### 物理代写|理论力学作业代写Theoretical Mechanics代考|Elements of Differential Calculus

statistics-lab™ 为您的留学生涯保驾护航 在代写理论力学Theoretical Mechanics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。我们的专家在代写理论力学Theoretical Mechanics代写方面经验极为丰富，各种代写理论力学Theoretical Mechanics相关的作业也就用不着说。

• 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

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Set of Numbers

One defines the following types of numbers:
$$\begin{array}{ll} \mathbb{N}={1,2,3, \ldots} & \text { natural numbers } \ \mathbb{Z}={\ldots,-2,-1,0,1,2,3, \ldots} & \text { integer numbers } \ \mathbb{Q}=\left{x ; x=\frac{p}{q} ; p \in \mathbb{Z}, q \in \mathbb{N}\right} & \text { rational numbers } \ \mathbb{R}={x ; \text { continuous number line }} & \text { real numbers. } \end{array}$$
Therefore
$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} .$$

The body of complex numbers $\mathbb{C}$ will be introduced and discussed later in Sect. 2.3.5. For the above-mentioned set of numbers the basic operations addition and multiplication are defined in the well-known manner. We will remind here only shortly to the process of raising to a power.
For an arbitrary real number $a$ the $n$-th power is defined as:
$$a^{n}=\underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}{n \text {-fold }} \quad n \in \mathbb{N} .$$ There are the following rules: 1 . $$(a \cdot b)^{n}=\underbrace{(a \cdot b) \cdot(a \cdot b) \cdot \ldots \cdot(a \cdot b)}{n \text {-fold }}=a^{n} \cdot b^{n}$$
$2 .$
$$a^{k} \cdot a^{n}=\underbrace{a \cdot a \cdot \ldots \cdot a}{k \text {-fold }} \cdot \underbrace{a \cdot a \cdot \ldots \cdot a}{n \text {-fold }}=a^{k+n}$$
$3 .$
$$\left(a^{n}\right)^{k}=\underbrace{a^{n} \cdot a^{n} \cdot \ldots \cdot a^{n}}_{k \text {-fold }}=a^{n \cdot k} .$$
Even negative exponents are defined as can be seen by the following consideration:
$$a^{n}=a^{n+k-k}=a^{n} \cdot a^{-k} \cdot a^{k} \quad \curvearrowright \quad a^{-k} \cdot a^{k}=1 .$$
Therefore we have:
$$a^{-k} \equiv \frac{1}{a^{k}} \quad \forall a \in \mathbb{R} \quad(a \neq 0) .$$
Furthermore, we recognize the important special case:
$$a^{k-k} \equiv a^{0}=1 \quad \forall a \in \mathbb{R} .$$
This relation is valid also for $a=0$.
Analogously and as an extension of (1.4) split exponents can be defined:
$$b^{n}=a=\left(a^{\frac{1}{n}}\right)^{n} \quad \curvearrowright \quad b=a^{\frac{1}{n}} .$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Sequence of Numbers and Limiting Values

By a sequence of numbers we will understand a sequence of (indexed) real numbers:
$$a_{1}, a_{2}, a_{3}, \cdots, a_{n}, \cdots \quad a_{n} \in \mathbb{R}$$
We have finite and infinite sequences of numbers. In case of a finite sequence the index $n$ is restricted to a finite subset of $\mathbb{N}$. The sequence is formally denoted by the symbol
$$\left{a_{n}\right}$$
and represents a mapping of the natural numbers $\mathbb{N}$ on the body of real numbers $\mathbb{R}$ :
$$f: n \in \mathbb{N} \longrightarrow a_{n} \in \mathbb{R} \quad\left(n \longrightarrow a_{n}\right)$$
Examples
$1 .$
$$a_{n}=\frac{1}{n} \quad \longrightarrow a_{1}=1, a_{2}=\frac{1}{2}, a_{3}=\frac{1}{3}, a_{4}=\frac{1}{4} \ldots$$

$$a_{n}=\frac{1}{n(n+1)} \quad \longrightarrow a_{1}=\frac{1}{1 \cdot 2}, a_{2}=\frac{1}{2 \cdot 3}, a_{3}=\frac{1}{3 \cdot 4}, \cdots$$
$3 .$
$$a_{n}=1+\frac{1}{n} \quad \longrightarrow a_{1}=2, a_{2}=\frac{3}{2}, a_{3}=\frac{4}{3}, a_{4}=\frac{5}{4}, \cdots$$
Now we define the
Limiting value (limit) of a sequence of numbers
If $a_{n}$ approaches for $n \rightarrow \infty$ a single finite number $a$, then $a$ is the limiting value (limes) of the sequence $\left{a_{n}\right}$ :
$$\lim {n \rightarrow \infty} a{n}=a ; a_{n} \stackrel{n \rightarrow \infty}{\longrightarrow} a$$
$$\begin{gathered} \left{a_{n}\right} \text { converges to } a \ \Longleftrightarrow \forall \varepsilon>0 \quad \exists n_{\varepsilon} \in \mathbb{N} \text { so that }\left|a_{n}-a\right|<\varepsilon \quad \forall n>n_{\varepsilon} \end{gathered}$$
Does such an $a$ not exist then the sequence is called divergent. In case $\left{a_{n}\right}$ converges to $a$, then for each $\varepsilon>0$ only a finite number of sequence elements has a distance greater than $\varepsilon$ to $a$.
Examples
$1 .$
$$\left.\left{a_{n}\right}=\left{\frac{1}{n}\right} \longrightarrow 0 \quad \text { (null sequence }\right)$$
$2 .$
$$\left{a_{n}\right}=\left{\frac{n}{n+1}\right} \rightarrow 1$$
because:
$$\frac{n}{n+1}=\frac{1}{1+\frac{1}{n}} \longrightarrow \frac{1}{1+0}=1$$
In anticipation, we have here already used the rule (1.22).

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Series and Limiting Values

Adding up the terms of an infinite sequence of numbers leads to what is called a series:
$$a_{1}, a_{2}, a_{3}, \cdots, a_{n}, \cdots \curvearrowright a_{1}+a_{2}+a_{3}+\cdots+a_{n}+\cdots=\sum_{m=1}^{\infty} a_{m}$$

Strictly, the series is defined as limiting value of a sequence of (finite) partial sums:
$$S_{r}=\sum_{m=1}^{r} a_{m}$$
The series converges to $S$ if
$$\lim {r \rightarrow \infty} S{r}=S$$
does exist. If not then it is called divergent.
A necessary condition for the series $\sum_{m=1}^{\infty} a_{m}$ to be convergent is
$$\lim {m \rightarrow \infty} a{m}=0$$
For, if $\sum_{m=1}^{\infty} a_{m}$ is indeed convergent then it must hold:
$$\lim {m \rightarrow \infty} a{m}=\lim {m \rightarrow \infty}\left(S{m}-S_{m-1}\right)=\lim {m \rightarrow \infty} S{m}-\lim {m \rightarrow \infty} S{m-1}=S-S=0 .$$
However, Eq. (1.26) is not a sufficient condition. A prominent counter-example represents the harmonic series:
$$\sum_{m=1}^{\infty} \frac{1}{m}=1+\frac{1}{2}+\frac{1}{3}+\cdots$$
It is divergent, although $\lim _{m \rightarrow \infty} \frac{1}{m}=0$ ! The proof of this is given as an Exercise 1.1.3. In mathematics (analysis) one learns of different necessary and sufficient conditions of convergence for infinite series:

In the course of this book we do not need these criteria explicitly and thus restrict ourselves to only making a remark.

The geometric series turns out to be an important special case of an infinite series being defined as
$$q^{0}+q^{1}+q^{2}+\cdots+q^{m}+\cdots=\sum_{m=1}^{\infty} q^{m-1}$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Set of Numbers

\begin{array}{ll} \mathbb{N}={1,2,3, \ldots} & \text { 自然数 } \ \mathbb{Z}={\ldots,-2,-1,0, 1,2,3, \ldots} & \text { 整数 } \ \mathbb{Q}=\left{x ; x=\frac{p}{q} ; p \in \mathbb{Z}, q \in \mathbb{N}\right} & \text { 有理数 } \ \mathbb{R}={x ; \text { 连续​​数线 }} & \text { 实数。} \结束{数组}\begin{array}{ll} \mathbb{N}={1,2,3, \ldots} & \text { 自然数 } \ \mathbb{Z}={\ldots,-2,-1,0, 1,2,3, \ldots} & \text { 整数 } \ \mathbb{Q}=\left{x ; x=\frac{p}{q} ; p \in \mathbb{Z}, q \in \mathbb{N}\right} & \text { 有理数 } \ \mathbb{R}={x ; \text { 连续​​数线 }} & \text { 实数。} \结束{数组}

ñ⊂从⊂问⊂R.

2.

3.
(一种n)ķ=一种n⋅一种n⋅…⋅一种n⏟ķ-折叠 =一种n⋅ķ.

bn=一种=(一种1n)n↷b=一种1n.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Sequence of Numbers and Limiting Values

\left{a_{n}\right}\left{a_{n}\right}

F:n∈ñ⟶一种n∈R(n⟶一种n)

1.

3.

If一种n方法n→∞单个有限数一种， 然后一种是序列的极限值 (limes)\left{a_{n}\right}\left{a_{n}\right}:

\begin{gathered} \left{a_{n}\right} \text { 收敛于 } a \ \Longleftrightarrow \forall \varepsilon>0 \quad \exists n_{\varepsilon} \in \mathbb{N} \text {所以 }\left|a_{n}-a\right|<\varepsilon \quad \forall n>n_{\varepsilon} \end{gathered}\begin{gathered} \left{a_{n}\right} \text { 收敛于 } a \ \Longleftrightarrow \forall \varepsilon>0 \quad \exists n_{\varepsilon} \in \mathbb{N} \text {所以 }\left|a_{n}-a\right|<\varepsilon \quad \forall n>n_{\varepsilon} \end{gathered}

1.
\left.\left{a_{n}\right}=\left{\frac{1}{n}\right} \longrightarrow 0 \quad \text { (空序列}\right)\left.\left{a_{n}\right}=\left{\frac{1}{n}\right} \longrightarrow 0 \quad \text { (空序列}\right)
2.
\left{a_{n}\right}=\left{\frac{n}{n+1}\right} \rightarrow 1\left{a_{n}\right}=\left{\frac{n}{n+1}\right} \rightarrow 1

nn+1=11+1n⟶11+0=1

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Series and Limiting Values

∑米=1∞1米=1+12+13+⋯

q0+q1+q2+⋯+q米+⋯=∑米=1∞q米−1

## 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。