## 数学代写|微分几何代写Differential Geometry代考|MATH 464

statistics-lab™ 为您的留学生涯保驾护航 在代写微分几何Differential Geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微分几何Differential Geometry代写方面经验极为丰富，各种代写微分几何Differential Geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Frames Associated to Coordinate Systems

Many problems in introductory mechanics involve finding the trajectory of a particle under the influence of various forces and/or subject to certain constraints. The first approach uses the coordinate functions and describes the trajectory as
$$\vec{r}(t)=(x(t), y(t), z(t))=x(t) \vec{\imath}+y(t) \vec{\jmath}+z(t) \vec{k} .$$
Newton’s equations of motion then lead to differential equations in the three coordinate functions $x(t), y(t)$, and $z(t)$. The velocity function is the derivative, namely
\begin{aligned} \vec{r}^{\prime}(t) &=\frac{d}{d t}(x(t) \vec{\imath})+\frac{d}{d t}(y(t) \vec{\jmath})+\frac{d}{d t}(z(t) \vec{k}) \ &=x^{\prime}(t) \vec{\imath}+x(t) \frac{d}{d t}(\vec{\imath})+y^{\prime}(t) \vec{\jmath}+y(t) \frac{d}{d t}(\vec{\jmath})+z^{\prime}(t) \vec{k}+z(t) \frac{d}{d t}(\vec{k}) \ &=x^{\prime}(t) \vec{\imath}+y^{\prime}(t) \vec{\jmath}+z^{\prime}(t) \vec{k} \end{aligned}
because $\frac{d}{d t} \vec{\imath}=0, \frac{d}{d t} \vec{\jmath}=0$, and $\frac{d}{d t} \vec{k}=0$. This last remark shows that the frame $(\vec{\imath}, \vec{\jmath}, \vec{k})$ associated to the Cartesian coordinate systems is a constant frame.

As we discuss variable frames, we introduce a nice way to describe the rate of change of a variable frame. Suppose that $\left{\vec{u}{1}, \vec{u}{2}, \vec{u}{3}\right}$ is a basis of $\mathbb{R}^{3}$ and let $\vec{a}$ and $\vec{b}$ be two other vectors with components $\vec{a}=a{1} \vec{u}{1}+a{2} \vec{u}{2}+a{3} \vec{u}_{3}$ and $\vec{b}=$ $b_{1} \vec{u}{1}+b{2} \vec{u}{2}+b{3} \vec{u}{3}$. Assuming that all vectors are column vectors, we can write these component definitions of $\vec{a}$ and $\vec{b}$ in the matrix expression $$\left(\begin{array}{ll} \vec{a} & \vec{b} \end{array}\right)=\left(\begin{array}{lll} \vec{u}{1} & \vec{u}{2} & \vec{u}{3} \end{array}\right)\left(\begin{array}{ll} a_{1} & b_{1} \ a_{2} & b_{2} \ a_{3} & b_{3} \end{array}\right)$$
Using this notation, we can express the relationships $\frac{d}{d t} \vec{\imath}=0, \frac{d}{d t} \vec{\jmath}=0$, and $\frac{d}{d t} \vec{k}=0$ by
$$\frac{d}{d t}\left(\begin{array}{lll} \vec{\imath} & \vec{\jmath} & \vec{k} \end{array}\right)=\left(\begin{array}{lll} \vec{\imath} & \vec{\jmath} & \vec{k} \end{array}\right)\left(\begin{array}{lll} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{array}\right) .$$

## 数学代写|微分几何代写Differential Geometry代考|Frames Associated to Trajectories

In the study of trajectories, whether in physics or geometry, it is often convenient to use a frame that is different from the Cartesian frame. Changing types of frames sometimes makes difficult integrals tractable or makes certain difficult differential equations manageable. In the particular context of special relativity, one talks about a momentarily comoving reference frame, abbreviated to MCRF. [50]

In the study of plane curves, it is common to use the frame ${\vec{T}, \vec{U}}$ to study the local properties of a plane curve $\vec{x}(t)$. (See [5, Chapter 1].) The vector $\vec{T}(t)$ is the unit tangent vector $\vec{T}(t)=\vec{x}^{\prime}(t) /\left|\vec{x}^{\prime}(t)\right|$, and the unit normal vector $\vec{U}(t)$, is the result of rotating $\vec{T}(t)$ by $\pi / 2$ in the counterclockwise direction. This is a moving frame that is defined in terms of a given regular curve $\vec{x}(t)$ and, at $t=t_{0}$, is viewed as based at the point $\vec{x}\left(t_{0}\right)$. To compare with applications in physics, it is important to note that the ${\vec{T}, \vec{U}}$ frame is not the same as the polar coordinate frame $\left{\vec{e}{r}, \vec{e}{\theta}\right}$. From Equation (2.4) (and ignoring the $z$-component), we know that
$$\vec{e}{T}=(\cos \theta, \sin \theta) \quad \text { and } \quad \vec{e}{\theta}=(-\sin \theta, \cos \theta) .$$
Assuming that $x, y, r$, and $\theta$ are functions of $t$ and since $x=r \cos \theta$ and $y=r \sin \theta$, we have
$$\vec{x}^{\prime}(t)=\left(x^{\prime}(t), y^{\prime}(t)\right)=\left(r^{\prime} \cos \theta-r \theta^{\prime} \sin \theta, r^{\prime} \sin \theta+r \theta^{\prime} \cos \theta\right)=r^{\prime} \vec{e}{r}+r \theta^{\prime} \vec{e}{\theta} .$$
We then calculate the speed function to be
$$s^{\prime}(t)=\left|\vec{x}^{\prime}(t)\right|=\sqrt{\left(r^{\prime}\right)^{2}+r^{2}\left(\theta^{\prime}\right)^{2}}$$
and find the unit tangent and unit normal vectors to be
\begin{aligned} &\vec{T}=\frac{1}{\sqrt{\left(r^{\prime}\right)^{2}+r^{2}\left(\theta^{r}\right)^{2}}}\left(r^{\prime} \vec{e}{r}+r \theta^{\prime} \vec{e}{\theta}\right), \ &\vec{U}=\frac{1}{\sqrt{\left(r^{\prime}\right)^{2}+r^{2}\left(\theta^{r}\right)^{2}}}\left(-r \theta^{\prime} \vec{e}{r}+r^{\prime} \vec{e}{\theta}\right) . \end{aligned}

## 数学代写|微分几何代写Differential Geometry代考| Frames Associated to Coordinate Systems

$$\vec{r}(t)=(x(t), y(t), z(t))=x(t) \vec{\imath}+y(t) \vec{\jmath}+z(t) \vec{k} .$$

$$\vec{r}^{\prime}(t)=\frac{d}{d t}(x(t) \vec{\imath})+\frac{d}{d t}(y(t) \vec{\jmath})+\frac{d}{d t}(z(t) \vec{k}) \quad=x^{\prime}(t) \vec{\imath}+x(t) \frac{d}{d t}(\vec{\imath})+y^{\prime}(t) \vec{\jmath}+y(t) \frac{d}{d t}(\vec{\jmath})+z^{\prime}(t)$$

## 数学代写|微分几何代写Differential Geometry代考| Frames Associated to Trajectories

$$\vec{e} T=(\cos \theta, \sin \theta) \quad \text { and } \quad \vec{e} \theta=(-\sin \theta, \cos \theta) .$$

$$\vec{x}^{\prime}(t)=\left(x^{\prime}(t), y^{\prime}(t)\right)=\left(r^{\prime} \cos \theta-r \theta^{\prime} \sin \theta, r^{\prime} \sin \theta+r \theta^{\prime} \cos \theta\right)=r^{\prime} \vec{e} r+r \theta^{\prime} \vec{e} \theta$$

$$s^{\prime}(t)=\left|\vec{x}^{\prime}(t)\right|=\sqrt{\left(r^{\prime}\right)^{2}+r^{2}\left(\theta^{\prime}\right)^{2}}$$

$$\vec{T}=\frac{1}{\sqrt{\left(r^{\prime}\right)^{2}+r^{2}\left(\theta^{r}\right)^{2}}}\left(r^{\prime} \vec{e} r+r \theta^{\prime} \vec{e} \theta\right), \quad \vec{U}=\frac{1}{\sqrt{\left(r^{\prime}\right)^{2}+r^{2}\left(\theta^{r}\right)^{2}}}\left(-r \theta^{\prime} \vec{e} r+r^{\prime} \vec{e} \theta\right) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|微分几何代写Differential Geometry代考|MATH4030

statistics-lab™ 为您的留学生涯保驾护航 在代写微分几何Differential Geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微分几何Differential Geometry代写方面经验极为丰富，各种代写微分几何Differential Geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Differentiation Rules; Functions of Class C

In a single-variable calculus course, one learns a number of differentiation rules. With functions $F$ from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$, one must use some caution since the matrix $[d F]$ of the differential $d F$ is not a vector function but a matrix of functions. (Again, we remind the reader that our notation for evaluating the matrix of functions $[d F]$ at a point $\vec{a}$ is $\left[d F_{\vec{a}}\right]$.)

Theorem 1.3.1. Let $U$ be an open set in $\mathbb{R}^{n}$. Let $F$ and $G$ be functions from $U$ to $\mathbb{R}^{m}$, and let $w: U \rightarrow \mathbb{R}$ be a scalar function. If $F, G$, and $w$ are differentiable at $\vec{a}$, then $F+G$ and $w F$ are differentiable at $\vec{a}$ and

1. $d(F+G){\vec{a}}=d F{\vec{a}}+d G_{\vec{a}}$;
2. $\left[d(w F){\vec{a}}\right]=w(\vec{a})\left[d F{\vec{a}}\right]+[F(\vec{a})]\left[d w_{\vec{a}}\right]$.
Proof. The proof for both parts follows from Proposition 1.2.17. Explicitly for the second part, the $i j$-entry of $\left[d(w F){\vec{a}}\right]$ is $$\frac{\partial\left(w F{i}\right)}{\partial x_{j}}=w(\vec{a}) \frac{\partial F_{i}}{\partial x_{j}}(\vec{a})+\frac{\partial w}{\partial x_{j}}(\vec{a}) F_{i}(\vec{a}) .$$
The first term on the right side is the $i j$-entry of $w(\vec{a})\left[d F_{\vec{a}}\right]$ while the second term is the $i j$-entry of $[F(\vec{a})]\left[d w_{\vec{a}}\right]$, which is the product of a columns by a row vector. The result follows.

Note that in Theorem 1.3.1(2), $[F(\vec{a})]$ is a column vector of dimension $m$, while $\left[d w_{\vec{a}}\right]$ is a row vector of dimension $n$. Hence $[F(\vec{a})]\left[d w_{\vec{a}}\right]$ is an $m \times n$ matrix of rank $1 .$

## 数学代写|微分几何代写Differential Geometry代考|Inverse and Implicit Function Theorems

In single- and multivariable calculus of a function $F: \mathbb{R}^{n} \rightarrow \mathbb{R}$, one defines a critical point as a point $\vec{a}=\left(a_{1}, \ldots, a_{n}\right)$ such that the gradient of $F$ at $\vec{a}$ is $\overrightarrow{0}$, i.e.,
$$\nabla F(\vec{a})=\left(\frac{\partial F}{\partial x_{1}}(\vec{a}), \ldots, \frac{\partial F}{\partial x_{n}}(\vec{a})\right)=\overrightarrow{0}$$
At such a point, $F$ is said to have a flat tangent line or tangent plane, and, according to standard theorems in calculus, $F(\vec{a})$ is either a local minimum, local maximum, or a “saddle point.” This notion is a special case of the following general definition.
Definition 1.4.1. Let $U$ be an open subset of $\mathbb{R}^{n}$ and $F: U \rightarrow \mathbb{R}^{m}$ a differentiable function. We call $q \in U$ a critical point of $F$ if $F$ is not differentiable at $q$ or if $d F_{q}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is not of maximum rank, i.e., if $\operatorname{rank}\left(d F_{q}\right)<\min (m, n)$. If $q$ is a critical point of $F$, we call $F(q)$ a critical value. If $p \in \mathbb{R}^{m}$ is not a critical value of $F$ (even if $p$ is not in the image of $F$ ), then we call $p$ a regular value of $F$.

We point out that this definition simultaneously generalizes the notion of a critical point for functions $F: U \rightarrow \mathbb{R}$, with $U$ an open subset of $\mathbb{R}^{n}$, and the definition for a critical point of a parametric curve in $\mathbb{R}^{n}$ (Definition $3.2 .1$ in $[5]$ ). If $m=n$, the notion of a critical point has a few alternate equivalent criteria.

## 数学代写|微分几何代写Differential Geometry代考| Differentiation Rules; Functions of Class C

1. $d(F+G) \vec{a}=d F \vec{a}+d G_{\vec{a}}$
2. $[d(w F) \vec{a}]=w(\vec{a})[d F \vec{a}]+[F(\vec{a})]\left[d w_{\vec{a}}\right]$.
证明。这两个部分的证明都来自命题1.2.17。明确地对于第二部分， $i j$-进入 $[d(w F) \vec{a}]$ 是
$$\frac{\partial(w F i)}{\partial x_{j}}=w(\vec{a}) \frac{\partial F_{i}}{\partial x_{j}}(\vec{a})+\frac{\partial w}{\partial x_{j}}(\vec{a}) F_{i}(\vec{a}) .$$
右侧的第一个术语是 $i j$-进入 $w(\vec{a})\left[d F_{\vec{a}}\right]$ 而第二个术语是 $i j$-进入 $[F(\vec{a})]\left[d w_{\vec{a}}\right]$ ，它是列乘以行向量的乘 积。结果如下。
请注意，在定理 $1.3 .1$ (2) 中， $[F(\vec{a})]$ 是维度的列向量 $m$ 而 $\left[d w_{\vec{a}}\right]$ 是维度的行向量 $n$. 因此 $[F(\vec{a})]\left[d w_{\vec{a}}\right]$ 是一个 $m \times n$ 等级矩阵 1 .

## 数学代写|微分几何代写Differential Geometry代考| Inverse and Implicit Function Theorems

$$\nabla F(\vec{a})=\left(\frac{\partial F}{\partial x_{1}}(\vec{a}), \ldots, \frac{\partial F}{\partial x_{n}}(\vec{a})\right)=\overrightarrow{0}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写微分几何Differential Geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微分几何Differential Geometry代写方面经验极为丰富，各种代写微分几何Differential Geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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)


## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。