Posted on Categories:Real analysis, 实分析, 数学代写

# 数学代写|实分析代写Real Analysis代考|MATH450

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|实分析代写Real Analysis代考|Definition and Properties of Riemann Integral

The present section extends that development to several variables. A certain amount of the theory parallels what happened in one variable, and proofs for that part of the theory can be obtained by adjusting the notation and words of Section I.4 in simple ways. Results of that kind are much of the subject matter of this section.

In later sections we shall take up results having no close analog in Section I.4. The main results of this kind are
(i) a necessary and sufficient condition for a function to be Riemann integrable,
(ii) Fubini’s Theorem, concerning the relationship between multiple integrals and iterated integrals in the various possible orders,
(iii) a change-of-variables formula for multiple integrals.
We begin a discussion of these in the next section.
The one-variable theory worked with a bounded function $f:[a, b] \rightarrow \mathbb{R}$, with domain a closed bounded interval, and we now work with a bounded function $f: A \rightarrow \mathbb{R}$ with domain $A$ a “closed rectangle” in $\mathbb{R}^n$. For this purpose a closed rectangle (or “closed geometric rectangle”) in $\mathbb{R}^n$ is a bounded set of the form
$$A=\left[a_1, b_1\right] \times \cdots \times\left[a_n, b_n\right]$$
with $a_j \leq b_j$ for all $j$. Let us abbreviate $\left[a_j, b_j\right]$ as $A_j$. In geometric terms the sides or faces are assumed parallel to the axes or coordinate hyperplanes. We shall use the notion of open rectangle in later sections and chapters, an open rectangle being a similar product of bounded open intervals $\left(a_j, b_j\right)$ for $1 \leq j \leq n$. However, in this section the term “rectangle” will always mean closed rectangle.

## 数学代写|实分析代写Real Analysis代考|Riemann Integrable Functions

Let $E$ be a subset of $\mathbb{R}^n$. We say that $E$ is of measure 0 if for any $\epsilon>0, E$ can be covered by a finite or countably infinite set of closed rectangles in the sense of Section 7 of total volume less than $\epsilon$. It is equivalent to require that $E$ can be covered by a finite or countably infinite set of open rectangles of total volume less than $\epsilon$. In fact, if a system of open rectangles covers $E$, then the system of closures covers $E$ and has the same total volume; conversely if a system of closed rectangles covers $E$, then the system of open rectangles with the same centers and with sides expanded by a factor $1+\delta$ covers $E$ as long as $\delta>0$.

Several properties of sets of measure 0 are evident: a set consisting of one point is of measure 0 , a face of a closed rectangle is a set of measure 0 , and any subset of a set of measure 0 is of measure 0 . Less evident is the fact that the countable union of sets of measure 0 is of measure 0 . In fact, if $\epsilon>0$ is given and if $E_1, E_2, \ldots$ are sets of measure 0 , find finite or countably infinite systems $\mathcal{R}_j$ of closed rectangles for $j \geq 1$ such that the total volume of the members of $\mathcal{R}_j$ is $<\epsilon / 2^n$. Then $\mathcal{R}=\bigcup_j \mathcal{R}_j$ is a system of closed rectangles covering $\bigcup_j E_j$ and having total volume $<\epsilon$.

The goal of this section is to prove the following theorem, which gives a useful necessary and sufficient condition for a function of several variables to be Riemann integrable. The theorem immediately extends from the scalar-valued case as stated to the case that $f$ has values in $\mathbb{R}^m$ or $\mathbb{C}^m$.

## 数学代写|实分析代写Real Analysis代考|Definition and Properties of Riemann Integral

(1)函数为Riemann可积的充分必要条件;
(ii)关于多重积分和不同阶次的迭代积分之间关系的富比尼定理;
(iii)多重积分的变量变换公式。

$$A=\left[a_1, b_1\right] \times \cdots \times\left[a_n, b_n\right]$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。