Extension and Application of Deleting Items and Disturbing Mesh Theorem of Riemann Integral
This is a theoretical extension of known theorems in integral calculus, likely of interest to mathematicians working in analysis and differential geometry, but it appears incremental as it generalizes existing results without introducing new concepts or applications.
The paper extends the deleting items and disturbing mesh theorems from Riemann integral to multiple, line, and surface integrals, and derives analogous formulae for Green's, Stokes', and divergence theorems, as well as the general Stokes' theorem on differential manifolds.
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.