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

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## 数学代考|微分流形代考Differential Manifold代写|Tensor fields

We now specialize the previous considerations to the special situation when $E$ is the tangent bundle of $M, E \cong T M$. The cotangent bundle is then
$$T^* M:=(T M)^* .$$
We define the tensor bundles of $M$
$$\mathcal{T}_s^r(M):=\mathcal{T}_s^r(T M)=(T M)^{\otimes r} \otimes\left(T^* M\right)^{\otimes s} .$$
Definition 2.3.4. (a) A tensor field of type $(r, s)$ over the open set $U \subset M$ is a section of $\mathcal{T}_s^r(M)$ over $U$.

(b) A degree $r$ differential form (r-form for brevity) is a section of $\Lambda^r\left(T^* M\right)$. The space of (smooth) $r$-forms over $M$ is denoted by $\Omega^r(M)$. We set
$$\Omega^{\bullet}(M):=\bigoplus_{r \geq 0} \Omega^r(M) .$$
(c) A Riemannian metric on a manifold $M$ is a metric on the tangent bundle. More precisely, it is a symmetric $(0,2)$-tensor $g$, such that for every $x \in M$, the bilinear map
$$g_x: T_x M \times T_x M \rightarrow \mathbb{R}$$
defines a Euclidean metric on $T_x M$.

## 数学代考|微分流形代考Differential Manifold代写|Fiber bundles

We consider useful at this point to bring up the notion of fiber bundle. There are several reasons to do this.

On one hand, they arise naturally in geometry, and they impose themselves as worth studying. On the other hand, they provide a very elegant and concise language to describe many phenomena in geometry.

We have already met examples of fiber bundles when we discussed vector bundles. These were “smooth families of vector spaces”. A fiber bundle wants to be a smooth family of copies of the same manifold. This is a very loose description, but it offers a first glimpse at the notion about to be discussed.

The model situation is that of direct product $X=F \times B$, where $B$ and $F$ are smooth manifolds. It is convenient to regard this as a family of manifolds $\left(F_b\right)_{b \in B}$. The manifold $B$ is called the base, $F$ is called the standard (model) fiber, and $X$ is called the total space. This is an example of trivial fiber bundle.

In general, a fiber bundle is obtained by gluing a bunch of trivial ones according to a prescribed rule. The gluing may encode a symmetry of the fiber, and we would like to spend some time explaining what do we mean by symmetry.

## 数学代考|微分流形代考Differential Manifold代写|Tensor fields

$$T^* M:=(T M)^* .$$

$$\mathcal{T}s^r(M):=\mathcal{T}_s^r(T M)=(T M)^{\otimes r} \otimes\left(T^* M\right)^{\otimes s} .$$ 定义 2.3.4。 (a) 尖型的张荲场 $(r, s)$ 在开集上 $U \subset M$ 是一部分 $\mathcal{T}_s{ }^r(M)$ 超过 $U$. (b) 学位 $r$ 微分形式 (为简洁起见, $r$ 形式) 是 $\Lambda^r\left(T^* M\right)$. (平滑) 的空间 $r$-表格结束 $M$ 表示为 $\Omega^r(M)$. 我们设置 $$\Omega^{\bullet}(M):=\bigoplus{r \geq 0} \Omega^r(M) .$$
(c) 流形上的黎曼度量 $M$ 是切从上的度量。更准确地说，它是一个对称的 $(0,2)$-张量 $g$, 这样对于每个 $x \in M$ ，双线性映射
$$g_x: T_x M \times T_x M \rightarrow \mathbb{R}$$

## 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。

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

In this subsection we want to explain rigorously a phenomenon with which the reader may already be intuitively acquainted. We describe it first in a special case.

Suppose $M$ is a submanifold of dimension 2 in $\mathbb{R}^3$. Then, a simple thought experiment suggests that most horizontal planes will not be tangent to $M$. Equivalently, if we denote by $f$ the restriction to $M$ of the function $(x, y, z) \mapsto z$, then for most real numbers $h$ the level set $f^{-1}(h)=M \cap{z=h}$ does not contain a point where the differential of $f$ is zero, so that most level sets $f^{-1}(h)$ are smooth submanifolds of $M$ of codimension 1 , i.e., smooth curves on $M$.
We can ask a more general question. Given two smooth manifolds $X, Y$, a smooth map $f: X \rightarrow Y$, is it true that for “most” $y \in Y$ the level set $f^{-1}(y)$ is a smooth submanifold of $X$ of codimension $\operatorname{dim} Y$ ? This question has a positive answer, known as Sard’s theorem.
Definition 2.1.14. Suppose that $Y$ is a smooth, connected manifold of dimension $m$.
(a) We say that a subset $S \subset Y$ is negligible if, for any coordinate chart of $Y, \Psi: U \rightarrow \mathbb{R}^m$, the set $\Psi(S \cap U) \subset \mathbb{R}^m$ has Lebesgue measure zero in $\mathbb{R}^m$.
(b) Suppose $F: X \rightarrow Y$ is a smooth map, where $X$ is a smooth manifold. A point $x \in X$ is called a critical point of $F$, if the differential $D_x F: T_x X \rightarrow T_{F(x)} Y$ is not surjective.

We denote by $C r_F$ the set of critical points of $F$, and by $\Delta_F \subset Y$ its image via $F$. We will refer to $\Delta_F$ as the discriminant set of $F$. The points in $\Delta_F$ are called the critical values of $F$.

## 数学代考|微分流形代考Differential Manifold代写|Vector bundles

The tangent bundle $T M$ of a manifold $M$ has some special features which makes it a very particular type of manifold. We list now the special ingredients which enter into this special structure of $T M$ since they will occur in many instances. Set for brevity $E:=T M$, and $F:=\mathbb{R}^m$ $(m=\operatorname{dim} M)$. We denote by $\operatorname{Aut}(F)$ the Lie group $\mathrm{GL}(n, \mathbb{R})$ of linear automorphisms of $F$. Then
(a) $E$ is a smooth manifold, and there exists a surjective submersion $\pi: E \rightarrow M$. For every $U \subset M$ we set $\left.E\right|U:=\pi^{-1}(U)$ (b) From (2.1.4) we deduce that there exists a trivializing cover, i.e., an open cover $\mathcal{U}$ of $M$, and for every $U \in \mathcal{U}$ a diffeomorphism $$\Psi_U:\left.E\right|_U \rightarrow U \times F, \quad v \mapsto\left(p=\pi(v), \Phi_p^U(v)\right)$$ (b1) $\Phi_p$ is a diffeomorphism $E_p \rightarrow F$ for any $p \in U$ (b2) If $U, V \in \mathcal{U}$ are two trivializing neighborhoods with non empty overlap $U \cap V$ then, for any $p \in U \cap V$, the $\operatorname{map} \Phi{V U}(p)=\Phi_p^V \circ\left(\Phi_p^U\right)^{-1}: F \rightarrow F$ is a linear isomorphism, and moreover, the map
$$p \mapsto \Phi_{V U}(p) \in \operatorname{Aut}(F)$$
is smooth.
In our special case, the map $\Phi_{V U}(p)$ is explicitly defined by the matrix (2.1.4)
$$A(p)=\left(\frac{\partial y^j}{\partial x^i}(p)\right)_{1 \leq i, j \leq m}$$

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

（b）假设 $F: X \rightarrow Y$ 是 个光滑的地图，其中 $X$ 是 个光滑的流形。一个点 $x \in X$ 称为临界点 $F$ ，如果微分 $D_x F: T_x X \rightarrow T_{F(x)} Y$ 不是主观的。

## 数学代考|微分流形代考Differential Manifold代写|Vector bundles

（一） $E$ 是一个光滑流形，并且存在一个满射浸没 $\pi: E \rightarrow M$. 对于每一个 $U \subset M$ 我们设置 $E \mid U:=\pi^{-1}(U)(\mathrm{b})$ 从(2.1.4)我 们推昌出存在一个平凡鞜盖，即一个开覆盖 $\mathcal{U}$ 的 $M$ ，并且对于每个 $U \in \mathcal{U}$ 微分同胚
$$\Psi_U:\left.E\right|U \rightarrow U \times F, \quad v \mapsto\left(p=\pi(v), \Phi_p^U(v)\right)$$ (b1) $\Phi_p$ 是微分同顺 $E_p \rightarrow F$ 对于任何 $p \in U(\mathrm{~b} 2)$ 如果 $U, V \in \mathcal{U}$ 是两个具有非空重殒的平凡邻域 $U \cap V$ 那么，对于佳何 $p \in U \cap V ，$ 这map $\Phi V U(p)=\Phi_p^V \circ\left(\Phi_p^U\right)^{-1}: F \rightarrow F$ 是 个线性同构，而且，映射 $$p \mapsto \Phi{V U}(p) \in \operatorname{Aut}(F)$$

$$A(p)=\left(\frac{\partial y^j}{\partial x^i}(p)\right)_{1 \leq i, j \leq m}$$

## 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。

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

## avatest™帮您通过考试

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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。