PHYSICAL CHEMISTRY IN BRIEF

CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 77

3.3 Some properties of the total differential

3.3.1 Total differential

Let us consider functionsM(x, y) andN(x, y) continuous and differentiable (to the second
order) on a simply connected region (for details, see a basic course of differential calculus). The
necessary and sufficient condition for the differential form

dz=M(x, y)dx+N(x, y)dy (3.23)

to be the total differential of the functionz=z(x, y) is the equality of the derivatives
(
∂M
∂y

)

x

=

(

∂N

∂x

)

y

, (3.24)

at all points of the region, where

M=

(

∂z

∂x

)

y

, N=

(

∂z

∂y

)

x

.

Hence for the total differential of the functionz=z(x, y) it holds

dz=

(

∂z

∂x

)

y

dx+

(

∂z

∂y

)

x

dy. (3.25)

Note:Equation (3.24) requires that the mixed second partial derivatives should be inde-

pendent of the order of differentiation, i.e. that

∂^2 z

∂x∂y

=

∂^2 z

∂y∂x

. (3.26)

Example

Is the differential form

dz= (10xy^3 + 7)dx+ 15x^2 y^2 dy

the total differential of functionz?