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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|Time-Harmonic Phenomena

We consider the time-dependent Maxwell equations in a homogeneous medium (for instance, vacuum), set in a bounded domain Dom, written as two second-order wave equations (see Eqs. (1.128)-(1.129)). Assuming that there is no charge, both electromagnetic fields are divergence-free. The wave equations for each of the fields being of the same nature, we will consider only one of them, for instance,
\begin{aligned} &\frac{\partial^2 \boldsymbol{E}}{\partial t^2}+c^2 \text { curl curl } \boldsymbol{E}=-\frac{1}{\varepsilon_0} \frac{\partial \boldsymbol{J}}{\partial t}, \ &\operatorname{div} \boldsymbol{E}-0, \end{aligned}
with the initial conditions
$$\boldsymbol{E}(0)=\boldsymbol{E}_0, \quad \frac{\partial \boldsymbol{E}}{\partial t}(0)=\boldsymbol{E}_1 .$$

Since the domain Dom is bounded, one has to add a boundary condition, such as the perfect conductor boundary condition (1.135). The problem to solve can be expressed as
$$\frac{d^2 \boldsymbol{U}}{d t^2}(t)+A \boldsymbol{U}(t)=\boldsymbol{F}(t) \text { for } t>0, \quad \boldsymbol{U}(0)=\boldsymbol{U}_0, \frac{d \boldsymbol{U}}{d t}(0)=\boldsymbol{U}_1,$$
Where:

• $\boldsymbol{U}(t)$ is the unknown, here the electric field;
• $A$ is the operator acting on the solution, here $c^2$ curl curl ;
• $\boldsymbol{F}(t)$ is the right-hand side, here $-\varepsilon_0^{-1} \partial_t \boldsymbol{J}$;
• $\boldsymbol{U}0, \boldsymbol{U}_1$ is the initial data. The problem is set in the vector space of divergence-free solutions with vanishing tangential components on the boundary, the so-called domain of the operator $A$. It can be proven that the operator $A$ is compact, self-adjoint and positive-definite, and that there exists an orthonormal basis of eigenmodes $\left(\boldsymbol{\mu}_k\right){k \geq 1}$ and a set of corresponding non-negative eigenvalues $\left(\lambda_k\right){k \geq 1}$ (counted with their multiplicity) such that $A \mu_k=\lambda_k \mu_k$ for all $k \geq 1$ (we refer the reader to Chap. 8 for details). Moreover, the multiplicities of all eigenvalues are finite, and furthermore, $\lim {k \rightarrow+\infty} \lambda_k=+\infty$. The set $\left{\lambda_k, k \geq 1\right}$ is the spectrum of the operator $A$.

## 物理代写|电磁学代写electromagnetism代考|Boundary Conditions

As we remarked at the beginning of this section, the differential Maxwell equations are insufficient to characterize the fields in a strict subset of $\mathbb{R}^3$. On the other hand, the integral Maxwell equations yield four interface conditions, respectively described by Eqs. (1.11) and (1.12). How can these conditions be used? Let us call $\mathcal{O}$ the domain of interest, and $\partial \mathcal{O}$ its boundary. Note that $\partial \mathcal{O}$ can alternatively be seen as the interface between $\mathcal{O}$ and $\mathbb{R}^3 \backslash \overline{\mathcal{O}}$, so the electromagnetic fields fulfill conditions (1.11-1.12) on $\partial \mathcal{O}$. In addition, the behavior of the electromagnetic fields is known in $\mathbb{R}^3 \backslash \mathcal{O}$ (otherwise, we would have to compute them!) or, more realistically, in an exterior domain $\mathcal{O}^{\prime}$ included in $\mathbb{R}^3 \backslash \overline{\mathcal{O}}$, such that $\overline{\mathcal{O}} \cap \overline{\mathcal{O}}-\partial \mathcal{O}$. As a consequence, one can gather some useful information as to the behavior of the fields in $\mathcal{O}$, on the boundary $\partial \mathcal{O}$.
For instance, let us assume now that the domain $\mathcal{O}$ is bounded, or partially bounded (i.e., along one direction, like the “pipe” in Fig. 1.1), and that it is encased (at least locally) in a perfect conductor. Then, as we saw in Sect. 1.1, the fields vanish outside $\mathcal{O}$ (cf. our discussion on skin depth and on the notion of perfect conductor). From condition (1.11 right), we infer that
$$\boldsymbol{B} \cdot \boldsymbol{n}=0 \text { on } \partial \mathcal{O},$$
with $\boldsymbol{n}$ the unit outward normal vector to $\partial \mathcal{O}$, with the convention that outward goes from $\mathcal{O}$ to $\mathcal{O}^{\prime}$. Likewise, from condition ( $1.12$ left), we get
$$\boldsymbol{E} \times \boldsymbol{n}=0 \text { on } \partial \mathcal{O} .$$

The conclusion is that the normal component $B_n=\boldsymbol{B} \cdot \boldsymbol{n}{\mid \partial \mathcal{O}}$ (respectively tangential components $\left.\boldsymbol{E}{\top}=\boldsymbol{n} \times(\boldsymbol{E} \times \boldsymbol{n})_{\mid \partial \mathcal{O}}\right)$ of $\boldsymbol{B}$ (respectively $\boldsymbol{E}$ ) uniformly vanish on $\partial \mathcal{O}$ : we call these conditions ${ }^{17}$ the perfect conductor boundary conditions.

From the physical point of view, these conditions are macroscopic, since they result from the idealization of quantities defined on surfaces. On the other hand, from a mathematical point of view, these conditions are sufficient to ensure the uniqueness of the solution, in the absence of topological considerations. As we shall see in Chap. 5, condition (1.134) can be rigorously inferred from condition (1.135), whereas the reciprocal assertion is not valid.

# 电磁学代考

## 物理代写|电磁学代写电磁学代考|时谐现象

.

$$\boldsymbol{E}(0)=\boldsymbol{E}_0, \quad \frac{\partial \boldsymbol{E}}{\partial t}(0)=\boldsymbol{E}_1 .$$ 时
\begin{aligned} &\frac{\partial^2 \boldsymbol{E}}{\partial t^2}+c^2 \text { curl curl } \boldsymbol{E}=-\frac{1}{\varepsilon_0} \frac{\partial \boldsymbol{J}}{\partial t}, \ &\operatorname{div} \boldsymbol{E}-0, \end{aligned}

$$\frac{d^2 \boldsymbol{U}}{d t^2}(t)+A \boldsymbol{U}(t)=\boldsymbol{F}(t) \text { for } t>0, \quad \boldsymbol{U}(0)=\boldsymbol{U}_0, \frac{d \boldsymbol{U}}{d t}(0)=\boldsymbol{U}_1,$$

• $\boldsymbol{U}(t)$ 这里的电场是未知的，
• $A$ 算符是否作用于解 $c^2$
• $\boldsymbol{F}(t)$ 是右边吗 $-\varepsilon_0^{-1} \partial_t \boldsymbol{J}$
• .$\boldsymbol{U}0, \boldsymbol{U}_1$ 是初始数据。这个问题被设定在无散度解的向量空间中边界上切向分量消失，也就是所谓的算子域 $A$。可以证明，经营者 $A$ 是紧的，自伴随的和确定的，并且存在特征模的标准正交基 $\left(\boldsymbol{\mu}_k\right){k \geq 1}$ 和一组对应的非负特征值 $\left(\lambda_k\right){k \geq 1}$ (以它们的多样性计算)如此 $A \mu_k=\lambda_k \mu_k$ 为所有人 $k \geq 1$ (详情请读者参阅第八章)。而且，所有特征值的多重度都是有限的，并且， $\lim {k \rightarrow+\infty} \lambda_k=+\infty$。布景 $\left{\lambda_k, k \geq 1\right}$ 是算子的频谱吗 $A$.

## 物理代写|电磁学代写电磁代考|边界条件

$$\boldsymbol{B} \cdot \boldsymbol{n}=0 \text { on } \partial \mathcal{O},$$
with $\boldsymbol{n}$ 单位向外的法向量 $\partial \mathcal{O}$和向外的约定 $\mathcal{O}$ 到 $\mathcal{O}^{\prime}$。同样，from condition ( $1.12$ 左)，我们得到
$$\boldsymbol{E} \times \boldsymbol{n}=0 \text { on } \partial \mathcal{O} .$$

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