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# 数学代写|实分析代写Real Analysis代考|MATH450

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## 数学代写|实分析代写Real Analysis代考|Partitions of Unity

In Section 10 we shall use a “partition of unity” in proving a change-of-variables formula for multiple integrals. As a general matter in analysis, a partition of unity serves as a tool for localizing analysis problems to a neighborhood of each point. The result we shall use in Section 10 is as follows.

Proposition 3.14. Let $K$ be a compact subset of $\mathbb{R}^n$, and let $\left{U_1, \ldots, U_k\right}$ be a finite open cover of $K$. Then there exist continuous functions $\varphi_1, \ldots, \varphi_k$ on $\mathbb{R}^n$ with values in $[0,1]$ such that
(a) each $\varphi_i$ is 0 outside of some compact set contained in $U_i$,
(b) $\sum_{i=1}^k \varphi_i$ is identically 1 on $K$.
REMARKS. The system $\left{\varphi_1, \ldots \varphi_k\right}$ is an instance of a “partition of unity.” For a general metric space $X$, a partition of unity is a family $\Phi$ of continuous functions from $X$ into $[0,1]$ with sum identically 1 such that for each point $x$ in $X$, there is a neighborhood of $x$ where only finitely many of the functions are not identically 0 . The side condition about neighborhoods ensures that the sum $\sum_{\varphi \in \Phi} \varphi(x)$ has only finitely many nonzero terms at each point and that arbitrary partial sums are well-defined continuous functions on $X$. If $\mathcal{U}$ is an open cover of $X$, the partition of unity is said to be subordinate to the cover $\mathcal{U}$ if each member of $\Phi$ vanishes outside some member of $\mathcal{U}$. Further discussion of partitions of unity beyond the present setting appears in the problems at the end of Chapter $\mathrm{X}$. The use of partitions of unity involving continuous functions tends to be good enough for applications to integration problems, but applications to partial differential equations and smooth manifolds are often aided by partitions of unity involving smooth functions, rather than just continuous functions. ${ }^1$

## 数学代写|实分析代写Real Analysis代考|Inverse and Implicit Function Theorems

The Inverse Function Theorem and the Implicit Function Theorem are results for working with coordinate systems and for defining functions by means of solving equations. Let us use the latter application as a device for getting at the statements of both the theorems.

In the one-variable situation we are given some equation, such as $x^2+y^2=$ $a^2$, and we are to think of solving for $y$ in terms of $x$, choosing one of the possible $y$ ‘s for each $x$. For example, one solution is $y=-\sqrt{a^2-x^2},-a<x<a$; unless some requirement like continuity is imposed, there are infinitely many such solutions. In one-variable calculus the terminology is that this solution is “defined implicitly” by the given equation. In terms of functions, the functions $F(x, y)=x^2+y^2-a^2$ and $y=f(x)=-\sqrt{a^2-x^2}$ are such that $F(x, f(x))$ is identically 0 . It is then possible to compute $d y / d x$ for this solution in two ways. Only one of these methods remains within the subject of one-variable calculus, namely to compute the “total differential” of $x^2+y^2-a^2$, however that is defined, and to set the result equal to 0 . One obtains $2 x d x+2 y d y=0$ with $x$ and $y$ playing symmetric roles. The declaration that $x$ is to be an independent variable and $y$ is to be dependent means that we solve for $d y / d x$, obtaining $d y / d x=-x / y$. The other way is more transparent conceptually but makes use of multivariable calculus: it uses the chain rule in two-variable calculus to compute $d / d x$ of $F(x, f(x))$ as the derivative of a composition, the result being set equal to 0 because $(d / d x) F(x, f(x))$ is the derivative of the 0 function. This second method gives $\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} f^{\prime}(x)=0$, with the partial derivatives evaluated where $(x, y)=(x, f(x))$. Then we can solve for $f^{\prime}(x)$ provided $\partial F / \partial y$ is not zero at a point of interest, again obtaining $f^{\prime}(x)=-x / y$. It is an essential feature of both methods that the answer involves both $x$ and $y$; the reason is that there is more than one choice of $y$ for some $x$ ‘s, and thus specifying $x$ alone does not determine all possibilities for $f^{\prime}(x)$.

In the general situation we have $m$ equations in $n+m$ variables. Some $n$ of the variables are regarded as independent, and we think in terms of solving for the other $m$. An example is
\begin{aligned} z^3 x+w^2 y^3+2 x y & =0, \ x y z w-1 & =0, \end{aligned}
with $x$ and $y$ regarded as the independent variables.

## 数学代写|实分析代写Real Analysis代考|Partitions of Unity

(a)每个$\varphi_i$在包含在$U_i$中的某个紧集之外为0;
(b) $\sum_{i=1}^k \varphi_i$与$K$相同。

## 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。