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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MATH5270 The Jordan curve theorem

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|The Jordan curve theorem

In this section we shall complete the proof of the Jordan curve theorem for regular curves, by showing that the complement of the support of a simple

closed regular plane curve of class $C^2$ has at least two components. To get there we need a new ingredient, which we shall construct by using the degree introduced in Section 2.1.

Given a continuous closed plane curve, there are (at least) two ways to associate with it a curve with values in $S^1$, and consequently a degree. In this section we are interested in the first way, while in next section we shall use the second one.

Definition 2.3.1. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a continuous closed plane curve. Given a point $p \notin \sigma([a, b])$ we may define $\phi_p:[a, b] \rightarrow S^1$ by setting
$$\phi_p(t)=\frac{\sigma(t)-p}{|\sigma(t)-p|} .$$
The winding number $\iota_p(\sigma)$ of $\sigma$ with respect to $p$ is, by definition, the degree of $\phi_p$; it measures the number of times $\sigma$ goes around the point $p$.

Fig. $2.4$ shows the winding number of a curve with respect to several points, computed as we shall see in Example 2.3.5.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|The turning tangents theorem

There is another very natural way of associating a $S^1$-valued curve (and consequently a degree) with a closed regular plane curve.

Definition 2.4.1. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a closed regular plane curve of class $C^1$, and let $\mathbf{t}:[a, b] \rightarrow S^1$ be its tangent versor, given by
$$\mathbf{t}(t)=\frac{\sigma^{\prime}(t)}{\left|\sigma^{\prime}(t)\right|} .$$
The rotation index $\rho(\sigma)$ of $\sigma$ is the degree of the map $\mathbf{t}$; it counts the number of full turns made by the tangent versor to $\sigma$.

Corollary 2.1.18 provides us with a simple formula to compute the rotation index:

Proposition 2.4.2. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a closed regular plane curve of class $C^1$ with oriented curvature $\tilde{\kappa}:[a, b] \rightarrow \mathbb{R}$. Then
$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \tilde{\kappa}\left|\sigma^{\prime}\right| \mathrm{d} t=\frac{1}{2 \pi} \int_a^b \frac{\operatorname{det}\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)}{\left|\sigma^{\prime}\right|^2} \mathrm{~d} t .$$
Proof. By Corollary 2.1.18,
$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \operatorname{det}\left(\mathbf{t}, \mathbf{t}^{\prime}\right) \mathrm{d} t$$

## 数学代写|曲线和曲面代写Curves And Surfaces代考|The Jordan curve theorem

$$\phi_p(t)=\frac{\sigma(t)-p}{|\sigma(t)-p|} .$$

## 数学代写|曲线和曲面代写Curves And Surfaces代考|The turning tangents theorem

$$\mathbf{t}(t)=\frac{\sigma^{\prime}(t)}{\left|\sigma^{\prime}(t)\right|} .$$

$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \tilde{\kappa}\left|\sigma^{\prime}\right| \mathrm{d} t=\frac{1}{2 \pi} \int_a^b \frac{\operatorname{det}\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)}{\left|\sigma^{\prime}\right|^2} \mathrm{~d} t .$$

$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \operatorname{det}\left(\mathbf{t}, \mathbf{t}^{\prime}\right) \mathrm{d} t$$

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