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数学代写|偏微分方程代考Partial Differential Equations代写|MATH3700 Classical Vector Analysis

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数学代写|偏微分方程代考Partial Differential Equations代写|Classical Vector Analysis

In this chapter, we review relevant concepts of multivariable analysis, basic geometric properties of curves and surfaces, and also some important theorems of classical vector calculus. The concepts reviewed in the first two sections are fundamental to every aspect of topics discussed in the subsequent chapters. We also need some variants of the fundamental divergence theorem to formulate differential equation models of certain important phenomena. In Sect. 2.1, we recall theorems related to three important analytical properties of a scalar or vector function defined on an open set in $\mathbb{R}^{n}$ and fix notations for use in the sequel. In Sect. 2.2, we recall some important geometric notions related to classical theory of curves and surfaces. In Sect. 2.3, we discuss three fundamental theorems related to the del operator
$$\nabla: \equiv\left(\frac{\partial}{\partial x_{1}}, \ldots, \frac{\partial}{\partial x_{n}}\right)$$
for the case when $n=3$. Throughout this book, a symbol in boldface font represents a vector quantity, which we may view as a column vector, whenever necessary.

数学代写|偏微分方程代考Partial Differential Equations代写|Multivariable Calculus

The set $\mathbb{R}^{n}$ of $n$-tuples of real numbers is a linear space over the field $\mathbb{R}$, where the addition and scalar multiplication are defined, respectively, as

\begin{aligned} \boldsymbol{x}+\boldsymbol{y} &=\left(x_{1}+y_{1}, \ldots, x_{n}+y_{n}\right) \ a \cdot \boldsymbol{x} &=\left(a x_{1}, \ldots, a x_{n}\right) \end{aligned}
for $\boldsymbol{x}=\left(x_{1}, \ldots, x_{n}\right), \boldsymbol{y}=\left(y_{1}, \ldots, y_{n}\right) \in \mathbb{R}^{n}$ and $a \in \mathbb{R}$. The usual dot product of vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ defines an inner product on $\mathbb{R}^{n}$, which is written as $\langle\boldsymbol{x}, \boldsymbol{y}\rangle$. That is,
$$\langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{x} \cdot \boldsymbol{y}=x_{1} y_{1}+\cdots+x_{n} y_{n} .$$
It can be shown that the function $\langle,\rangle: \mathbb{R}^{n} \rightarrow \mathbb{R}$, given by (2.1.1) is a positive definite, symmetric, bilinear functional (Exercise 2.1). Therefore, $\mathbb{R}^{n}$ is an inner product space. In general, a linear space $X$ over the field $\mathbb{R}$ is called an inner product space if there exists a positive definite, symmetric, bilinear functional $b: X \times X \rightarrow \mathbb{R}$.

Further, a linear space $X$ over the field $\mathbb{R}$ is called a normed space if there exists a positive definite, absolute homogeneous, subadditive function $p: X \rightarrow \mathbb{R}$, where $p(x)$ is called the norm of $x \in X$. In particular, it can be shown that the function ||$: \mathbb{R}^{n} \rightarrow \mathbb{R}$ given by
$$|\boldsymbol{x}|:=\sqrt{\langle\boldsymbol{x}, \boldsymbol{y}\rangle}=\sqrt{x_{1}^{2}+\cdots+x_{n}^{2}}$$

偏微分方程代写

数学代写|偏微分方程代考Partial Differential Equations代写|Classical Vector Analysis

$$\nabla: \equiv\left(\frac{\partial}{\partial x_{1}}, \ldots, \frac{\partial}{\partial x_{n}}\right)$$

数学代写|偏微分方程代考Partial Differential Equations代写|Multivariable Calculus

$$\boldsymbol{x}+\boldsymbol{y}=\left(x_{1}+y_{1}, \ldots, x_{n}+y_{n}\right) a \cdot \boldsymbol{x} \quad=\left(a x_{1}, \ldots, a x_{n}\right)$$

$$\langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{x} \cdot \boldsymbol{y}=x_{1} y_{1}+\cdots+x_{n} y_{n} .$$

$$|\boldsymbol{x}|:=\sqrt{\langle\boldsymbol{x}, \boldsymbol{y}\rangle}=\sqrt{x_{1}^{2}+\cdots+x_{n}^{2}}$$

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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。