An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 175.

Condition for an exact differential in two variables.

319

Differentiating this equation with respect to y, since we must have:
F(x,y) (x, ), al(a,y) = (
ay Q (X y), ay Q (, Y)
we find the relation:
x
(1) Q (x, y) - Q (a, y) - -fP(X, y) dx.
a
We find in like manner the analogous equation:
(2) P(x, y) - P(x, a) -= -J Q (, y) dy.
a
Therefore the functions P and Q cannot be independent; they
must satisfy these equations, of which one is a consequence of the
other. These conditions are necessary. Does it then follow, that,
since the integrals can be differentiated, the functions P and Q also
can be differentiated respectively for y and x? Instead of equation (1)
in which a and x signify arbitrary values we can write:
x+h x+h
Q(x-thy)-Qxy) a 1 "ID, -s * 1 PIT(x,yAy), —P(x,y)
h1 a y)-^^ = j P(x, ) dx == Lim j - y d
h2 aS aOy dy=O h A
x x
This expression on the right side has therefore for arbitrarily small
values of h a limiting value for Ay — 0. But at each point at which P
is a continuous function of x, the integral on the right can be replaced
by its mean value, so that we obtain the equation:
Q(x+h,y) - Q(x,y ) = Lil P(x + 07,y O + Ay)-P(Sx+iO,y)
h Ay-=0 Ay
Here 0 is a function depending on Ay, and the interval Oh can be
diminished arbitrarily by choice of h. Nevertheless we must not argue
that the determinate limiting value which is found on the right,
is the differential quotient with respect to y of the function P(x, y)
at a determinate point. For, that limiting value only arises by the
argument Oh also varying, whereas the derived function with respect
to y is defined as
LimP(Xy -+ AMy) - P( x,y)
Jy —0 Ay
It is only in case the derived function P- exists for all points within
the domain, and is an integrable function with respect to both the
variables that we can infer by the proof in ~ 169 that differentiation
and integration are interchangeable in equation (1), and thence the
existence of, as well as the equality Cases may be
assigned i whi it becomes necessary to formulate the condition
assigned in which it becomes necessary to formulate the condition