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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MM512 How to define a surface

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|How to define a surface

As we did for curves, we begin by discussing the question of the correct definition of what a surface is. Our experience from the one-dimensional case suggests two possible approaches: we might define surfaces as subsets of the space with some properties, or we can define them as maps from an open subset of the plane to the space, satisfying suitable regularity properties.
Working with curves we preferred this second approach, since the existence of parameterizations by arc length allowed us to directly relate the geometric properties of the support of the curve with the differential properties of the curve itself.

As we shall see, in the case of surfaces the situation is significantly more complex. The approach that emphasizes maps will be useful to study local questions; but from a global viewpoint it will be more effective to privilege the other approach.

But let us not disclose too much too soon. Let us instead start by introducing the obvious generalization of the notion of a regular curve:

Definition 3.1.1. An immersed (or parametrized) surface in space is a map $\varphi: U \rightarrow \mathbb{R}^3$ of class $C^{\infty}$, where $U \subseteq \mathbb{R}^2$ is an open set, such that the differential d $\varphi_x: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ is injective (that is, has rank 2) in every point $x \in U$. The image $\varphi(U)$ of $\varphi$ is the support of the immersed surface.

Remark 3.1.2. For reasons that will become clear in Section $3.4$ (see Remark 3.4.20), when studying surfaces we shall only use $C^{\infty}$ maps, and we shall not discuss lower regularity issues.

Remark 3.1.3. The differential $\mathrm{d} \varphi_x$ of $\varphi=\left(\varphi_1, \varphi_2, \varphi_3\right)$ in $x \in U$ is represented by the Jacobian matrix
$$\operatorname{Jac} \varphi(x)=\left|\begin{array}{ll} \frac{\partial \varphi_1}{\partial x_1}(x) & \frac{\partial \varphi_1}{\partial x_2}(x) \ \frac{\partial \varphi_2}{\partial x_1}(x) & \frac{\partial \varphi_2}{\partial x_2}(x) \ \frac{\partial \varphi_3}{\partial x_1}(x) & \frac{\partial \varphi_3}{\partial x_2}(x) \end{array}\right| \in M_{3,2}(\mathbb{R})$$
As for curves, in this definition the emphasis is on the map rather than on its image. Moreover, we are not asking for the immersed surfaces to be a homeomorphism with their images or to be injective (see Example 3.1.6); both these properties are nevertheless locally true. To prove this, we need a lemma, somewhat technical but extremely useful.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|Smooth functions

Local parametrizations are the tool that allows us to give concrete form to the idea that a surface locally resembles an open subset of the plane; let us see how to use them to determine when a function defined on a surface is smooth.
Definition 3.2.1. Let $S \subset \mathbb{R}^3$ be a surface, and $p \in S$. A function $f: S \rightarrow \mathbb{R}$ is of class $C^{\infty}$ (or smooth) at $p$ if there exists a local parametrization $\varphi: U \rightarrow S$ at $p$ such that $f \circ \varphi: U \rightarrow \mathbb{R}$ is of class $C^{\infty}$ in a neighborhood of $\varphi^{-1}(p)$. We shall say that $f$ is of class $C^{\infty}$ (or smooth) if it is so at every point. The space of $C^{\infty}$ functions on $S$ will be denoted by $C^{\infty}(S)$.

Remark 3.2.2. A smooth function $f: S \rightarrow \mathbb{R}$ is automatically continuous. Indeed, let $I \subseteq \mathbb{R}$ be an open interval, and $p \in f^{-1}(I)$. By assumption, there is a local parametrization $\varphi: U \rightarrow S$ at $p$ such that $f \circ \varphi$ is of class $C^{\infty}$ (and thus continuous) in a neighborhood of $\varphi^{-1}(p)$. Then $(f \circ \varphi)^{-1}(I)=\varphi^{-1}\left(f^{-1}(I)\right)$ is a neighborhood of $\varphi^{-1}(p)$. But $\varphi$ is a homeomorphism with its image; so $f^{-1}(I)$ has to be a neighborhood of $\varphi\left(\varphi^{-1}(p)\right)=p$. Since $p$ was arbitrary, it follows that $f^{-1}(I)$ is open in $S$, and so $f$ is continuous.

A possible problem with this definition is that it might depend on the particular local parametrization we have chosen: a priori, there might be another local parametrization $\psi$ at $p$ such that $f \circ \psi$ is not smooth in $\psi^{-1}(p)$. Luckily, the following theorem implies that this cannot happen.

## 数学代写曲线和曲面代写CURVES AND SURFACES代考|HOW TO DEFINE A SURFACE

$$\operatorname{Jac} \varphi(x)=\left|\frac{\partial \varphi_1}{\partial x_1}(x) \quad \frac{\partial \varphi_1}{\partial x_2}(x) \frac{\partial \varphi_2}{\partial x_1}(x) \quad \frac{\partial \varphi_2}{\partial x_2}(x) \frac{\partial \varphi_3}{\partial x_1}(x) \quad \frac{\partial \varphi_3}{\partial x_2}(x)\right| \in M_{3,2}(\mathbb{R})$$

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