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.

296

General theorems concerning the double integral. Bk. III. cb. VIII.

Second: when f(x, y) in separate points or in separate lines
(oc1 places) is discontinuous or indeterminate between finite limits,
or again, when it has infinitely many maxima and minima with finite
fluctuations.
Third: when f(x, y) in infinitely many lines (oo2 places) is
finitely discontinuous or indeterminate or fluctuates infinitely often, but
when the sum of the superficial elements in which the fluctuations
D,, exceed an arbitrarily small number 6, can be made arbitrarily small.
This third requirement, which embraces the first two, suggests an
extension of the distinction established by Cantor of sets of points
in a domain of two dimensions, rising from linear sets to plane
sets. Infinitely many lines do not give rise to a plane set of points
when their initial elements form a discrete set of points; on the other
hand we do obtain a plane set of points, when the initial elements
of the lines belong to a linear set or mass. The following investigation, however, is restricted to functions that satisfy the first
or second requirement and the third possibility will only be cited
incidentally.
The limiting value of the sum:
Lim tf(r1) ~ r +f(v2) ' 2 + * * + -f(rn).n } for n == o
is usually denoted by:
(2)
f (x, y)dxddy, or: f(x, y)dT,
and is called the definite double integral in the domain T.
The definite double integral, as well as the simple integral, admits
of a geometrical interpretation. If we lay off the value of the function
z = f(x, y) perpendicular to the plane xy, the integral:
r(2)
f(x, y)d T
expresses the volume of the cylinder whose base is the area enclosed
by the curve (p(x, y) and that is bounded above the plane xy by the
surface z = f(x, y). )
Among the theorems resulting from this definition of the double
integral we only notice specially the following.
1. When (p(x, y) and ip(x, y) are any finite integrable functions
within a certain domain, their product also is integrable within the same
domain. (Proof as in ~ 146.)
2. The First Theorem of the Mean Value: When the
*) By the geometrical problems: the determination of volume and the
measurement of curved surfaces, the analytical conception of the double integral
was introduced. Riemann's investigations on the definite integral established
the fundamental principle both for the double and also for multiple integrals.