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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MATH5437 Tangent plane

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|Tangent plane

We have seen that tangent vectors play a major role in the study of curves. In this section we intend to define the notion of a tangent vector to a surface at a point. The geometrically simplest way is as follows:

Definition 3.3.1. Let $S \subseteq \mathbb{R}^3$ be a set, and $p \in S$. A tangent vector to $S$ at $p$ is a vector of the form $\sigma^{\prime}(0)$, where $\sigma:(-\varepsilon, \varepsilon) \rightarrow \mathbb{R}^3$ is a curve of class $C^{\infty}$ whose support lies in $S$ and such that $\sigma(0)=p$. The set of all possible tangent vectors to $S$ at $p$ is the tangent cone $T_p S$ to $S$ at $p$.

Remark 3.3.2. A cone (with the origin as vertex) in a vector space $V$ is a subset $C \subseteq V$ such that $a v \in C$ for all $a \in \mathbb{R}$ and $v \in C$. It is not difficult to verify that the tangent cone to a set is in fact a cone in this sense. Indeed, first of all, the zero vector is the tangent vector to a constant curve, so $O \in T_p S$ for all $p \in S$. Next, if $a \in \mathbb{R}^*$ and $O \neq v \in T_p S$, if we choose a curve $\sigma:(-\varepsilon, \varepsilon) \rightarrow S$ with $\sigma(0)=p$ and $\sigma^{\prime}(0)=v$, then the curve $\sigma_a:(-\varepsilon /|a|, \varepsilon /|a|) \rightarrow S$ given by $\sigma_a(t)=\sigma(a t)$ is such that $\sigma_a(0)=p$ and $\sigma_a^{\prime}(0)=a v$; so $a v \in T_p S$ as required by the definition of cone.

Example 3.3.3. If $S \subset \mathbb{R}^3$ is the union of two straight lines through the origin, it is straightforward to verify (check it) that $T_O S=S$.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|Tangent vectors and derivations

Definition 3.3.6 of tangent plane is not completely satisfactory: it strongly depends on the fact that the surface $S$ is contained in $\mathbb{R}^3$, while it would be nice to have a notion of tangent vector intrinsic to $S$, independent of its embedding in the Euclidean space. In other words, we would like to have a definition of $T_p S$ not as a subspace of $\mathbb{R}^3$, but as an abstract vector space, depending only on $S$ and $p$. Moreover, since we are dealing with “differential geometry”, sooner or later we shall have to find a way to differentiate on a surface.

Surprisingly enough, we may solve both these problems at the same time. The main idea is contained in the following example.

Example 3.4.1. Let $U \subseteq \mathbb{R}^2$ be an open set, and $p \in U$. Then we can associate with each tangent vector $v \in T_p U=\mathbb{R}^2$ a partial derivative:
$$v=\left.\left(v_1, v_2\right) \mapsto \frac{\partial}{\partial v}\right|_p=\left.v_1 \frac{\partial}{\partial x_1}\right|_p+\left.v_2 \frac{\partial}{\partial x_2}\right|_p,$$
and all partial derivatives are of this kind. So, in a sense, we may identify $T_p U$ with the set of partial derivatives.

## 数学代写|曲线和曲面代写CURVES AND SURFACES代 考|TANGENT VECTORS AND DERIVATIONS

$$v=\left.\left(v_1, v_2\right) \mapsto \frac{\partial}{\partial v}\right|_p=\left.v_1 \frac{\partial}{\partial x_1}\right|_p+\left.v_2 \frac{\partial}{\partial x_2}\right|_p$$

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