Modern Engineering Thermodynamics

3.3 Fun with Mathematics 59
CRITICAL THINKING
If you have a composite function of the form f(x, y, z) = 0, where x = y + z, y = xz, and z = y/x, then are the following chain
rule differentials correct or not?
1. dx = dy + dz.
2. dy = zdx + xdz.
3. dz = dy/x + dx/y.
these three functions has a different form, even though they all yield x. Since these functions are not the same, if
follows that their partial derivatives are not equal, or
∂x
≠ ∂x
≠ ∂x
∂y
z
∂y
w
∂y
q
That is why we always indicate which variables are held constant in partial differentiation. This also informs the
reader as to which independent variables are being used in a functional relation.
Equation (3.1) tells us that two of the three variables are independent. If we choose the independent variables
to be x and z, then the preceding equation, which relates x and z, is valid only if the coefficients of both dx and
dz are equal to zero (otherwise, they would not be independent). Then we have
or
so
We also must have
or
1 − ∂x
∂y
∂x
∂y
z
z
∂x
=
∂y
z
∂x
∂y
z
∂x
∂y
z
∂y
∂z
∂y
∂x
∂y
∂x
= 0
z
z
= 1
−1
∂y
(3.2)
∂x
x
∂y
∂z
x
z
+ ∂x
= 0
∂z y
= − ∂x
∂z y
Then, using the results of Eq. (3.2), we can write
∂x ∂y
∂z
= −1 (3.3)
∂y ∂z ∂x y
z
x
EXAMPLE 3.1
Show that, if the pressure p of a substance is a function of its temperature T and its density ρ, we can write
∂p ∂ρ
= − ∂ρ
∂T ∂p ∂T
Solution
From Eq. (3.3) with x = p, y = T, and z = ρ, we have
∂p
∂T
ρ
p
∂T
∂ρ
Τ
∂ρ
∂p
p
T
p
= −1
(Continued )