Extension and Application of Deleting Items and Disturbing Mesh Theorem of Riemann Integral

TL;DR

This paper extends the deleting items and disturbing mesh theorems of Riemann integrals to multiple and line integrals, and generalizes these concepts to theorems on differential manifolds, broadening their applicability.

Contribution

It introduces new incomplete sum sequences for various integrals and extends deleting items and disturbing mesh theorems to multiple integrals, line integrals, surface integrals, and general Stokes' theorem.

Findings

Extended theorems to multiple, line, and surface integrals.

Derived deleting items and disturbing mesh formulas for classical theorems.

Generalized theorems to differential manifolds.

Abstract

The deleting items and disturbing mesh theorems of Riemann Integral are extended to multiple integral,line integral and surface integral respectively by constructing various of incomplete Riemann sum and non-Riemann sum sequences which converge to the same limit of classical Riemann sum. And, the deleting items and disturbing mesh formulae of Green's theorem, Stokes' theorem and divergence theorem (Gauss's or Ostrogradsky 's theorem) are also deduced. Then, the deleting items and disturbing mesh theorems of general Stokes' theorem on differential manifold are also derived.