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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|微分几何代写Differential Geometry代考|Manifolds with Boundaries

Despite the flexibility of the definition of a differentiable manifold, it does not allow for a boundary. In many applications, it is useful to have the notion of a manifold with a boundary. This notion relies on the concept of a Euclidean half-space.
Definition 3.5.1. Let $\vec{a}$ be a unit vector in $\mathbb{R}^{n}$, i.e., $\vec{a} \in \mathcal{S}^{n}$. The half-space $H_{\vec{a}}$ is
$$H_{\vec{a}}=\left{\vec{x} \in \mathbb{R}^{n} \mid \vec{x} \cdot \vec{a} \geq 0\right} .$$
The boundary of the half-space is $\partial H_{\vec{a}}=\left{\vec{x} \in \mathbb{R}^{n} \mid \vec{x} \cdot \vec{a}=0\right}$.
Note that for distinct unit vectors $\vec{a}$ and $\vec{b}$, the half-spaces $H_{\vec{a}}$ and $H_{\vec{b}}$ are not equal.

Since the topology on $H_{\vec{a}}$ is the subset topology inherited from $\mathbb{R}^{n}$, a set is open in $H_{\vec{a}}$ if and only if it is equal to $U \cap H_{\vec{a}}$ for some open set $U \subset \mathbb{R}^{n}$. Figure $3.9$ depicts a Euclidean half-space of of $\mathbb{R}^{2}$ along with two open subsets. One open set arises as an open square which is already a subset of $H_{\vec{a}}$ and so does not include any point of its boundary. The other open set arises as the intersection of an open disk with $H_{\vec{a}}$. This intersection includes the segment along $\partial H_{\vec{a}}$, the line perpendicular to $\vec{a}$.

Definition 3.5.2. A differentiable $n$-manifold $M$ with boundary has the same definition as in Definition 3.1.3 except that the ranges for the charts are open subsets of a half-space $H$ of $\mathbb{R}^{n}$. The boundary of the manifold, written $\partial M$, is the set of points $p$ such that in some coordinate chart $\phi: U \rightarrow H$, where $H$ is a half-space, $\phi(p) \in \partial H$

The most commonly used Euclidean half-spaces in $\mathbb{R}^{n}$ are the upper half-space and the lower half-space, defined respectively as
\begin{aligned} &\mathbb{R}{+}^{n}=\left{\left(x{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n} \mid x_{n} \geq 0\right}=H_{(0, \ldots, 0,1)} \ &\mathbb{R}{-}^{n}=\left{\left(x{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n} \mid x_{n} \leq 0\right}=H_{(0, \ldots, 0,-1)} . \end{aligned}

## 数学代写|微分几何代写Differential Geometry代考|Multilinear Algebra

Many of the objects of interest in differential geometry on manifolds are expressed properly in the context of multilinear algebra. Consequently, this chapter introduces linear algebraic concepts that are not commonly included in a first linear algebra course. The underlying field for all objects outside this chapter is the set of reals $\mathbb{R}$, but this chapter introduces the concepts for an arbitrary field $K$ of characteristic 0 (e.g., $\mathbb{Q}, \mathbb{R}$, or $\mathbb{C}$ ).

Before jumping in, we mention our habit of notation for components associated to certain linear algebraic objects. Let $V$ be a vector space over $K$ with $\operatorname{dim} V=n$. If $\mathcal{B}=\left(e_{1}, e_{2}, \ldots, e_{n}\right)$ is an ordered basis of $V$, the coordinates of $v \in V$ with respect to $\mathcal{B}$ are
$$[v]{\mathcal{B}}=\left(\begin{array}{c} v^{1} \ v^{2} \ \vdots \ v^{n} \end{array}\right), \quad \text { where } \quad v=v^{1} e{1}+v^{2} e_{2}+\cdots+v^{n} e_{n}$$
If the basis of $V$ is understood from the problem or if we use a standard basis of $V$, we write $[v]$. It is common to abuse the notation and say that a vector is equal to the $n \times 1$ matrix of its coordinates but we must always be careful to understand that components are given with respect to some basis.

If $V$ is a vector space of dimension $n$ and if $\mathcal{B}=\left{e_{1}, e_{2}, \ldots, e_{n}\right}$ and $\mathcal{B}^{\prime}=$ $\left{f_{1}, f_{2}, \ldots, f_{n}\right}$ are two bases, there is an $n \times n$ matrix $P_{\mathcal{B}^{\prime}}^{\mathcal{B}}$ that converts the $\mathcal{B}$ coordinates of a vector to $\mathcal{B}^{\prime}$-coordinates. In particular, for all $v \in V$,
$$[v]{\mathcal{B}^{\prime}}=P{\mathcal{B}^{\prime}}^{\mathcal{B}}[v]{\mathcal{B} .} .$$ This matrix is found by $P{\mathcal{B}^{\prime}}^{\mathcal{B}}=\left(\left[e_{1}\right]{\mathcal{B}^{\prime}} \quad\left[e{2}\right]{\mathcal{B}^{\prime}} \quad \cdots,\left[e{n}\right]{\mathcal{B}^{\prime}}\right)$. Writing the components of $P{\mathcal{B}^{\prime}}^{\mathcal{B}}$ as $\left(p_{j}^{i}\right)$, where $i$ is the row index and $j$ is the column index, and the $\mathcal{B}^{\prime}$ coordinate of $v$ as $\left(\bar{v}^{i}\right)$, we can write (4.1) as
$$\bar{v}^{i}=\sum_{j=1}^{n} p_{j}^{i} v^{j} .$$ As we introduce constructions in multilinear algebra and refer to how the components of objects change under a change of basis, we will refer to (4.2) repeatedly to see appropriate generalizations.

# 微分几何代考

## 数学代写|微分几何代写Differential Geometry代考| Multilinear Algebra

$$[v] \mathcal{B}=\left(v^{1} v^{2}: v^{n}\right), \quad \text { where } \quad v=v^{1} e 1+v^{2} e_{2}+\cdots+v^{n} e_{n}$$

$$[v] \mathcal{B}^{\prime}=P \mathcal{B}^{\prime \mathcal{B}}[v] \mathcal{B} . .$$
$$\bar{v}^{i}=\sum_{j=1}^{n} p_{j}^{i} v^{j} .$$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师