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.

318 General theorems concerning the double integral. Bk. III. ch. VIII.

x,y
(Pdx + Qdy)= F(x, y) + C = F(x, y)- F(xo, y,),
Xo, Yo
because the left side vanishes for x — x, y = y, therefore C must
be equal to - F(xo, y,).
Thus also the connexion between the definite and the indefinite
integral in two variables is developed subject to all the hypotheses
here necessary.
Note: It is to be observed, that this deduction of a continuous
function F(x,y) from its partial derived functions P and Q requires
not only that these derived functions be continuous but also that
they admit of being differentiated respectively for y and x
within the domainS); while there is no analogue to this with functions
of a single variable.
It is therefore not unimportant for us to realize that the condition
for an exact differential must be modified in certain cases. Let us
formulate the problem as follows:
Given in a rectangular domain, from x a to x==b,
y- a to y- 3, two continuous functions P and Q. What
further conditions must they fulfil, in order that there may
be a continuous function F(x,y) in the domain, for which:
g- P(x, Q(y),
Ox~y '
and how is this function determined?
All continuous functions whose partial derived function with respect
to x coincides with P are collectively included in the form:
x
F(x,y) =JP(x,y)dx + Y,
a
where Y is a continuous function of y only. Making x - a, we find:
x
F(x, y) - F(a, y) = fP(x, y)dx.
__..~~~~~~ a
*) There may be points at which the functions P and Q cease to be
continuous and finite. But these points can be enclosed within arbitrarily small
neighbourhoods; the enclosing curves are then counted among the boundaries
of the domain, rendering it multiply connected. But they do not ultimately
come into consideration 'in the formation of the simple integral, if for the interior
of such a region the double integral aj (- -a d )dxdy must vanish. The
Jr J,, \y x X
equationa _ aQ) d dy = 0, formed for every arbitrary part of the
entire domain, is the necessary and sufficient hypothesis on which the derived
theorems are based.