Page 144 - Foundations Of Differential Calculus

7. On the Differentiation of Functions of Two or More Variables 127
VIII. If V = sin x cos y, then

dV = dx cos x cos y − dy sin x sin y.

x
e y
, then
IX. If V = 2 2
x + y
2
z
e ydz e z x dy − yx dx
+ .
dV = 2 2 2 2 2 2
x + y (x + y ) x + y

2 2
z
X. If V = e arcsin x − x − y , the result is
2
x + x − y 2

2
x − y 2
z
dV = e dz arcsin x −
2
x + x − y 2
2
xy dy − y dx
z
+ e √ .
2
2
2
3/4
x + x − y 2 (x − y ) x
217. We have seen that if V is a function of two variables x and y, its
differential will have the form dV = Pdx + Qdy, in which P and Q are
functions that depend on V and are determined by it. It follows that these
two quantities P and Q in some certain way depend on each other, since
each depends on the function V . Whatever this connection between the
finite quantities P and Q may be, which we will have to investigate, it is
clear that not all differential formulas with the form Pdx+Qdy, in which P
and Q are arbitrarily chosen, can be the differential of some finite function
V of x and y. Unless this relationship between the functions P and Q is
present, which the nature of differentiation requires, a differential of the
type Pdx + Qdy clearly cannot arise from differentiation, and in turn will
have no integral.
218. Therefore, in integration it is of great interest to know this rela-
tionship between the quantities P and Q in order that we may distinguish
between those that really arose from the differentiation of some finite func-
tion and those that were formed arbitrarily and have no integral. Although
we are not going to take up the business of integration here, still this is
a convenient time to investigate this relationship by looking more deeply
into the nature of real differentials. Not only is this knowledge extremely
necessary for integral calculus, for which we are preparing the way, but
also it will cast significant light on differential calculus itself. First, then,
it is clear that if V is a function of two variables x and y, then both the
differentials dx and dy must be present in the differential Pdx + Qdy.It