1236 B Some Useful Mathematics
partial derivative with the same variables fixed in each “quotient.” The result is, holding
Pandnfixed:
(
∂U
∂T)
P,n(
∂U
∂T
)
V,n(
∂T
∂T
)
P,n+
(
∂U
∂V
)
T,n(
∂V
∂T
)
P,n+
(
∂U
∂n)
T,V(
∂n
∂T)
P,n
(B-6)
The derivative ofTwith respect toTis equal to unity, and the derivative ofnwith
respect to anything is equal to zero ifnis fixed, so that:
(
∂U
∂T)
P,n(
∂U
∂T
)
V,n+
(
∂U
∂V
)
T,n(
∂V
∂T
)
P,n(B-7)
Equation (B-7) is an example of thevariable-change identity. The version for any
particular case can be obtained by systematically replacing each letter by the letter for
any desired variable.
The Reciprocal Identity. If the roles of the independent and dependent variables
are reversed, keeping the same variables held constant, the resulting derivative is the
reciprocal of the original derivative. An example of this identity is:
(
∂V
∂P)
T,n1
(∂P/∂V)T,n(B-8)
This identity has the same form as though the derivatives were simple quotients, instead
of limits of quotients.
TheChainRule. If the independent variable of a function is itself a function of another
variable, the chain rule can be used to obtain the derivative of the first dependent variable
with respect to the second independent variable. For example, ifUis considered to be
a function ofP,V, andn, whilePis considered to be a function ofT,V, andn, then
(
∂U
∂T)
V,n(
∂U
∂P
)
V,n(
∂P
∂T
)
V,n(B-9)
The same quantities must be held fixed in all of the derivatives in the identity.
We can also obtain the differential of a quantity which is expressed as a function
of one variable which is in turn given as a function of other variables. For example, if
ff(u) anduu(x,y,z):
(
∂f
∂x)
y,z(
df
du)(
∂u
∂x)
y,z(B-10)
The differential offcan be written:dfdf
du[(
∂u
∂x)
y,zdx+(
∂u
∂y)
x,zdy+(
∂u
∂z)
x,ydz]
(B-11)
Euler’s Reciprocity Relation. A second derivative is the derivative of a first deriva-
tive. Iffis a differentiable function of two independent variables,xandy, there are
four second derivatives:∂^2 f
∂y∂x(
∂
∂y(
∂f
∂x)
y)
x(B-12a)