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The present paper concerns
the difficulty which one encounters in text books of Advanced Calculus of giving a
simple and elementary definition of area of a smooth nonparametric surface in En+1
such that, within the same elmentary framework, one can then prove that the area so
defined is equal to the classical area integral.
The authors were first made aware of the considerable interest of such a
task in 1955 with the publication of Angus Taylor’s now classic textbook
“Advanced Calculus”. The following statement is taken from page 384 of this
book:
“It is logically and aesthetically desirable to have a definition of surface area
which is directly geometric, and which does not put too many restrictions on the
surface. A good definition ought not to depend upon the method of representing the
surface analytically, and should not be limited to smooth surfaces. The demand for
such a definition poses a very difficult problem, however. It may surprise the student
to know that the problem has occupied the attention of many able mathematicians
over the last fifty years, and that the end of research on the question is not yet in
sight.”
In the present paper we present an idea which seems to answer the questions
raised by Angus Taylor for surfaces S : z = f(x1,⋯,xn), which are continuous with
their first order partial derivatives. The idea is to develop a scheme for the
construction of sequences of suitably chosen polyhedra inscribed within the given
surface, such that the corresponding sequences of the polyhedral areas converge to
the classical area integral for the surface, and hence to the Lebesgue area of
S.
In previous papers [1], [7] we discussed our definition of area for surfaces
S : z = f(x1,x2). In [7] we took in consideration surfaces z = f(x1,x2) with f
continuous with its first order partial derivatives. In [1] we gave a necessary and
sufficient condition in order that for a surface z = f(x1,x2) there are sequences
of inscribed polyhedra satisfying the requirements of our definitions (see
[1]).
J. A. Serret [6] in 1868 proposed a geometric definition of area, but H. A.
Schwartz [5] in 1882 proved that Serret’s definition was incorrect. Other geometric
definitions of area and constructions have been proposed, and we mention here for
example the ones of S. Kempisty [3] for surfaces S: z = f(x1,x2) with f absolutely
continuous in the sense of Tonelli. For general expositions concerning area, in
particular, Le261 besgue area, we refer to the well known texts of T. Rado [4] and L.
Cesari [2].
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