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

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.

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## 1 Introduction

The deleting items theorem and disturbing mesh theorem of Riemann Integral and the combination of deleting items and disturbing theorem of Riemann Integral are proposed for the one-dimensional Riemann Integral in [1]. We aim to extend the deleting item and disturbing mesh theorems to multiple integral,line integral and surface integral in this paper. Then, we develop the deleting items and disturbing mesh formulae for Green’s theorem, Stokes’ theorem and divergence theorem (Gauss’s or Ostrogradsky ’s theorem). At last, we derive the deleting items and disturbing mesh theorem of general Stokes’ theorem in manifold analysis.

As Green’s theorem, Stokes’ theorem and divergence theorem have strong physics and engineering background. Our researches not only discuss the integral based on Riemann sum in mathematical analysis, but also show the uncertainty of physical measurement based on integral in the sense of limit theory especially in computation simulation with finite partition, and reveal the complexity of mathematical modeling from the existences of numerous different physical formulas in finite status approaching the same classical physical limit formulas and obeying same theory.

## 2 Deleting Items and Disturbing Mesh Theorem of Multiple Integral

There is no uniform expression on multiple integral, the integral intervals can be , , general intervals (where is bounded, unbounded, open, closed, half open in ), or Jordan measurable set [2,3,4,5]. Definition 1 refers to [3].

Definition 1 (Multiple Integral) Let , . Let be a real-valued and bounded function. Let intervals be partition of induced by partitions of the coordinate intervals , such that , and no two of the intervals have common interior points. Denote , The mesh (the maximum among the diameters of the intervals of the partition P). For any distinguished points , , . The sum

 σ(f,P,ξ):=m∑k=1f(ξk)|Ik|=m∑k=1f(ξk)m(Ik) (1)

is called Riemann sum of corresponding to the partition of interval with distinguished points , where is the measure of .

 ∫If(x)dx:=limλ(P)→0σ(f,P,ξ), (2)

Provides the limit exists, it is called the multiple Riemann integral of function over the interval and is integrable. Or,

 ∫If(x)dx=∫⋯∫If(x1,x2,…,xn)dx1⋯dxn:=limλ(P)→0m∑k=1f(ξk)m(Ik). (3)

Particularly, if , is called equal partition, though may not be congruent figures.

The deleting items and disturbing mesh theorems of Multiple Integral are as follows.

Theorem 1 (Deleting Items Theorem of Multiple Integral) Let , . Let be a real-valued and bounded function. Let intervals be partition of induced by partitions of the coordinate intervals , such that , and no two of the intervals have common interior points. Denote , the mesh (the maximum among the diameters of the intervals of the partition P). For any distinguished points , , . Denote . Then,

• For a fixed natural number , denote , and .222Given , there are index sets of . Then,

 limλ(P)→0∑k∈J∖JKf(ξk)m(Ik)=∫If(x)dx. (4)

where is the measure of .

• If is an equal partition, for any natural number , satisfying , denote , and . 333Given and , there are index sets of . Then,

 limλ(P)→0∑k∈J∖JK(m)f(ξk)m(Ik)=∫If(x)dx. (5)

where is the measure of .

Proof Since is bounded, , so that . As is integrable over , , , while ,

 |m∑k=1f(ξk)m(Ik)−∫If(x)dx|<ϵ.

Since . For above , , while ,

 |max1≤k≤m{m(Ik)}|<ϵ.

(1) Since

 |∑k∈J∖JKf(ξk)m(Ik)−∫If(x)dx|=|m∑k=1f(ξk)m(Ik)−∫If(x)dx−∑k∈JKf(ξk)m(Ik)|≤|m∑k=1f(ξk)m(Ik)−∫If(x)dx|+|∑k∈JKf(ξk)m(Ik)|≤ε+KMλ(P)≤ε+KMε=(1+KM)ε

Then,

 limλ(P)→0∑k∈J∖JKf(ξk)m(Ik)=∫If(x)dx

(2) As

 |∑k∈J∖JK(m)f(ξk)m(Ik)−∫If(x)dx|=|m∑k=1f(ξk)m(Ik)−∫If(x)dx+∑k∈JK(m)f(ξk)m(Ik)|≤|m∑k=1f(ξk)m(Ik)−∫If(x)dx|+|∑k∈JK(m)f(ξk)m(Ik)|≤ε+M∑k∈JK(m)m(Ik)=ε+Mm(I)∑k∈JK(m)m(Ik)m(I)≤ε+Mm(I)K(m)m=(1+Mm(I))ε

Then,

 limλ(P)→0∑k∈J∖JK(m)f(ξk)m(Ik)=∫If(x)dx

Theorem 2 (Disturbing Mesh Theorem of Multiple Integral) Let , . Let be a real-valued and bounded function. Let intervals be partition of induced by partitions of the coordinate intervals , such that , and no two of the intervals have common interior points. Denote , The mesh (the maximum among the diameters of the intervals of the partition P). For any distinguished points , , .

Let intervals be distorted or inaccuracy grid of respectively, and , , . If,

 limm→+∞m∑k=1m(Ik△~Ik)=0 (6)

Then,

 limλ(~P)→0m∑k=1f(ξk)m(~Ik)=∫If(x)dx. (7)

where is the measure of .

Proof Since

 |m∑k=1f(ξk)m(~Ik)−∫If(x)dx|=|m∑k=1f(ξk)m(~Ik)−m∑k=1f(ξk)m(Ik)+m∑k=1f(ξk)m(Ik)−∫If(x)dx|≤|m∑k=1f(ξk)m(~Ik)−m∑k=1f(ξk)m(Ik)|+|m∑k=1f(ξk)m(Ik)−∫If(x)dx|=|m∑k=1f(ξk)(m(~Ik)−m(Ik))|+|m∑k=1f(ξk)m(Ik)−∫If(x)dx|≤Mm∑k=1(m(~Ik△Ik))+|m∑k=1f(ξk)m(Ik)−∫If(x)dx|

and

 limm→+∞m∑k=1m(Ik△~Ik)=0,  limm→+∞m∑k=1f(ξk)m(Ik)=∫If(x)dx

Then,

 limm→+∞m∑k=1f(ξk)m(~Ik)=∫If(x)dx

This ends the proof.

Note that may not be defined on , though can be extended on it according to the values on intersect boundary. To avoid overelaborate discussion, we only consider . For convenience, we assume that is defined on . And the distorted grid may not satisfy the two conditions: no two of the intervals have common interior points and . In addition,

 limm→+∞m∑k=1m(Ik△~Ik)=0

have concrete background. For example,

Example 1. , and , .

Example 2. , and , .

Theorem 3 (Deleting Items and Disturbing Mesh Theorem of Multiple Integral) Let , . Let be a real-valued and bounded function. Let intervals be partition of induced by partitions of the coordinate intervals , such that , and no two of the intervals have common interior points. Denote , The mesh (the maximum among the diameters of the intervals of the partition P). For any distinguished points , , .

Let intervals be disturbed or inaccuracy grid of respectively, and , , . If,

 limm→+∞m∑k=1m(Ik△~Ik)=0 (8)

Denote . Then,

• For a fixed natural number , denote , and . Then,

 limλ(~P)→0∑k∈J∖JKf(ξk)m(~Ik)=∫If(x)dx. (9)

where is the measure of .

• If is an equal partition. For any natural number , satisfying , denote , and . Then,

 limλ(~P)→0∑k∈J∖JK(m)f(ξk)m(~Ik)=∫If(x)dx. (10)

where is the measure of .

Proof (1) Since

 limm→+∞m∑k=1m(Ik△~Ik)=0

Then,

 limλ(P)→0λ(~P)=0

According to Theorem 1 and 2, we obtain

 limλ(~P)→0∑k∈J∖JKf(ξk)m(~Ik)=limλ(P)→0∑k∈J∖JKf(ξk)m(Ik)=limλ(P)→0m∑k=1f(ξk)m(Ik)=∫If(x)dx

(2) According to Theorem 1 and 2, we obtain

 limλ(~P)→0∑k∈J∖JK(m)f(ξk)m(~Ik)=limλ(P)→0∑k∈J∖JK(m)f(ξk)m(Ik)=limλ(P)→0m∑k=1f(ξk)m(Ik)=∫If(x)dx.

Note: Theorem 1,2,3 can be easily extended to the integral domain of Jordan-measurable set in .

## 3 Deleting Items and Disturbing Mesh Theorem of Line Integral

There are two types of linear integral: first type linear integral and second type linear integral, or scalar linear integral and vector linear integral [5,7]. Definition 2 and 3 refer to [7].

Definition 2 (Scalar Line Integral) Let be a path of class , and be a continuous function whose domain contains the image of (so that the composite is defined). Let be a partition of . Let be an arbitrary points in the -th subinterval of the partition, . Then we consider the Riemann sum

 m∑k=1f(x(t∗k))△sk, (11)

where is the length of the -th segment of (i.e. the portion of defined for ). If the limit of Riemann sum exists as , we define the limit as the scalar line integral

 ∫xfds=limmax{△tk}→0m∑k=1f(x(t∗k))△sk=∫baf(x(t))∥x′(t)∥dt. (12)

where .

Definition 3 (Vector Line Integral) Let be a path of class (). Let be a vector field defined on such that contains the image of . Assume that varies continuously along . Let be a partition of . Let be an arbitrary points in the -th subinterval of the partition, . Then we consider the Riemann sum

 m∑k=1F(x(t∗k))⋅△sk, (13)

where . If the limit of Riemann sum exists as , we define the limit as vector line integral

 (14)

where .

The deleting items and disturbing mesh theorems of scalar line integral are as follows.

Theorem 4 (Deleting Items Theorem of Scalar Line Integral) Let be a path of class . be a continuous function whose domain contains the image of ,and the scalar line integral of over exists. Let be a partition of , , . Let be an arbitrary points in . . Denote . Then,

• For a fixed natural number , denote , and . Then,

 limmax{△tk}→0∑k∈J∖JKf(x(t∗k))△sk=∫xfds. (15)
• If is an equal partition, for any natural number , satisfying , denote , and . Then,

 limmax{△tk}→0∑k∈J∖JK(m)f(x(t∗k))△sk=∫xfds. (16)

where , .

Theorem 5 (Disturbing Mesh Theorem of Scalar Line Integral) Let be a path of class . be a continuous function whose domain contains the image of , and the scalar line integral of over exists.

Let be a partition of , , . Let be an arbitrary points in . .

Let be a distorted partition of , , , , satisfying

 limm→+∞m∑k=1m(Ik△~Ik)=0

Then,

 limmax{△~tk}→0m∑k=1f(x(t∗k))△~sk=∫xfds. (17)

where , .

Theorem 6 (Deleting Items and Disturbing Mesh Theorem of Scalar Line Integral) Let be a path of class . be a continuous function whose domain contains the image of , and the scalar line integral of over exists.

Let be a partition of , , . Let be an arbitrary points in . .

Let be a distorted partition of , , , , satisfying

 limm→+∞m∑k=1m(Ik△~Ik)=0

Denote .Then,

• For a fixed natural number , denote , and . Then,

 limmax{△~tk}→0∑k∈J∖JKf(x(t∗k))△~sk=∫xfds. (18)
• If is an equal partition. For any natural number , satisfying , denote , and . Then,

 limmax{△~tk}→0∑k∈J∖JK(m)f(x(t∗k))△~sk=∫xfds. (19)

where , , .

The deleting items and disturbing mesh theorems of vector line integral are as follows.

Theorem 7 (Deleting Items Theorem of Vector Line Integral) Let be a path of class (). Let be a vector field defined on such that contains the image of . Assume that varies continuously along .

Let be a partition of , , . Let be an arbitrary points in . . Denote . Then,

• For a fixed natural number , denote , and . Then,

 (20)
• If is an equal partition, for any natural number , satisfying , denote , and . Then,

 (21)

where , .

Theorem 8 (Disturbing Mesh Theorem of Vector Line Integral) Let be a path of class (). Let be a vector field defined on such that contains the image of . Assume that varies continuously along ,and the vector line integral of over exists.

Let