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.

314 General theorems concerning the double integral. Bk. Ill. ch. VIII.
extended along a closed curve in a positive direction, signifies, in terms
of a single variable, the value of
F(x, y)d y,
when in the first place the entrances and exits of parallels to the axis of x:
X -= V1(y) < xA = 2 (y) < X3 == P 3() < X4 ==- 4() (.*
are calculated for each value of y from p(x, y) = 0 the equation of
the boundary curve, and then the sum of integrals:
J( F{ ydy — j (i(Y) 4J ((' )) O — i M (y), s) y
is formed for increasing values of y.
Employing this definition, the following is the statement of
Gre en's theorem:
TWhen the function f(x, y) is integrable woithin a domain, the integral:
jf (x, y) dx = F(x, y)
is in general for each value of y a continuous function of x and for
each value of x an integrable function with, respect to y. The double
integral:
f (x, y) dx dy
extended throughout the entire domain is equal to the simople integral:
fF(x, y) sin cpds
extended along the boundary curves of the domain in a positive circuit.
This equality still holds good when the values of the functions
f and F are altered arbitrarily at infinitely many points, provided the
integrals are not thereby changed.
Therefore the value of the definite double integral depends only
on the values of F at the points of the boundary curves.
We can also frame the theorem thus, reversing the order of ideas:
When the partial derivate F of a function F(x, y) is integrable over
the entire domain, even admitting that there are points or curves at
which it is discontinuous or infinite, we have always:
J)rr lVd l xdy = f F(x, y) sin Y ds.
Likewise, interchanging the letters x and y, and denoting the angle
that the inward direction of the normal makes with the positive axis
of x by 4, so that at the entrances of parallels to the axis of y