Physical Chemistry Third Edition

1238 B Some Useful Mathematics

whereL,M, andNare some functions ofx,y, andz. A general differential form like

that of Eq. (B-19) is sometimes called aPfaffian form. If the functionsL,M, and

N are not the appropriate partial derivatives of the same function, then the differ-

entialduis aninexact differential, and has some different properties from an exact

differential.

To test the differentialdufor exactness, we can see if the appropriate derivatives

ofL,M, andNare mixed second derivatives of the same function and obey the Euler

reciprocity relation:

(

∂L

∂y

)

x,z



(

∂M

∂x

)

y,z

(exact differential) (B-20a)

(

∂L

∂z

)

x,y



(

∂N

∂x

)

y,z

(exact differential) (B-20b)

(

∂M

∂z

)

x,y



(

∂N

∂y

)

x,z

(exact differential) (B-20c)

If any one of the conditions of Eq. (B-20) is not obeyed thenduis an inexact differential,

and if all of them are obeyed thenduis an exact differential.

B.2 Integral Calculus with Several Variables

There are two principal types of integrals of functions of several variables, the line

integral and the multiple integral.

Line Integrals For a differential with two independent variables,

duM(x,y)dx+N(x,y)dy

a line integral is denoted by

∫

c

du

∫

c

[

M(x,y)dx+N(x,y)dy

]

(B-21)

where the lettercdenotes a curve in thexyplane. This curve givesyas a function

ofxandxas a function ofy, as in Figure B.1. We say that the integral is carried out

along this curve (or path). To carry out the integral, we replaceyinMby the function

ofxgiven by the curve and replacexinNby the function ofygiven by the curve. If

x these functions are represented byy(x) andx(y):

y

Curve giving y 5 y(x)

orx 5 x(y)

Figure B.1 A Curve Givingyas a
Function ofxor givingxas a Func-
tion ofy.

∫

c

du

∫x 2

x 1

M(x,y(x))dx+

∫y 2

y 1

N(x(y),y)dy (B-22)

wherex 1 andy 1 are the coordinates of the initial point of the line integral andx 2 and

y 2 are the coordinates of the final point. Each integral is an now an ordinary integral

and can be carried out in the usual way. If the differential form has three or more

independent variables, the procedure is analogous. The curve must be a curve in a

space of all independent variables, giving each one of the other independent variables

as a function of one variable.

There is an important theorem of mathematics concerning the line integral of an

exact differential: Ifdzis an exact differential it is the differential of a functionz.