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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MATH322 Schonflies’ theorem

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|Schonflies’ theorem

In this section we give an elementary proof due to Thomassen (see [23]) of the Schönflies theorem for Jordan curves mentioned in Remark 2.3.7. Along the way we shall also give a proof of the Jordan curve theorem for continuous curves.

Remark 2.8.1. In this section, with a slight abuse of language, we shall call Jordan arcs and curves what we have been calling supports of Jordan arcs and curves.

Definition 2.8.2. A simple polygonal arc in the plane is a Jordan arc consisting of finitely many line segments. Analogously, a simple polygonal closed curve is a plane Jordan curve consisting of finitely many line segments.

We begin by proving the Jordan curve theorem for simple polygonal closed curves.

Lemma 2.8.3. If $C \subset \mathbb{R}^2$ is a simple polygonal closed curve, then $\mathbb{R}^2 \backslash C$ consists of exactly two components having $C$ as their common boundary.
Proof. We begin by showing that $\mathbb{R}^2 \backslash C$ has at most two components. Assume, by contradiction, that $p_1, p_2, p_3 \in \mathbb{R}^2 \backslash C$ belong to distinct components of $\mathbb{R}^2 \backslash C$, and choose an open disk $D \subset \mathbb{R}^2$ such that $D \cap C$ is a line segment (so that $D \backslash C$ has just two components). Since each component of $\mathbb{R}^2 \backslash C$ has $C$ as its boundary, for $j=1,2,3$ we may find a curve starting from $p_j$, arriving as close to $C$ as we want, and then going parallel to $C$ till it meets $D$. But $D \backslash C$ has just two components; so at least two of the $p_j$ ‘s can be connected by a curve, against the hypothesis that they belong to distinct components.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|Local theory of surfaces

The rest of this book is devoted to the study of surfaces in space. As we did for the curves, we shall begin by trying to understand how best define a surface; but, unlike what happened for curves, for surfaces it will turn out to be more useful to work with subsets of $\mathbb{R}^3$ that locally look like an open subset of the plane, instead of working with maps from an open subset of $\mathbb{R}^2$ to $\mathbb{R}^3$ having an injective differential.

When we say that a surface locally looks like an open subset of the plane, we are not (only) talking about its topological structure, but (above all) about its differential structure. In other words, it must be possible to differentiate functions on a surface exactly as we do on open subsets of the plane: computing a partial derivative is a purely local operation, so it is has to be possible to perform similar operation in every object that locally looks (from a differential viewpoint) like an open subset of the plane.

To carry out this program, after presenting in Section $3.1$ the official definition of what a surface is, in Section $3.2$ we shall define precisely the family of functions that are smooth on a surface, that is, the functions we shall be able to differentiate; in Section $3.4$ we shall show how to differentiate them, and we shall define the notion of differential of a smooth map between surfaces. Furthermore, in Sections $3.3$ and $3.4$, we shall introduce the tangent vectors to a surface and we shall explain why they are an embodiment of partial derivatives. Finally, in the supplementary material we shall prove (Section 3.5) Sard’s theorem, an important result about critical points of smooth functions, and we shall see (Section 3.6) how to extend smooth functions from a surface to the whole of $\mathbb{R}^3$.

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