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数学代写|偏微分方程代考Partial Differential Equations代写|APPMTH3023 Parametrised Surfaces

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数学代写|偏微分方程代考Partial Differential Equations代写|Parametrised Surfaces

We recall some basic concepts related to regular surfaces. Informally, an $(n+1)-$ dimensional regular surfaces is the trace of a scalar-valued function $\varphi \in C^{2}(\Omega)$, where $\Omega \subset \mathbb{R}^{n}$ is an open (connected) set. In particular, the concept of a parametrised surface in $\mathbb{R}^{3}$ is a 2-dimensional analogue of parametrised curves.

Definition 2.26 Let $I$ and $J$ be any types of intervals, and take $\Omega=I \times J$. For a continuous function $\mathbf{r}: I \times J \rightarrow \mathbb{R}^{3}$, let

$$S_{\mathbf{r}}=\left{\boldsymbol{x} \in \mathbb{R}^{3}: \boldsymbol{x}=\mathbf{r}(u, v), \text { for some }(u, v) \in I \times J\right}$$
Then $S_{\mathbf{r}}$ is called a parametrised surface in $\mathbb{R}^{3}$, with $(u, v) \in I \times J$ being the parameters, and $\mathbf{r}$ a parametrisation. We say a parametrised surface $S_{\mathbf{r}}$ is regular if the parametrisation $\mathbf{r}$ is a continuously differentiable function.

For $(u, v) \in I \times J$, the three coordinate functions $x_{i}: I \times J \rightarrow \mathbb{R}$ for a point $\mathbf{r}(u, v) \in S_{\mathbf{r}}$ can be written as
$$x_{1}=x_{1}(u, v), \quad x_{2}=x_{2}(u, v), \quad x_{3}=x_{3}(u, v) .$$

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

The development of vector analysis is primarily due to English mathematician Oliver Heaviside (1850-1925), and independently by American mathematician Josiah Gibbs (18391903). Heaviside published his work in 1893 as part of the book “The Elements of Vectorial Algebra and Analysis”, whereas Gibbs’ work first appeared as the book “Elements of Vector Analysis”, published in 1901 as the compilation of the his lectures delivered in 1881 at Yale University. Heaviside applied vector analysis tools to reformulate the twelve of twenty equations related to electromagnetic radiations in vector form, which were originally proposed by Scottish mathematician and scientist James Maxwell (1831-1879) during 1861-62. The twelve equations are recognised in modern physics as the Maxwell’s four fundamental equations (see Appendix A.2 for details). Most notations and terminology introduced in this section are due to Gibbs.

In this section, we discuss the three fundamental theorems due to Gauss, Stokes, and Helmholtz that are applied in the next two chapters to derive differential equation models for some important practical problems related to physical phenomena such as fluid flow, heat conduction, mechanical vibrations, and electromagnetic waves. In all that follows, the 3-dimensional del operator as introduced earlier plays the lead role. We shall use shorthand operator notations as given below:
$$\partial_{x} \equiv \frac{\partial}{\partial x}, \quad \partial_{y} \equiv \frac{\partial}{\partial y}, \quad \partial_{x x} \equiv \frac{\partial^{2}}{\partial x^{2}}, \quad \partial_{y x} \equiv \frac{\partial^{2}}{\partial y \partial x}, \quad \text { etc. }$$

偏微分方程代写

数学代写|偏微分方程代考Partial Differential Equations代写|Parametrised Surfaces

\left 的分隔符缺失或无法识别

$$x_{1}=x_{1}(u, v), \quad x_{2}=x_{2}(u, v), \quad x_{3}=x_{3}(u, v) .$$

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

Heaviside 于 1893 年将他的作品作为”矢量代数与分析的元龶”一书的一部分出版，而吉布斯的作品首先作为”矢量分析的元表”一

$$\partial_{x} \equiv \frac{\partial}{\partial x}, \quad \partial_{y} \equiv \frac{\partial}{\partial y}, \quad \partial_{x x} \equiv \frac{\partial^{2}}{\partial x^{2}}, \quad \partial_{y x} \equiv \frac{\partial^{2}}{\partial y \partial x}, \quad \text { etc. }$$

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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。