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• 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

## 物理代写|电磁学代写electromagnetism代考|Traces of Vector Fields

In order to define properly the trace on $\Gamma$ of elements of $\boldsymbol{H}($ curl, $\Omega$ ) or of $\boldsymbol{H}$ (div, $\Omega$ ), it is convenient to have integration-by-parts formulas at one’s disposal. As a matter of fact, one can proceed by duality, with respect to the spaces $\boldsymbol{H}^{1 / 2}(\Gamma)$ and $H^{1 / 2}(\Gamma)$, respectively, that is, those trace spaces that originate from $\boldsymbol{H}^1(\Omega)$ and $H^1(\Omega)$.

From now on, let $\Omega$ be a domain. As far as notations are concerned, one notices that in a domain, which is bounded by definition, the index (for compact support) $^{\text {(for }}$ of the set $C_c^{\infty}(\bar{\Omega})$ of Definition $2.1 .27$ can be dropped.

Let us begin with density results (cf. [117, Chapter I] and Amrouche, 2011, Private communication).
Proposition 2.2.12 One has the following:

• $\boldsymbol{C}^{\infty}(\bar{\Omega})$ is dense in $\boldsymbol{H}($ curl, $\Omega)$;
• $\boldsymbol{C}^{\infty}(\bar{\Omega})$ is dense in $\boldsymbol{H}$ (div, $\left.\Omega\right)$;
• for $s \in] 0,1 / 2\left[, \boldsymbol{C}^{\infty}(\bar{\Omega})\right.$ is dense in $\boldsymbol{H}_{-s}(\operatorname{div}, \Omega)$.
With the help of Proposition 2.2.4, one easily infers other results.
Corollary 2.2.13 Under the assumptions (2.10) about $\S$, one concludes that:
• $\oint^{-1} \boldsymbol{C}^{\infty}(\bar{\Omega})$ is dense in $\boldsymbol{H}(\mathbf{c u r l} \xi, \Omega)$;
• $\xi^{-1} \boldsymbol{C}^{\infty}(\bar{\Omega})$ is dense in $\boldsymbol{H}(\operatorname{div} \S, \Omega)$.
One can define the unit outward normal vector $\boldsymbol{n}=n_1 \boldsymbol{e}1+n_2 e_2+n_3 \boldsymbol{e}_3$ to its boundary, almost everywhere (cf. Proposition 2.1.25). It is well-known that it holds that, for two functions $f$ and $g$ of $C^1(\overline{S 2})$, $$\int{\Omega}\left{f \frac{\partial g}{\partial x_i}+\frac{\partial f}{\partial x_i} g\right} d x=\int_{\Gamma} f g n_i d \Gamma, \quad i=1,2,3 .$$
What can be deduced from this formula?

## 物理代写|电磁学代写electromagnetism代考|Practical Function Spaces in the (t, x) Variable

To solve some time-dependent problems, in particular, the time-dependent Maxwell equations, one needs to introduce function spaces depending both on the time variable $t$ and on the space variable $\boldsymbol{x}$. Indeed, in that case, the unknowns, i.e., the electromagnetic fields, depend on the $(t, x)$ variable. Obviously, one can consider distributions in space and time, that is, on $\mathbb{R} \times \mathbb{R}^3$. However, one generally distinguishes between the variables $t$ and $\boldsymbol{x}$, since they do not play the same role. Classically, one deals with the values of a field at a given time $t$. Hence, for a function $f$ depending on both $x$ and $t$, we are interested in $\boldsymbol{x} \mapsto f\left(t_0, \boldsymbol{x}\right)$, for a given $t_0$.

More precisely, let $T_{-} \in\left[-\infty,+\infty\left[\right.\right.$ and $\left.\left.T_{+} \in\right]-\infty,+\infty\right]$ with $T-<T_{+}$ respectively denote the initial and final times, and let $\Omega$ denote the subset of $\mathbb{R}^3$ of interest. With respect to distributions in space and time, the corresponding space of distributions is simply $\mathcal{D}^{\prime}(] T_{-}, T_{+}[\times \Omega)$. A classical result that allows one to go back and forth from distributions in the $(t, \boldsymbol{x})$ variable to continuous functions of the variable $t$, with values in function spaces of the variable $\boldsymbol{x}$, is that
the tensor product space $\mathcal{D}(] T_{-}, T_{+}[) \otimes \mathcal{D}(\Omega)$ is dense in $\mathcal{D}(] T_{-}, T_{+}[\times \Omega)$.
Next, consider the function
\begin{aligned} f: T_{-}, T_{+}[\times \Omega& \mapsto \mathbb{R} \ (t, \boldsymbol{x}) & \mapsto f(t, \boldsymbol{x}) . \end{aligned}
For any time $t \in] T_{-}, T_{+}[$, one can introduce the function $f(t)$
\begin{aligned} f(t): \Omega & \rightarrow \mathbb{R} \ \boldsymbol{x} & \mapsto f(t, \boldsymbol{x}), \end{aligned}
so that the function $f$ can be identified with the function
\begin{aligned} ] T_{-}, T_{+}[& \rightarrow{\Omega \rightarrow \mathbb{R}} \ t & \mapsto f(t) . \end{aligned}
In what follows, we will define the function spaces in the $(t, x)$ variable, which will be useful for the weak formulations in the subsequent chapters.

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Traces of Vector Fields

• $\boldsymbol{C}^{\infty}(\bar{\Omega}$ )密集在 $\boldsymbol{H}$ (卷曲, $\Omega$ );
• $\boldsymbol{C}^{\infty}(\bar{\Omega})$ 密集在 $\boldsymbol{H}$ (分区， $\Omega$ );
• 为了 $s \in] 0,1 / 2\left[, \boldsymbol{C}^{\infty}(\bar{\Omega})\right.$ 密集在 $\boldsymbol{H}_{-s}(\operatorname{div}, \Omega)$.
在命题 2.2.4 的帮助下，人们很容易推断出其他结果。
推论 2.2.13 在假设 (2.10) 下§，一个结论是:
• $\oint^{-1} \boldsymbol{C}^{\infty}(\bar{\Omega})$ 密集在 $\boldsymbol{H}(\operatorname{curl} \xi, \Omega)$;
• $\xi^{-1} \boldsymbol{C}^{\infty}(\bar{\Omega})$ 密集在 $\boldsymbol{H}(\operatorname{div} \S, \Omega)$.
可以定义单位外向法向量 $n=n_1 e 1+n_2 e_2+n_3 e_3$ 到它的边界，几乎无处不在（参见命题 2.1.25) 。众所周知，它认为，对于两个函数 $f$ 和 $g$ 的 $C^1(\overline{S 2})$,
$\backslash$ left 的分隔符玦失或无法识别
从这个公式可以推导出什么?

## 物理代写|电磁学代写electromagnetism代考|Practical Function Spaces in the (t, x) Variable

$$f: T_{-}, T_{+}[\times \Omega \mapsto \mathbb{R}(t, \boldsymbol{x}) \quad \mapsto f(t, \boldsymbol{x})$$

$$f(t): \Omega \rightarrow \mathbb{R} \boldsymbol{x} \quad \mapsto f(t, \boldsymbol{x}),$$

$$] T_{-}, T_{+}[\rightarrow \Omega \rightarrow \mathbb{R} t \quad \mapsto f(t) .$$

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师