APPM 2350
HOMEWORK 07
FALL 2010
Due Friday, October 15, 2010 at 4 pm under your TA’s door.
1. Previously in the semester, we derived equations for the distance between a point and a line, and
a point and a plane.
Your task is to derive the same results, but using the method of Lagrange
multipliers.
(a) Use Lagrange multipliers to establish the formula
D
=

ax
0
+
by
0
−
d

√
a
2
+
b
2
for the distance
D
from the point (
x
0
, y
0
) to the line
ax
+
by
=
d
.
(b) Use Lagrange multipliers to establish the formula
D
=

ax
0
+
by
0
+
cz
0
−
d

√
a
2
+
b
2
+
c
2
for the distance
D
from the point (
x
0
, y
0
, z
0
) to the line
ax
+
by
+
cz
=
d
.
2. Suppose you have three identical boxes, and some total number of particles, say
N
. Let
n
1
,
n
2
, and
n
3
denote the number of particles in each box. Of course, we have that
3
j
=1
n
j
=
N.
(1)
We can then define the probability of a particle being in the
j
th
box as
P
j
=
n
j
N
. Then clearly,
3
j
=1
P
j
= 1
.
(2)
Now we define the entropy,
S
, of our system of boxes and particles as
S
=
−
3
j
=1
P
j
ln
P
j
.
(3)
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 Fall '07
 ADAMNORRIS
 Thermodynamics, Statistical Mechanics, Entropy, pj, lagrange multipliers

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