PHZ 3113 Fall 2010
Homework #3, Due Monday, September 20
1. (a) Let f (x, y) be a function of x and y, with y a function of x and r. Write (∂f /∂x)r in
terms of (∂f /∂x)y , (∂f /∂y)x , and (∂y/∂x)r . Let f (x, r) be a function of x and r, with r a
function of x and y. Write (∂f /∂x)y in terms of (∂f /∂x)r , (∂f /∂r)x , and (∂r/∂x)y .
(b) Let
x2 − y 2
,
f= 2
a + x2 + y 2
where a is a constant. Compute (∂f /∂x)y . Let r 2 = x2 + y 2 . Write f as a function of x and
r. Compute (∂f /∂x)r . Compare your results with the expressions found in (a).
2. The variables x and y are related to the variables u and v by x =
Write the Laplacian operator ∇2 f
∇2 f =
√
2u cos v, y =
√
2u sin v.
∂2f
∂2f
+
,
∂x2
∂y 2
in the variables u and v.
3. The free energy F (T, V ) of an ideal gas is
"
F = −NkT ln
V
N
mkT
2πh̄
3/2 #
,
where N, k, and h̄ are constants.
(a) From dF = −S dT − p dV , Compute S and p. Do you recognize the pressure you obtain?
(b) The internal energy is given by U = F + T S. Compute u. Your answer might appear to
be simple, but show from dU that U should be a function of S and V . Write U as a function
of S and V . Compute the pressure from U.
4. Find the point quadrant on the curve defined by
5 2
5
x + 3xy + y 2 = 1
2
2
that is closest to the point (x, y) = (1, 1).