Physical Chemistry Third Edition

B Some Useful Mathematics 1237

∂^2 f

∂x∂y



(

∂

∂x

(

∂f

∂y

)

x

)

y

(B-12b)

(

∂^2 f

∂x^2

)

y



(

∂

∂x

(

∂f

∂x

)

y

)

y

(B-12c)

(

∂^2 f

∂y^2

)

x



(

∂

∂y

(

∂f

∂y

)

x

)

x

(B-12d)

We refer to the second partial derivatives in Eqs. (B-12a) and (B-12b) asmixed second

partial derivatives. TheEuler reciprocity relationis a theorem of mathematics: Iffis

differentiable, then the two mixed second partial derivatives in Eq. (B-12a) and (B-12b)

are the same function:

∂^2 f

∂y∂x



∂^2 f

∂x∂y

(B-13)

For a function of three variables, there are nine second partial derivatives, six

of which are mixed derivatives. The mixed second partial derivatives obey relations

exactly analogous to Eq. (B-13). For example,

(

∂^2 V

∂T ∂P

)

n



(

∂^2 V

∂P∂T

)

n

(B-14)

The same third independent variable is held fixed in both derivatives, as shown by the

subscript.

The Cycle Rule.Ifx,y, andzare related so that any two of them can be considered

as independent variables we can write the cycle rule:

(

∂z

∂x

)

y

(

∂x

∂y

)

z

(

∂y

∂z

)

x

− 1 (B-15)

We obtain this identity in a nonrigorous way. The differentialdzcan be written

dz

(

∂z

∂x

)

y

dx+

(

∂z

∂y

)

x

dy (B-16)

We consider the special case in whichzis held fixed so thatdz0, and “divide”

Eq. (B-16) nonrigorously bydy. The “quotient”dx/dyat constantzis interpreted as a

partial derivative at constantz, and the “quotient”dy/dyequals unity. We obtain

0 

(

∂z

∂x

)

y

(

∂x

∂y

)

z

+

(

∂z

∂y

)

x

(B-17)

We multiply by (∂y/∂z)xand apply the reciprocal identity to obtain

(

∂z

∂x

)

y

(

∂x

∂y

)

z

(

∂y

∂z

)

x

− 1 (B-18)

which is equivalent to Eq. (B-15).

Exact and Inexact Differentials Equation (B-1) gives the differential of a function,

which is called anexact differential. We can also write a general differential in terms

ofdx,dy, anddz:

duL(x,y,z)dx+M(x,y,z)dy+N(x,y,z)dz (B-19)