## 数学代写|多变量微积分代写multivariable calculus代考|МАТН280

statistics-lab™ 为您的留学生涯保驾护航 在代写多变量微积分multivariable calculus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。我们的专家在代写多变量微积分multivariable calculus代写方面经验极为丰富，各种代写多变量微积分multivariable calculus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|多变量微积分代写multivariable calculus代考|Limits and continuity.

In the next chapter we introduce and explore the concept of partial differentiation. In the lead up to that discussion it will be necessary to explain a number of concepts we shall then take for granted. Most importantly there is the notion of function continuity. For multivariable functions this will be discussed in detail in Section 2.B, but we can set the stage here with a short review of the subject as it relates to functions of one variable.

Function continuity is defined in terms of limiting processes. Mention has already been made of limit points of closed sets. We said that a point $\boldsymbol{a}$ is a limit point if any open sphere centred on $\boldsymbol{a}$, no matter how small in radius, contains points other than $\boldsymbol{a}$.

Similarly, segments of the real line possess the property that any open interval $I$, no matter how small, centred on a point $a$, contain points $x$ in $I$ different from $a$. The real line and any of its finite segments are therefore said to be complete: containing no gaps. This conjures up the notion of a set continuum, moving smoothly from one real value to another, never meeting any holes.
This notion gives critical meaning to the formalism $x \rightarrow a$ as the process of approaching a real value $a$ along the real line. To be even more precise, we specify $x \rightarrow a^{-}$and $x \rightarrow a^{+}$as meaning the respective approaches to $a$ along the real line from “below” $a(xa)$.

Now with thought given to single-variable functions defined on a domain $D_f \subset \mathbb{R}$, the different approaches $x \rightarrow a^{-}$and $x \rightarrow a^{+}$for $a, x \in D_f$ can have all manner of implications for the function. Assuming $a, x \in D_f$ we define the process of taking a limit of a function, which we denote either by
$$\lim {x \rightarrow a^{-}} f(x), \lim {x \rightarrow a^{+}} f(x) \text {, or } \lim _{x \rightarrow a} f(x)$$
as considering the sequence of values $f$ progressively takes as $x \rightarrow a^{-}$, $x \rightarrow a^{+}$, or in their combination. These considerations are of course separate to the question of what value $f$ actually takes at $a$. To summarize all of these ideas we have the following definition.

## 数学代写|多变量微积分代写multivariable calculus代考|Coordinate systems

Up until now we have represented points in $\mathbb{R}^2$ and $\mathbb{R}^3$ in terms of Cartesian coordinates, $(x, y)$ as in Example $1.1$ and $(x, y, z)$ as in Example 1.2. However, problems arise that are better described in other coordinate systems. Such problems arise in both the differential and integral calculus (Sections 3.E, 4.E, and $4 . \mathrm{H})$ and are usually associated with the geometry of the region under consideration. The most common coordinate systems that we will encounter are the polar coordinate system in $\mathbb{R}^2$, and the cylindrical and spherical coordinate systems in $\mathbb{R}^3$. Note that there are other standard systems that can be useful in specific cases (see [15]) and even non-standard systems may be needed to solve some problems (see Section 4.E).

There are three general features to note. First, the 2D Cartesian and polar coordinate systems have the same origin. Similarly, the 3D Cartesian and cylindrical or spherical coordinate systems have a common origin. Second, the non-Cartesian coordinates are designed to uniquely identify and represent every point in $\mathbb{R}^2$ or $\mathbb{R}^3$, as do their Cartesian counterparts. That is, these coordinate systems span the whole of $\mathbb{R}^2$ and $\mathbb{R}^3$, respectively. Finally, the individual coordinate variables within a given non-Cartesian system are independent of each other, just as the individual Cartesian coordinates are independent variables in the Cartesian system.

# 多变量微积分代写

## 数学代写|多变量微积分代写multivariable calculus代考|Limits and continuity.

$$\lim x \rightarrow a^{-} f(x), \lim x \rightarrow a^{+} f(x), \text { or } \lim _{x \rightarrow a} f(x)$$

## 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|多变量微积分代写multivariable calculus代考|МАTH263

statistics-lab™ 为您的留学生涯保驾护航 在代写多变量微积分multivariable calculus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。我们的专家在代写多变量微积分multivariable calculus代写方面经验极为丰富，各种代写多变量微积分multivariable calculus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|多变量微积分代写multivariable calculus代考|Introduction to sets

It is not possible to provide a picture of $\boldsymbol{x} \in \mathbb{R}^n$ for $n>3$ (nor $\mathbb{R}^n$ itself). However, there should be no cause for concern as points in $\mathbb{R}^n$ behave the same as points in $\mathbb{R}^2$ and $\mathbb{R}^3$. That is, they follow the same set of rules. So it is enough to be familiar with points and point operations in $\mathbb{R}^2$ and $\mathbb{R}^3$, and then being able to generalize their properties. The most important point operations are listed below.
Vector algebra laws.
Let $\boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^n$ and $\lambda \in \mathbb{R}$.
At its most. basic description, $\mathbb{R}^n$ is an example of a linear vector space which is characterized by the two properties of addition and scalar multiplication:

(a) $\boldsymbol{x}+\boldsymbol{y}=\left(x_1+y_1, x_2+y_2, \ldots, x_n+y_n\right) \in \mathbb{R}^n$.
(b) $\lambda \boldsymbol{x}=\left(\lambda x_1, \lambda x_2, \ldots, \lambda x_n\right) \in \mathbb{R}^n$.
As a direct generalization of the scalar product of $2(\mathrm{c})$, points in $\mathbb{R}^n$ satisfy the so-called inner product,
(c) $\boldsymbol{x} \cdot \boldsymbol{y}=x_1 y_1+\ldots+x_n y_n \in \mathbb{R} \quad-\mathbb{R}^n$ is called an inner product space.
Finally, there are the following generalizations to $\mathbb{R}^n$ of the two fundamental geometric measures:
(d) $|x|=\sqrt{\boldsymbol{x} \cdot \boldsymbol{x}}=\sqrt{x_1^2+\cdots+x_n^2}$

• the length of $\boldsymbol{x}$.
(e) $|\boldsymbol{x}-\boldsymbol{y}|=\sqrt{\left(x_1-y_1\right)^2+\cdots+\left(x_n-y_n\right)^2}$
• the distance between points.
With this distance property $\mathbb{R}^n$ is also a so-called metric space, since the distance between points is one measure or metric that allows a geometric characterization of a space.

## 数学代写|多变量微积分代写multivariable calculus代考|Real-valued functions

In Chapters 2,3 , and 4 , we focus attention almost exclusively on scalarvalued functions of many variables, while in Chapter 5 we extend the ideas to vector-valued functions. In both contexts the following introduction to fundamental properties of multi-valued functions is invaluable. To start, we introduce some more notation and a pictorial view of what functions do.
In single-variable calculus we have the following scenario:
Let $y=f(x)$. The “graph” of $f$ is the set of ordered pairs ${(x, f(x))} \in \mathbb{R}^2$. This is shown graphically in Figure $1.11$ where the independent variable $x$ and dependent variable $y$ are plotted on mutually orthogonal axes.

This way of visualizing functions of one variable was introduced in the early $17^{\text {th }}$ century by René Descartes [17], and is named the Cartesian representa tion in recognition. It is quite a useful means of illustrating function dependence and function properties, especially for functions of one or two variables.
It ceases to be as useful, however, for functions of more than two variables. For the latter cases one resorts to simply considering a set-mapping picture. For the case $y=f(x)$ this is a simple interval-to-interval map as shown in Figure $1.12$.

# 多变量微积分代写

## 数学代写|多变量微积分代写multivariable calculus代考|Introduction to sets

(一个) $\boldsymbol{x}+\boldsymbol{y}=\left(x_1+y_1, x_2+y_2, \ldots, x_n+y_n\right) \in \mathbb{R}^n$.
(二) $\lambda \boldsymbol{x}=\left(\lambda x_1, \lambda x_2, \ldots, \lambda x_n\right) \in \mathbb{R}^n$.

(c) $\boldsymbol{x} \cdot \boldsymbol{y}=x_1 y_1+\ldots+x_n y_n \in \mathbb{R} \quad-\mathbb{R}^n$ 称为内积空间。

(d) $|x|=\sqrt{\boldsymbol{x} \cdot \boldsymbol{x}}=\sqrt{x_1^2+\cdots+x_n^2}$

• 的长度 $x$ 囯
(和) $|\boldsymbol{x}-\boldsymbol{y}|=\sqrt{\left(x_1-y_1\right)^2+\cdots+\left(x_n-y_n\right)^2}$
• 点之间的距离。
有了这个距离属性 $\mathbb{R}^n$ 也是所谓的度量空间，因为点之间的距离是允许对空间进行几何表征的一种度量或 度量。

## 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|多变量微积分代写multivariable calculus代考|MAT201

statistics-lab™ 为您的留学生涯保驾护航 在代写多变量微积分multivariable calculus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。我们的专家在代写多变量微积分multivariable calculus代写方面经验极为丰富，各种代写多变量微积分multivariable calculus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|多变量微积分代写multivariable calculus代考|Vectors and functions

Many mathematical properties possessed by functions of several variables are couched in geometric terms and with reference to elementary set theory. In this introductory chapter I will revisit some of the concepts that will be needed in later chapters. For example, vector calculus springs naturally from vector algebra so it is appropriate to begin the review with the latter topic. This is followed by a short review of elementary set theory, which will be referred to throughout the book and will indeed help establish many foundation concepts in both the differential and integral calculus. Coordinate systems and the notion of level sets are also discussed. Once again, both topics find application in differential and integral multivariable calculus, as well as in vector calculus.

It goes without saying that a review of single-variable functions is helpful. This begins in this chapter (Section 1.C), but continues in Chapters 2, 3 and 4 as needed.

To help appreciate the behaviour of multivariable functions defined on two or higher dimensional domains, It is useful to at least visualize their domains of definition. Sometimes, though, it is possible, as well as necessary, to visualize the entire graph of a function, or some approximation to it. Some people are more hard-wired to visual cues and visual information, while others are more comfortable with abstract ideas. Whatever your preference, being able to draw figures is always useful. Consequently, in this chapter we also review some basic 3D structures and show how to draw them using MatLAB ${ }^{\circledR}$. Of course, other software will serve equally well. In the event of the reader being unable to access software solutions, there is included a subsection which may hopefully illustrate, by example, how one can obtain a picture of a region or of a function graph directly from a mathematical formula or equation. Although it is not possible to offer a general procedure that works in all cases, some of the steps may be applicable in other instances.

## 数学代写|多变量微积分代写multivariable calculus代考|Some vector algebra essentials

Let $a>0$ be a scalar, and let
\begin{aligned} v &=(\alpha, \beta, \gamma) \ &=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k} \ &=\alpha \mathbf{e}_1+\beta \mathbf{e}_2+\gamma \mathbf{e}_3 \end{aligned}
be a vector in $\mathbb{R}^3$ (see Section 1.B) with $x-, y$-, and $z$-components $\alpha, \beta$, and $\gamma$.

This vector has been written in the three most common forms appearing in current texts. The sets ${\mathbf{i}, \mathbf{j}, \mathbf{k}}$ and $\left{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\right}$ represent the same set of unit vectors in mutually orthogonal directions in $\mathbb{R}^3$. The first form simply shows the components along the three orthogonal directions without reference to the unit vectors themselves, although the unit vectors and the coordinate system are implicit in this notation. The reader should be aware that we shall have occasion to refer to vectors using any of the three formats. The choice will depend on what is most convenient at that time without compromising understanding.

Multiplying a vector $\boldsymbol{v}$ with a scalar will return a new vector with either the same direction if the scalar is positive or the opposite direction if the scalar is negative. In either case the resulting vector has different magnitude (Figure 1.1). This re-scaling will be a feature in Chapter 5 where we will need vectors of unit magnitude. For $a \boldsymbol{v}$, with $a \in \mathbb{R}$, to be a unit vector we must have
$$|a \boldsymbol{v}|=|a||\boldsymbol{v}|=a \sqrt{\alpha^2+\beta^2+\gamma^2}=1 \text {, i.e., } a=\frac{1}{\sqrt{\alpha^2+\beta^2+\gamma^2}} .$$
Therefore, to construct a unit vector in the direction of a specific vector $v$ we simply divide $v$ by its length:
$$N=\frac{v}{|v|} .$$

# 多变量微积分代写

## 数学代写|多变量微积分代写multivariable calculus代考|Some vector algebra essentials

$$v=(\alpha, \beta, \gamma) \quad=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k}=\alpha \mathbf{e}1+\beta \mathbf{e}_2+\gamma \mathbf{e}_3$$ 成为向量 $\mathbb{R}^3$ (见第 1.B 节) 与 $x-, y$ ，和 $z$-成分 $\alpha, \beta$ ，和 $\gamma$. 该向量以当前文本中出现的三种最常见的形式编写。套装i, j, $\mathbf{k}$ 和 Vleft $\backslash$ mathbf $\left.{e}{-} 1, \backslash \operatorname{mathbf}{e}_{-} 2, \backslash \operatorname{mathbf}{e}_{-} 3 \backslash r i g h t\right}$ 在相互正交的方向上表示同一组单位向量 $\mathbb{R}^3$. 第一种形式简 单地显示了沿三个正交方向的分量，而不参考单位向量本身，尽管单位向量和坐标系在这种表示法中是隐含 的。读者应该知道我们将有机会使用这三种格式中的任何一种来引用向量。在不影响理解的情况下，选择将取 决于当时最方便的选择。

$$|a \boldsymbol{v}|=|a||\boldsymbol{v}|=a \sqrt{\alpha^2+\beta^2+\gamma^2}=1 \text {, i.e., } a=\frac{1}{\sqrt{\alpha^2+\beta^2+\gamma^2}} .$$

$$N=\frac{v}{|v|} \text {. }$$

## 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。