Green’s Theorem with No Differentiability

Isidore Fleishcher, Centre de recherches mathematiques, Universite de Montreal, C.P. 6128, succ. Centre-ville, Montreal, QC H3C 3J7,Canada. Green’s Theorem with NO Differentiability. The result is established for a Jordan measurable region with locally connected rectifiable boundary. The integrand F for the new plane integral to be used is a function of axisparallel rectangles, finitely additive on non-overlapping ones, hence unambiguously defined and additive on ”figures” (i.e. finite unions of axis-parallel rectangles). Define its integral over Jordan measurable S as the limit of its value on the figures, which contain a subfigure of S and are contained in a figure containing S, as the former/complements of the latter expand directedly to fill out S/the complement of S. The integral over every Jordan measurable region exists when additive F is ”absolutely continuous” in the sense of converging to zero as the area enclosed by its argument does, or with F the circumferential line integral ∮ P dx + Q dy for P , Q continuous at the locally connected rectifiable boundary of S and integrable along axis parallel line segments. Thus the equality of this area integral with the line integral around the boundary, to be proved, follows for the various integrals of divergence presented in: Pfeffer, W.F. The Riemann Approach to Integration, Cambridge Univ. Press, New York, 1993. In Advanced Calculus texts Green’s Theorem is presented for continuous vector fields with continuous first partial derivatives, defined in a region containing a simple piecewise smooth curve enclosing an area of not too complicated shape. More careful treatments— e.g. [A, §10–14]—dispense with the continuity of the derivatives in favour of their (bounded existence and) integrability over the interior; recently this requirement has been successively weakened further to integrability of the partials in the “generalized Riemann” sense [McL, §7.12] and beyond to “gage integrability” [Pf] which even follows from the mere existence of the derivative. By modifying this last integral further, it proves possible to obtain the theorem for a continuous vector field with no differentiability assumption whatsoever. 1991 Mathematics Subject Classification. 26B20 26A39 28A75. Typeset by AMS-TEX 1 The “integral” of an additive rectangle function over a Jordan measurable set For plane Jordan content, see [A, §10-4], [K]: Inner Jordan content of a bounded set S is the sup of areas of finite unions of axis-parallel rectangles in the interior of S; outer Jordan content, the inf of areas of such which meet closure of S. Their equality, “Jordan measurability”, comes to the boundary of S having Jordan content (equivalently, by compactness, Lebesgue measure) zero. Let F be a function of axis-parallel rectangles, finitely additive on non-overlapping ones, hence unambiguously defined and additive on “figures” (i.e. finite unions of such). We propose to define its integral over S as the common limit of its values, on partition’s (into axis-parallel rectangles of a rectangle containing S) smallest figure which contains the closure of S or largest contained in its interior, as the partition is refined. The integral over every Jordan measurable set exists when additive F is “absolutely continuous” in the sense of converging to zero as the area enclosed by its argument does—indeed, the difference between a containing and a contained figure of S eventually has small area and so the value of F on the difference would be small, hence the values of F on these figures eventually close. This obtains for the more usual kinds of area integral of a bounded integrand. However, independent of this continuity, the circumferential line integral ∮ P dx + Q dy, for P , Q integrable along axis-parallel line segments and continuous at the boundary if rectifiable and locally connected, construed as a rectangle function, will now be shown integrable in this sense to the usual line integral around the boundary as value. equality of this area integral with the line integral rectifiable boundary, about to be proved, can be used to 2 Green’s Theorem for Jordan regions with locally connected rectifiable boundaries Local connectedness is postulated to enable one to cover the boundary Γ with nonoverlapping axis-parallel rectangles of sufficiently small side length (finitely many by compactness) each of which encloses only a connected piece of Γ which enters and leaves the rectangle through the shorter opposite sides (if part of Γ is a horizontal or vertical interval, count this as a degenerate rectangle). Thus the sum of their perimeters is at most four times the length of Γ. Construe the boundary rectangle sides in the bounded component of S as the boundary of a figure in the interior of S, those in the unbounded component as the boundary of a cover. It is classical that the line integral of a continuous vector-valued function along a rectifiable curve Γ exists [A, §10-10]. Since a constant vector integrates around a closed curve to zero, every circumferential line integral is bounded by the oscillation of the vector integrand times the arc length of the circumference. integrand, Adding up the line integrals around the circumferences (of pieces of Γ and adjacent rectangle sides) shows the line integral along Γ close to that along each of the boundaries — and of course along boundaries derived from any smaller rectangle cover of Γ, which thus bound larger interior figures and smaller covering figures of S — hence equal to the above defined surface integral for the “flux”: i.e. the circumferential line integral construed as a rectangle function.