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# 数学代写|常微分方程代考Ordinary Differential Equations代写|Regular functions and maps in Euclidean space

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Regular functions and maps in Euclidean space

So far, the concepts introduced and the results proved relate to objects defined in open subsets of Euclidean space. In other words we have assumed that the coordinates in our analysis are kept unchanged throughout. But the study of a PDE frequently requires changing coordinates. The natural framework is then that of manifolds and fiber bundles. Here we are faced with an exposition problem: almost every concept introduced in this chapter and used in the sequel can be formulated within one of three categories: smooth, real-analytic, complex-analytic. It is convenient to follow a unified approach: throughout this chapter we shall use the adjective “regular” and the notation $\mathcal{R}$ for either $C^{\infty}, C^\omega$ or $O$ (the latter meaning holomorphic). When $\mathcal{R}=O$ the base field is $\mathbb{K}=\mathbb{C}$; in the other two cases $\mathbb{K}=\mathbb{R}$. Depending on the meaning of “regular” one often says that we are reasoning within the smooth (i.e., $C^{\infty}$ ), or the real-analytic (i.e., $C^\omega$ ) or the complex-analytic category.

In this section we recall the definition of a regular manifold and some of the terminology associated with such a structure.

Remark 9.1.1 It should be made clear at the outset that practically everything that will be said in this chapter about our “regular” structures, i.e., either $C^{\infty}$, real- or complex-analytic structures, applies as well to many other structures, for instance Gevrey structures or quasi-analytic structures, intermediate between $C^{\infty}$ and $C^\omega$. These generalizations are self-evident and will not be discussed here.

In $\mathbb{K}^n$ we make use of the Cartesian coordinates which, until specified otherwise, shall be denoted by $x_1, \ldots, x_n$ even when $\mathbb{K}=\mathbb{C}$.

Notation 9.1.2 Let $\Omega$ be an open subset of $\mathbb{K}^n$. We shall denote by $\mathcal{R}(\Omega)$ the ring of $\mathbb{K}$-valued regular (i.e., $C^{\infty}, C^\omega$ or holomorphic) functions in $\Omega$.
With our choice of the meaning of “regular” the following can be asserted:
(1) The polynomial functions in $\mathbb{K}^n$ are regular.
(2) If $\Omega \subset \mathbb{K}^n$ is an open set then $\mathcal{R}(\Omega)$ can be identified with a subring of the ring of $\mathbb{K}$-valued smooth functions, $C^{\infty}(\Omega$; $\mathbb{K})$.
(3) If $\Omega_1 \subset \Omega_2 \subset \mathbb{K}^n$ the restriction of functions from $\Omega_2$ to $\Omega_1$ induces a ring homomorphism of $\mathcal{R}\left(\Omega_2\right)$ into $\mathcal{R}\left(\Omega_1\right)$.

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Regular manifolds

For us a regular manifold will be a topological space $\mathcal{M}$ (always countable at infinity, i.e., equal to the union of a sequence of compact subsets) equipped with an assignment $\mathcal{U} \mapsto \mathcal{R}(\mathcal{U}), \mathcal{U}$ being an arbitrary open subset of $\mathcal{M}$ and $\mathcal{R}(\mathcal{U})$ a special subring of the ring $C(\mathcal{U} ; \mathbb{K})$ of the $\mathbb{K}$-valued continuous functions in $\mathcal{U}$. The elements of $\mathcal{R}(\mathcal{U})$ are the regular functions (i.e., $C^{\infty}, C^\omega$ or $\left.O\right)$ in $\mathcal{U}(\mathcal{U}$ itself regarded as a regular manifold). The correspondence $\mathcal{U} \mapsto \mathcal{R}(\mathcal{U})$ must satisfy the following three “axioms”:

(M1) Restriction to any open set $\mathcal{V} \subset \mathcal{U}$ defines a ring homomorphism $\mathcal{R}(\mathcal{U}) \longrightarrow$ $\mathcal{R}(\mathcal{V})$.
(M2) Let $\mathcal{U}=\bigcup_{\iota \in I} \mathcal{U}\iota$ be the union of a family of open subsets of $\mathcal{M}$. If the restriction to $\mathcal{U}\iota$ of a continuous function $f$ in $\mathcal{U}$ belongs to $\mathcal{R}\left(\mathcal{U}_l\right)$ for every $\iota \in I$ then $f \in \mathcal{R}(\mathcal{U})$
(M3) Every point $\wp \in \mathcal{M}$ is contained in an open set $\mathcal{U}$ having the following property: there is a regular isomorphism $\varphi$ of $\mathcal{U}$ onto an open subset of $\mathbb{K}^n$ such that the pullback map $\mathcal{R}(\varphi(\mathcal{U})) \ni f \mapsto f \circ \varphi$ is an algebra isomorphism onto $\mathcal{R}(\mathcal{U})$. The integer $n$ is independent of the point $\varphi$.

Simply phrased, this definition states that we know what the regular (i.e., $C^{\infty}, C^\omega$ or $O$ ) functions in our manifold are. According to (M2)-(M3) the regularity of a function is completely determined by its regularity in a neighborhood of each point of its domain of definition.

The number $n$ is the dimension of the manifold $\mathcal{M}$ and is denoted by $\operatorname{dim}_{\mathbb{K}} \mathcal{M}$ or simply by $\operatorname{dim} \mathcal{M}$ if there is no danger of confusion.

Property (M3) states that the topological space $\mathcal{M}$ is locally Euclidean. If $\mathcal{U}$ and $\varphi$ are as in (M3) the pair $(\mathcal{U}, \varphi)$ is called a local chart in $\mathcal{M}$. We can make use of the map $\varphi$ to pullback from $\mathbb{K}^n$ to $\mathcal{U}$ the Cartesian coordinates $x_j$ : if $\wp \in \mathcal{U}$ we write $x_j(\wp)=x_j(\varphi(\wp))$. This defines a system $\left(x_1, \ldots, x_n\right)$ of local coordinates in $\mathcal{U}$; it is then customary to write $\left(\mathcal{U}, x_1, \ldots, x_n\right)$ rather than $(\mathcal{U}, \varphi)$; we shall often refer to $\left(\mathcal{U}, x_1, \ldots, x_n\right)$ as a coordinate chart (or patch). Note that this convention allows us to use the same notation $f\left(x_1, \ldots, x_n\right)$ for a function in $\varphi(\mathcal{U})$ and for its pullback $f \circ \varphi$ in the local coordinates $\left(x_1, \ldots, x_n\right)$. A coordinate change in $\mathcal{U}$ is equivalent to a modification of the $\operatorname{map} \varphi$.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Regular functions and maps in Euclidean space

(3)如果$\Omega_1 \子集\Omega_2\子集\mathbb{K}^n$，则函数从$\Omega_2$到$\Omega_1$的限制导出$\mathcal{R}\left(\Omega_2\right)$到$\mathcal{R}\left(\Omega_1\right)$的环同态。

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Regular manifolds

(M1)对任意开集$\mathcal{V} \子集\mathcal{U}$的约束定义了一个环同态$\mathcal{R}(\mathcal{U}) \ longightarrow$ $\mathcal{R}(\mathcal{V})$。
(M2)设$\mathcal{U}=\bigcup_{\iota \in I} \mathcal{U}\iota$是$\mathcal{M}$的一组开子集的并集。如果连续函数$f$在$\mathcal{U}$中对$\mathcal{U}$的限制属于$\mathcal{R}\左(\mathcal{U}_l\右)$对于每一个$\iota \在$ I$中，则$f \在\mathcal{R}(\mathcal{U})$(M3) \mathcal{M}$中的每一个点$\wp \都包含在一个开集合$\mathcal{U}$中，它具有以下性质:$\mathcal{U}$的$\varphi$到$\mathbb{K}^n$的一个正则同构，使得回拉映射$\mathcal{R}(\ mathphi (\mathcal{U})) \ni f \mapsto f \circ \varphi$是到$\mathcal{R}(\mathcal{U})$的代数同构。整数$n$与点$\varphi$无关。 简单地说，这个定义表明我们知道流形中的正则函数(即C^{\ inty}， C^\ ω$或O$)是什么。根据(M2)-(M3)，一个函数的正则性完全取决于它在其定义域的每个点的邻域内的正则性。 数字$n$是流形$\mathcal{M}$的维数，表示为$\operatorname{dim}_{\mathbb{K}} \mathcal{M}$，如果没有混淆的危险，表示为$\operatorname{dim} \mathcal{M}$。 性质(M3)表明拓扑空间$\mathcal{M}$是局部欧几里德的。如果$\mathcal{U}$和$\v

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