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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MATH322 Local theory of curves

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

Elementary geometry gives a fairly accurate and well-established notion of what is a straight line, whereas is somewhat vague about curves in general. Intuitively, the difference between a straight line and a curve is that the former is, well, straight while the latter is curved. But is it possible to measure how curved a curve is, that is, how far it is from being straight? And what, exactly, is a curve? The main goal of this chapter is to answer these questions. After comparing in the first two sections advantages and disadvantages of several ways of giving a formal definition of a curve, in the third section we shall show how Differential Calculus enables us to accurately measure the curvature of a curve. For curves in space, we shall also measure the torsion of a curve, that is, how far a curve is from being contained in a plane, and we shall show how curvature and torsion completely describe a curve in space. Finally, in the supplementary material, we shall present (in Section 1.4) the local canonical shape of a curve; we shall prove a result (Whitney’s Theorem 1.1.7, in Section 1.5) useful to understand what cannot be the precise definition of a curve; we shall study (in Section 1.6) a particularly well-behaved type of curves, foreshadowing the definition of regular surface we shall see in Chapter 3 ; and we shall discuss (in Section 1.7) how to deal with curves in $\mathbb{R}^n$ when $n \geq 4$.

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

What is a curve (in a plane, in space, in $\mathbb{R}^n$ )? Since we are in a mathematical textbook, rather than in a book about military history of Prussian light cavalry, the only acceptable answer to such a question is a precise definition, identifying exactly the objects that deserve being called curves and those that do not. In order to get there, we start by compiling a list of objects that we consider without a doubt to be curves, and a list of objects that we consider without a doubt not to be curves; then we try to extract properties possessed by the former objects and not by the latter ones.

Example 1.1.1. Obviously, we have to start from straight lines. A line in a plane can be described in at least three different ways:

• as the graph of a first degree polynomial: $y=m x+q$ or $x=m y+q$;
• as the vanishing locus of a first degree polynomial: $a x+b y+c=0$;
• as the image of a map $f: \mathbb{R} \rightarrow \mathbb{R}^2$ having the form $f(t)=(\alpha t+\beta, \gamma t+\delta)$.
A word of caution: in the last two cases, the coefficients of the polynomial (or of the map) are not uniquely determined by the line; different polynomials (or maps) may well describe the same subset of the plane.

Example 1.1.2. If $I \subseteq \mathbb{R}$ is an interval and $f: I \rightarrow \mathbb{R}$ is a (at least) continuous function, then its graph
$$\Gamma_f={(t, f(t)) \mid t \in I} \subset \mathbb{R}^2$$
surely corresponds to our intuitive idea of what a curve should be. Note that we have
$$\Gamma_f={(x, y) \in I \times \mathbb{R} \mid y-f(x)=0},$$
that is a graph can always be described as a vanishing locus too. Moreover, it also is the image of the map $\sigma: I \rightarrow \mathbb{R}^2$ given by $\sigma(t)=(t, f(t))$.

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

• 作为一次㝖项式的图: $y=m x+q$ 或者 $x=m y+q$;
• 作为一次多项式的消失轨迹: $a x+b y+c=0$;
• 作为地图的图像 $f: \mathbb{R} \rightarrow \mathbb{R}^2$ 有形式 $f(t)=(\alpha t+\beta, \gamma t+\delta)$.
需要注意的是: 在最后两种情况下，多项式 (或映射) 的条数不是由线唯一确定的; 不同的多项式（或映射）可以很好地描 述平面的同一子集。
示例 1.1.2。如果 $I \subseteq \mathbb{R}$ 是 个区间并且 $f: I \rightarrow \mathbb{R}$ 是 个 (至少) 连紏函数，那么它的图
$$\Gamma_f=(t, f(t)) \mid t \in I \subset \mathbb{R}^2$$
肯定符合我们对曲线应该是什么的直观想法。请注意，我们有
$$\Gamma_f=(x, y) \in I \times \mathbb{R} \mid y-f(x)=0,$$
那是一个图也总是可以描述为一个消失的轨迹。而且，它也是地图的图像 $\sigma: I \rightarrow \mathbb{R}^2$ 由 $\sigma(t)=(t, f(t))$.

## MATLAB代写

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