
3
2-
2
1I
Radhika Gupta
2
Math 2210-005, Fall 2014
December 5
Change of Variables [10 + 15 points]
1. [10 points] Solve the system
u
=
x
—
y, v
2x + y
for x and y in terms of u and v. Then find the value of the Jacobian J(u, v). Hint
0x
8x
8u
8v
J(u,v)=
3
H
/3
2
lI
2
-
/3
=
(l/)(ii)
-
(1)1-2i3)
-
3
Radhika Gupta
Math 2210-005, Fall 2014
December 5
2. [15 points] Let R be the region in the first quadrant bounded by the lines
2x+y4,2x+y=7,x—y2.x—y—1.
(a) Draw the region R.
(b) Draw the region G in the
Evaluate ff(2x +
y)(x
ut
—
plane. [use part (a) for
u, v]
y)dA by channg the variables with the equation in part (a) and
integrating over a region C in the
ut
plane.
(cx)
)
rnL&
2
u
9-
2.
(2
) (j
.
.
‘
ow chfi
3
(
q
LZL3’
——
Radhika Gupta
3
Math 2210-005, Fall 2014
December 5
Independence of Path [15 points]
(a) Show that the work done by P(x, y)
pendent of path. That is, show that
P
=
+ 2xg3 in moving a particle from (1,1) to (3,4) is inde
is a conservative vector field.
(b) Find a function
that
(c) Use the
found above to find the work done in moving the particle from (1,1) to (3,4).
f such
function f you
Vf.
tt2-jJ
1
,ChV
A
/
:/
2
‘
) W
C
o’
pOCJ
-
I
1 1)
(L
l..)
—
(
Ci)
I
)-
6)
j
c]
CbU
(l)
-
2
-k rAeJreyre,
gu_1-
_)r
U
2
f
Radhika Gupta
4
Math 2210-005, Fall 2014
December 5
True/False [6 points]
Justify your answer.
2
(a)
I I
Jo
1
5 +2
(x
)
5
x’
d
xdy
y
=
0.
i—i
TRUJ.
c hofl
(b) If f(x, y)
0 on a region R and ‘JR f(x, y)dA
0, then f(x, y)
0 for all (x, y) in R.
P-A
Th
b
S
,C’-rJ
[DI
$
1i P (
c’
0
(c) The divergence of a vector field is a vector field.
PLc.
1’l—t
rjtP
6
Radhika Gupta
5
Math 2210-005, Fall 2014
December 5
Inventions [4 points]
To invent a vector field, you have to give the vector field as a function, for example F(x. y)
x
y
2
i
+ 3xj.
(a) Invent a force (vector) field in the xy-plane such that the work done by this force in moving a particle
along any line parallel to the p-axis is zero.
oLL’ e
-
—
p
(JLL
(t
—
----z;-
0,
toiu
u&tk 0
-
IA(
(b) Invent a vector field that has zero divergence at every point in the xy-plane but non-zero curl at
some point.
,i
rC)
j
0
0+0
r\
A
L’J
(t1)k
cr
2
(c) Invent a vector field that has zero curl at every point in the xy-plane but non-zero divergence at
some point.
I
1”
FCn)
(d) Invent a vector field whose curl and divergence are both zero everywhere in the xy-plane.
FC)
i)
L(j
th
,
7
•o’-fl,
€
iF
/1i€