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.

~ 172-174.

Product of two simple integrals.

311

j7(x)dxi. p(Y)}dY - f(x) )(y)(x^t+i - X)(Y+i - YV)
s ax + A' -t s' + AA'.
Now since as the values of m and n increase, the quantities A and A'
converge to zero, it is evident that the left side actually represents
the limiting value of the double sum. This theorem is also true,
when the functions f(x) and (p(y) become infinite within the interval
of integration, provided each alone remains integrable. For if f(x)
become infinite for x = c, and p(y) for y = c', then by the theorem
just proved we have:
c- ce'-d'
jfi d( xf (y) dy = fJf(x) g (y) dxdy.
a a
As d and 6' converge to zero, the product on the left side passes
over, by hypothesis, into a determinate finite value that at the same
time represents the value of the double integral in the rectangle up
to the limits x= c, y = c'.
17i. For the simple definite integral the following theorem holds:
When F(x) is a known continuous function whose derived function
is integrable and coincides generally with a function f(x), we have:
f(x)dx = F(x) -F(a).
The value of the definite integral depends therefore only on the values
of the function F at the limits of integration. An analogue for the
double integral is presented in the Theorem of Green (1793 —1841))):
Concerning the reduction of a double integral ofa unique
function to simple integrals along the boundary curve.
In the plane xy let there be given a finite connected domain,
bounded by one or more closed curves. In Fig. 16 ex. gr. we suppose
the domain to consist of the part of the plane enclosed by the external
curve omitting the two areas bounded by the ovals; in it there are
therefore three closed boundary curves. Let a function f(x, y) be
given for all points within and on the boundaries of this domain and
let it be integrable within the domain. Such a function may, as in
~ 167, be continuous, although it is also possible that in a 'discrete"
multiplicity of curves it may become discontinuous, indeterminate or
I) An Essay on the application of mathematical analysis to the theories of
Electricity and Magnetism. Nottingham 1828; reprinted in Crelle Journ. f. Math.,
Vol. 39, 44, 47; and again in Mathematical papers of the late George Green.
London 1871. - RIiemann: Grundlagen fir eine allgemeine Theorie etc. ~ 7-9.