Page 150 - Foundations Of Differential Calculus

7. On the Differentiation of Functions of Two or More Variables 133
228. From this equality we deduce the following conclusion. If any function
V of two variables x and y has a differential dV = Pdx + Qdy, then the
differential of Pdx, which comes from letting only y be variable while x
is held constant, is equal to the differential of Qdy, which comes from
letting only x be variable while y is held constant. For instance, if only
y is variable, then dP = Zdy and the differential of Pdx, taken in the
prescribed way, will be equal to Zdx dy. Now, if we let only x be variable,
then also dQ = Zdx. Thus the differential of Qdy, taken in the prescribed
way, will be Zdx dy. In this way we understand the relationship between P
and Q. In short, the differential of Pdx when x is constant must be equal
to the differential of Qdy when y is constant.

229. This remarkable property will become clearer if we illustrate it with
a few examples.

I. Let V = yx. Then

dV = ydx + xdy,

so that P = y and Q = x. When we keep x constant,
d.P dx = dx dy,

and when y is kept constant,

d.Q dy = dx dy,
so that the two differentials are equal.
2

II. Let V = x − 2xy. Then
xdx + ydx + xdy
dV = ,
2
x +2xy
so that
x + y x
and ,
P = 2 Q = 2
x +2xy x +2xy
so that when x is kept constant,

xy dx dy
d.P dx = 3/2 ,
2
(x +2xy)
and when y is kept constant,
xy dx dy
d.Q = 3/2 .
2
(x +2xy)