Page 145 - Foundations Of Differential Calculus

128 7. On the Differentiation of Functions of Two or More Variables
follows that it is not possible for P to be equal to zero, nor for Q to be
equal to zero. Hence, if P is a function of x and y, the formula Pdx cannot
be the differential of some finite quantity, that is, there is no finite quantity
whose differential is Pdx.
219. Thus there is no finite quantity V , whether algebraic or transcen-
dental, whose differential is yx dx, provided that y is a variable quantity
which is independent of x. If we should suppose that there did exist such a
finite quantity V , since y is part of its differential, it would necessarily be
the case that y would also be in the quantity V . But if V contained y, due
to the variability of y, necessarily dy would have to be in the differential
of V . However, since it is not present, it is not possible that yx dx could
arise from the differentiation of some finite quantity. It is equally clear that
the formula Pdx + Qdy,if Q is equal to zero and P contains y, cannot
be a real differential. At the same time we understand that the quantity Q
cannot be chosen arbitrarily, but that it depends on P.

220. In order to investigate this relationship between P and Q in the
differential dV = Pdx + Qdy, we first suppose that V is a function of
zero dimension in x and y. We proceed from particular cases to a general
relation. Suppose that we let y = tx. Then the quantity x vanishes from the
function V , and we have a function of t alone, which we call T, and whose
differential is Θ dt, where Θ is a function of t. We also substitute everywhere
y = tx and dy = tdx + xdt. From this we obtain Pdx + Qt dx + Qx dt.
Since dx is really not contained in this, we necessarily have P + Qt =0, so
that
−P −Px
Q = =
t y
and
Px + Qy =0.
Hence in this case we have found the relation between P and Q. Further-
more, it is necessary that Θ = Qx so that Qx is equal to a function of t,
that is, a function of zero dimension in x and y. Since Q =Θ/x,wehave
2
P = −Θy/x , and both Px and Qy are functions of zero dimension in x
and y.

221. If a function V is a function of zero dimension in x and y, and it
is differentiated, then its differential Pdx + Qdy will always be such that
Px + Qy = 0. That is, if in the differential instead of the differentials dx
and dy we write x and y, the result will be equal to zero, as will appear
from the following examples.
x
I. Let V = . Then
y
ydx − xdy
dV = ,
y 2