602
B Review of Traditional Calculus
this is also an important relation amongst partial derivatives. Notice, however,
the distinction between this relation and the permutation rule (B.1.13)
obtained earlier.
(vi) Some Properties of Exact Differentials.
Mathematically, if a differential df can be written in the form
df = P (x, y) dx + Q(x, y) dy
for all (x, y) in some region of the xy-plane and for all values of dx and dy,
we say that P (x, y) dx + Q(x, y) dy is an exact differential provided that
P (x, y), Q(x, y),
∂P
∂y
x
, and
∂Q
∂x
y
are all continuous on a rectangular
region S of the xy-plane and that
∂P
∂y
x
=
∂Q
∂x
y
for all (x, y) contained
in S. This requirement turns out to be both necessary and sufficient. An
immediate consequence of this condition is that if for a given function
f (x, y), both its two first partial derivatives and its two mixed second partial
derivatives are continuous at (x 0 , y 0 ), then
∂ 2 f
∂x∂y
=
∂ 2 f
∂y∂x
at (x 0 , y 0 ). This equality of the mixed second partial derivatives has very
important consequences for thermodynamics, as it is the source of the famous
Maxwell relations.
(vii) Homogeneous Function Theorem.
Let us consider briefly the Euler theorem for homogeneous functions, which
can be stated as follows. If λf (x, y, z) = f (λx, λy, λz), then
f = x
∂f
∂x
+ y
∂f
∂y
+ z
∂f
∂z
.
This particular theorem is relatively easy to prove, and it is perhaps useful to
see how the proof works.
We begin by defining a function F (x, y, z) via
λf (x, y, z) = F (x, y, z) ≡ f (λx, λy, λz)
or F (x, y, z) = f (x , y , z ), in which x = λx, y = λy, z = λz. Then, the
derivative of F (x, y, z) with respect to λ is given by
f (x, y, z) =
∂
∂λ
[λf (x, y, z)]
=
∂F
∂λ