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.

320 General theorems concerning the double integral. Bk. III. ch. VIII.
generally. Let (z) be a function, that within a determinate interval
is continuous, but has no second derived function. Substituting then
z = (x, y), a function Q () (x, y)) = F(x, y) is obtained, for which no
mixed differential coefficient 3-x exists, although this function has a
total first differential whose integral remains always independent of
the path of integration.
The condition -P Q was essential for the proof we have given
Dy Dx
of theorem 1. in this section, that the sum of the integrals:
(Pdx + Qdy)
formed in a positive circuit for all the boundary curves of a multiply
connected domain is zero. Whether this condition is necessary, we
have not investigated.
176. The conditions of integrability still hold, when instead of
the variables x and y two new variables u and v are introduced, whose
first and second derived functions exist.
Let x = p (u, v), y = *-(u, v), then by the equations:
dx a c+ a du ldv, dy -_ dnu+ ddv,
a u a (v a du a dv
the differential Pdx + Qdy is converted into the form P1 du + Q1 dv,
where:
P1 aP6c + Q W1 Pv + - P g O
and we have:
p) cp - a9
api ^ a^i ad}, a{a^
aV + a(V ap
aQ -m DP w + + a+
-- av \tx % ~y Du av Dx Dtu D y au
+ p D2 P D+ Q 2
Accordingly:
a. Qa
Dv aDu
and:
J(Pidu + Q dv)) = (Pdx + Qdy) + C.
In the deduction of this formula the existence of all the second
differential quotients of the original function is presupposed.