Posted on Categories:Differential Manifold, 微分流形, 数学代写

# 数学代考|微分流形代考Differential Manifold代写|MATH625 Preliminaries

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## 数学代考|微分流形代考Differential Manifold代写|Space and Coordinatization

Mathematics is a natural science with a special modus operandi. It replaces concrete natural objects with mental abstractions which serve as intermediaries. One studies the properties of these abstractions in the hope they reflect facts of life. So far, this approach proved to be very productive.

The most visible natural object is the Space, the place where all things happen. The first and most important mathematical abstraction is the notion of number. Loosely speaking, the aim of this book is to illustrate how these two concepts, Space and Number, fit together.

It is safe to say that geometry as a rigorous science is a creation of ancient Greeks. Euclid proposed a method of research that was later adopted by the entire mathematics. We refer of course to the axiomatic method. He viewed the Space as a collection of points, and he distinguished some basic objects in the space such as lines, planes etc. He then postulated certain (natural) relations between them. All the other properties were derived from these simple axioms.

Euclid’s work is a masterpiece of mathematics, and it has produced many interesting results, but it has its own limitations. For example, the most complicated shapes one could reasonably study using this method are the conics and/or quadrics, and the Greeks certainly did this. A major breakthrough in geometry was the discovery of coordinates by René Descartes in the 17th century. Numbers were put to work in the study of Space.

Descartes’ idea of producing what is now commonly referred to as Cartesian coordinates is familiar to any undergraduate. These coordinates are obtained using a very special method (in this case using three concurrent, pairwise perpendicular lines, each one endowed with an orientation and a unit length standard. What is important here is that they produced a one-to-one mapping
Euclidian Space $\rightarrow \mathbb{R}^3, \quad P \longmapsto(x(P), y(P), z(P))$

## 数学代考|微分流形代考Differential Manifold代写|The implicit function theorem

We gather here, with only sketchy proofs, a collection of classical analytical facts. For more details one can consult [34].

Let $X$ and $Y$ be two Banach spaces and denote by $L(X, Y)$ the space of bounded linear operators $X \rightarrow Y$. For example, if $X=\mathbb{R}^n, Y=\mathbb{R}^m$, then $L(X, Y)$ can be identified with the space of $m \times n$ matrices with real entries. For any set $S$ we will denote by $\mathbb{1}_S$ the identity map $S \rightarrow S$.

Definition 1.1.1. Let $F: U \subset X \rightarrow Y$ be a continuous function ( $U$ is an open subset of $X$ ). The map $F$ is said to be (Fréchet) differentiable at $u \in U$ if there exists $T \in L(X, Y)$ such that
$$\left|F\left(u_0+h\right)-F\left(u_0\right)-T h\right|_Y=o\left(|h|_X\right) \text { as } h \rightarrow 0,$$
i.e.,
$$\lim _{h \rightarrow 0} \frac{1}{|h|_X}\left|F\left(u_0+h\right)-F\left(u_0\right)-T h\right|_Y=0 .$$
Loosely speaking, a continuous function is differentiable at a point if, near that point, it admits a “best approximation” by a linear map.

When $F$ is differentiable at $u_0 \in U$, the operator $T$ in the above definition is uniquely determined by
$$T h=\left.\frac{d}{d t}\right|{t=0} F\left(u_0+t h\right)=\lim {t \rightarrow 0} \frac{1}{t}\left(F\left(u_0+t h\right)-F\left(u_0\right)\right) .$$
We will use the notation $T=D_{u_0} F$ and we will call $T$ the Fréchet derivative of $F$ at $u_0$.
Assume that the map $F: U \rightarrow Y$ is differentiable at each point $u \in U$. Then $F$ is said to be of class $C^1$, if the map $u \mapsto D_u F \in L(X, Y)$ is continuous. $F$ is said to be of class $C^2$ if $u \mapsto D_u F$ is of class $C^1$. One can define inductively $C^k$ and $C^{\infty}$ (or smooth) maps.

## 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。