### 数学代写|微分几何代写Differential Geometry代考|MATH3405

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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|Analysis of Multivariable Functions

Let $U$ be a subset of $\mathbb{R}^{n}$ and let $f: U \rightarrow \mathbb{R}^{m}$ be a function from $U$ to $\mathbb{R}^{m}$. Writing the input variable as
$$\vec{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right),$$
we denote the output assigned to $\vec{x}$ by $f(\vec{x})$ or $f\left(x_{1}, \ldots, x_{n}\right)$. Since the codomain of $f$ is $\mathbb{R}^{m}$, the images of $f$ are $m$-tuples so we can write
\begin{aligned} f(\vec{x}) &=\left(f_{1}(\vec{x}), f_{2}(\vec{x}), \ldots, f_{m}(\vec{x})\right) \ &=\left(f_{1}\left(x_{1}, x_{2}, \ldots, x_{n}\right), f_{2}\left(x_{1}, x_{2}, \ldots, x_{n}\right), \ldots, f_{m}\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right) . \end{aligned}
The functions $f_{i}: U \rightarrow \mathbb{R}$, for $i=1,2, \ldots, m$, are called the component functions of $f$.

We sometimes use the notation $\vec{f}(\vec{x})$ to emphasize the fact that the codomain $\mathbb{R}^{m}$ is a vector space and that any operation on $m$-dimensional vectors is permitted on functions $\vec{f}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$. Therefore, some authors call such functions vector functions of a vector variable.

In any Euclidean space $\mathbb{R}^{n}$, the standard basis is the set of vectors written as $\left{\vec{e}{1}, \vec{e}{2}, \ldots, \vec{e}{n}\right}$, where $$\vec{e}{i}=\left(\begin{array}{c} 0 \ \vdots \ 1 \ \vdots \ 0 \end{array}\right)$$
with the only nonzero entry 1 occurring in the $i$ th coordinate. If no basis is explicitly specified for $\mathbb{R}^{n}$, then it is assumed that one uses the standard basis.

At this point, a remark is in order concerning the differences in notations between calculus and linear algebra. In calculus, one usually denotes an element of $\mathbb{R}^{n}$ as an $n$-tuple and writes this element on one line as $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$. On the other hand, in order to reconcile vector notation with the usual manner we multiply a matrix by a vector, in linear algebra we denote an element of $\mathbb{R}^{n}$ as a column vector
$$\vec{x}=\left(\begin{array}{c} x_{1} \ x_{2} \ \vdots \ x_{n} \end{array}\right)$$

## 数学代写|微分几何代写Differential Geometry代考|Continuity, Limits, and Differentiability

Intuitively, a function is called continuous if it preserves “nearness.” A rigorous mathematical definition for continuity for functions from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ is hardly any different for functions from $\mathbb{R} \rightarrow \mathbb{R}$.

In calculus of a real variable, one does not study functions defined over a discrete set of real values because the notions behind continuity and differentiability do not make sense over such sets. Instead, one often assumes the function is defined over some interval. Similarly, for the analysis of functions $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$, one does not study functions defined from any subset of $\mathbb{R}^{n}$ into $\mathbb{R}^{m}$. One typically considers functions defined over what is called an open set in $\mathbb{R}^{n}$, a notion we define now.
Definition 1.2.1. The open ball around $\vec{x}{0}$ of radius $r$ is the set $$B{r}\left(\vec{x}{0}\right)=\left{\vec{x} \in \mathbb{R}^{n}:\left|\vec{x}-\vec{x}{0}\right|0 such that B_{r}(\vec{x}) \subset U. Intuitively speaking, the definition of an open set U in \mathbb{R}^{n} implies that at every point p \in U it is possible to “move” in any direction by at least a little amount \epsilon and still remain in U. This means that in some sense U captures the full dimensionality of the ambient space \mathbb{R}^{n}. This is why, when studying the analysis of functions from \mathbb{R}^{n} to \mathbb{R}^{m}, we narrow our attention to functions F: U \rightarrow \mathbb{R}^{m}, where U is an open subset of \mathbb{R}^{n}. The reader is encouraged to consult Subsection A.1.2 in Appendix A for more background on open and closed sets. The situation in which we need to consider an open set U and a point \vec{x}_{0} in U is so common that another terminology exists for U in this case. Definition 1.2.2. Let \vec{x}{0} \in \mathbb{R}^{n}. Any open set U in \mathbb{R}^{n} such that \vec{x}{0} \in U is called an open neighborhood, or more simply, a neighborhood, of \vec{x}{0}. We are now in a position to formally define continuity. Definition 1.2.3. Let U be an open subset of \mathbb{R}^{n}, and let F be a function from U into \mathbb{R}^{m}. The function F is called continuous at the point \vec{x}{0} \in U if F\left(\vec{x}{0}\right) exists and if, for all \varepsilon>0, there exists a \delta>0 such that for all \vec{x} \in \mathbb{R},$$ \left|\vec{x}-\vec{x}{0}\right|<\delta \Longrightarrow\left|F(\vec{x})-F\left(\vec{x}_{0}\right)\right|<\epsilon .
$$The function F is called continuous on U if it is continuous at every point of U. ## 微分几何代考 ## 数学代写|微分几何代写Differential Geometry代考| Analysis of Multivariable Functions 让 U 是 的子集 \mathbb{R}^{n} 并让 f: U \rightarrow \mathbb{R}^{m} 是来自 U 自 \mathbb{R}^{m}. 将输入变量编写为$$
\vec{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right),
$$我们表示分配给 \vec{x} 由 f(\vec{x}) 或 f\left(x_{1}, \ldots, x_{n}\right). 自共域以来 f 是 \mathbb{R}^{m} ，图像 f 是 m-元组，所以我们可以写$$
f(\vec{x})=\left(f_{1}(\vec{x}), f_{2}(\vec{x}), \ldots, f_{m}(\vec{x})\right) \quad=\left(f_{1}\left(x_{1}, x_{2}, \ldots, x_{n}\right), f_{2}\left(x_{1}, x_{2}, \ldots, x_{n}\right), \ldots, f_{m}\left(x_{1}, x_{2},\right.\right.
$$功能 f_{i}: U \rightarrow \mathbb{R} 为 i=1,2, \ldots, m ，称为 的组件函数 f. 我们有时使用符号 \vec{f}(\vec{x}) 强调共域的事实 \mathbb{R}^{m} 是一个向量空间，并且对 m-函数上允许使用维向量 \vec{f}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}. 因此，一些作者将此类函数称为向量变量的向量函数。 在任何欧几里得空间中 \mathbb{R}^{n} ，标准基础是向量的集合，写为 Veft {lvec{e}{1}，Ivec{e}2}，Vdots，Ivec{e}n}\right } } 哪里$$
\vec{e} i=(0 \vdots 1 \vdots 0)
$$唯一的非零条目 1 出现在 i th 坐标。如果没有明确指定基础 \mathbb{R}^{n} ，则假定使用标准基。 在这一点上，关于微积分和线性代数之间符号的差异，有必要进行一些评论。在微积分中，一个通常表示 \mathbb{R}^{n} 作 为 n-元组，并将此元素写在一行上 \left(x_{1}, x_{2}, \ldots, x_{n}\right). 另一方面，为了调和向量符号与通常的方式，我们将矩阵 乘以向量，在线性代数中，我们表示 \mathbb{R}^{n} 作为列向量$$
\vec{x}=\left(x_{1} x_{2} \vdots x_{n}\right)


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## MATLAB代写

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