B. Some Useful Mathematics
B.1 Differential Calculus with Several Variables
The fundamental equation of differential calculus for a functionff(x,y,z)is
df
(
∂f
∂x
)
y,z
dx+
(
∂f
∂y
)
x,z
dy+
(
∂f
∂z
)
x,y
dz (B-1)
The coefficients are partial derivatives. Each is obtained by the ordinary process of
differentiation, treating the other independent variables as constants. The symbols for
the variables that are held constant are written as subscripts to remind us what they are.
IfPis a function ofT,V, andn,
dP
(
∂P
∂T
)
V,n
dT+
(
∂P
∂V
)
T,n
dV+
(
∂P
∂n
)
T,V
dn (B-2)
This represents an infinitesimal change inPproduced by the corresponding infinites-
imal changes in the independent variablesT,V, andn. If finite changes are not too
large, we can write an approximate version of this equation:
∆P≈
(
∂P
∂T
)
V,n
∆T+
(
∂P
∂V
)
T,n
∆V+
(
∂P
∂n
)
T,V
∆n (B-3)
Some Mathematical Identities An equation that is valid for any values of the vari-
ables involved is called an identity. The first task of this section is to obtain several
identities involving partial derivatives.
An Identity for a Change of Variables. The expression for the differential of a
functionUU(T,V,n)is
dU
(
∂U
∂T
)
V,n
dT+
(
∂U
∂V
)
T,n
dV+
(
∂U
∂n
)
T,V
dn (B-4)
IfT,P, andnare used as the independent variables, thendUis given by
dU
(
∂U
∂T
)
P,n
dT+
(
∂U
∂P
)
T,n
dP+
(
∂U
∂n
)
T,P
dn (B-5)
To obtain our identity, we “divide” Eq. (B-4) bydTand specify thatPandnare fixed.
Of course, you cannot correctly do this, sincedTis infinitesimal, but it gives the correct
relationship between the derivatives. Each “quotient” such asdU/dTis interpreted as a
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