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.

Integration of an exact differential in two variables.

317

therefore:
J Q(x, y f 1Q(x,y Y) _-=dy dy 1(x, y ) —P(X1,y0)
Yo z - X1 SYo X = X
and consequently:
aF(x1, ) __ p
a P(X1,y1);
likewise:
aF(x,, yi) _ Q(x1, y1)
X1,YI
The integral function (Pdx + Qdy) can therefore be defined as that
Xo, Yo
continuous function of the variables x,, y,, which vanishes for the
values x0 yo, and whose partial derived functions with respect to
x, and y, are the functions P and Q.
By this enunciation the function F(x, y1) is completely defined.
For, all continuous functions of two variables whose partial
derived functions within a domain respectively coincide, can differ
only by an additive constant. Suppose, in fact, that F and jq are
two distinct functions for which:
F(x, y) a (x, y) aF(x, y) a (x, y)
- x x a y ay
then in consequence of this first equation we have for every value of y:
F(, y))= (X, ) + C,
where C being a quantity independent of x, can be only a continuous
function of y (~ 100); denoting this by,. we have:
aF(xy) _ aq (x, y) a _y
ay ay ay
but in consequence of the second equation y -- 0. Accordingly
ay Ac
(~ 100) Y is a constant also with respect to y.
The problem therefore is solved: When two continuous functions P
and Q are given which satisfy the equation a- a - within a simply
ay ax
connected domain; it is required to fiznd those continuous functions
whose partial derived ftnctions with respect to x and y coincide with
the values P and Q. All such functions are collectively expressed by:
xoYo
Conversely, knowing beforehand such a continuous function F(x, y),
the definite integral is thereby ascertained; for we have: