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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MA3205 Whitney’s Theorem

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|Whitney’s Theorem

The goal of this section is to give a proof of Whitney’s Theorem 1.1.7. Let us start with some preliminary results.

Lemma 1.5.1. There exists a function $\alpha: \mathbb{R} \rightarrow[0,1)$ which is monotonic, of class $C^{\infty}$ and such that $\alpha(t)=0$ if and only if $t \leq 0$.
Proof. Set
$$\alpha(t)= \begin{cases}\mathrm{e}^{-1 / t} & \text { if } t>0 \ 0 & \text { if } t \leq 0\end{cases}$$
see Fig. 1.9.(a). Clearly, $\alpha$ takes values in $[0,1)$, is monotonic, is zero only in $\mathbb{R}^{-}$, and is of class $C^{\infty}$ in $\mathbb{R}^*$; we have only to check that it is of class $C^{\infty}$ in the origin too. To verify this, it suffices to prove that the right and left limits of all derivatives in the origin coincide, that is, that
$$\lim _{t \rightarrow 0^{+}} \alpha^{(n)}(t)=0$$
for all $n \geq 0$. Assume we have proved the existence, for all $n \in \mathbb{N}$, of a polynomial $p_n$ of degree $2 n$ such that
$$\forall t>0 \quad \alpha^{(n)}(t)=\mathrm{e}^{-1 / t} p_n(1 / t)$$

## 数学代写|曲线和曲面代写Curves And Surfaces代考|Classification of 1-submanifolds

As promised at the end of Section 1.1, we want to discuss now another possible approach to the problem of defining what a curve is. As we shall see, even if in the case of curves this approach will turn out to be too restrictive, for surfaces it will be the correct way to follow (as you shall learn in Section 3.1).
The idea consists in concentrating on the support. The support of a curve has to be a subset of $\mathbb{R}^n$ that looks (at least locally) like an interval of the real line. What we have seen studying curves suggests that a way to give concrete form to the concept of “looking like” consists in using homeomorphisms with the image that are regular curves of class at least $C^1$ too. So we introduce:
Definition 1.6.1. A 1-submanifold of class $C^k$ in $\mathbb{R}^n$ (with $k \in \mathbb{N}^* \cup{\infty}$ and $n \geq 2$ ) is a connected subset $C \subset \mathbb{R}^n$ such that for all $p \in C$ there exist a neighborhood $U \subset \mathbb{R}^n$ of $p$, an open interval $I \subseteq \mathbb{R}$, and a map $\sigma: I \rightarrow \mathbb{R}^n$ (called local parametrization) of class $C^k$, such that:
(i) $\sigma(I)=C \cap U$;
(ii) $\sigma$ is a homeomorphism with its image;
(iii) $\sigma^{\prime}(t) \neq O$ for all $t \in I$.
If $\sigma(I)=C$, we shall say that $\sigma$ is a global parametrization. A periodic parametrization is a map $\sigma: \mathbb{R} \rightarrow \mathbb{R}^n$ of class $C^k$ which is periodic of period $\ell>0$, with $\sigma(\mathbb{R})=C$, and such that for all $t_0 \in \mathbb{R}$ the restriction $\left.\sigma\right|_{\left(t_0, t_0+\ell\right)}$ is a local parametrization of $C$ having image $C \backslash\left{\sigma\left(t_0\right)\right}$.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|Whitney’s Theorem

$$\alpha(t)= \begin{cases}\mathrm{e}^{-1 / t} & \text { if } t>00 \quad \text { if } t \leq 0\end{cases}$$

## 数学代写曲线和曲面代写Curves And Surfaces代考|Classification of 1submanifolds

，开区间 $I \subseteq \mathbb{R}$, 和一张地图 $\sigma: I \rightarrow \mathbb{R}^n$ 类的（称为局部参数化) $C^k$ ，这样:
(i) $\sigma(I)=C \cap U$
(二) $\sigma$ 是与其象的同胚;
(E) $\sigma^{\prime}(t) \neq O$ 对所有人 $t \in I$.

$\sigma(\mathbb{R})=C$ ，这样对于所有人 $t_0 \in \mathbb{R}$ 限制 $\left.\sigma\right|{(t 0, t 0+\ell)}$ 是一个局部参数化 $C$ 有形象 $\backslash$ left 的分隔符缺失或无法识别

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