## 物理代写|电动力学代写electromagnetism代考|FYS3500

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

## 物理代写|电动力学代写electromagnetism代考|The Principal Value Integral and Plemelj Formula

The Cauchy principal value is a generalized function defined by its action under an integral with an arbitrary function $f(x)$, namely,
$$\mathcal{P} \int_{-\infty}^{\infty} d x \frac{f(x)}{x-x_0}=\lim {\epsilon \rightarrow 0}\left[\int{-\infty}^{x_0-\epsilon} d x \frac{f(x)}{x-x_0}+\int_{x_0+\epsilon}^{\infty} d x \frac{f(x)}{x-x_0}\right] .$$
An important application where the principal value plays a role is the Plemelj formula:
$$\lim _{\epsilon \rightarrow 0} \frac{1}{x-x_0 \pm i \epsilon}=\mathcal{P} \frac{1}{x-x_0} \mp i \pi \delta\left(x-x_0\right) .$$
This expression is symbolic in the sense that it gains meaning when we multiply every term by an arbitrary function $f(x)$ and integrate over $x$ from $-\infty$ to $\infty$.
The correctness of (1.105) can be appreciated from Figure 1.4 and the identity
$$\frac{1}{x-x_0 \pm i \epsilon}=\frac{x-x_0}{\left(x-x_0\right)^2+\epsilon^2} \mp i \frac{\epsilon}{\left(x-x_0\right)^2+\epsilon^2} .$$
The real part of (1.106) generates the principal value in (1.105) because it is a symmetrically cut-off version of $1 /\left(x-x_0\right)$. The imaginary part of (1.106) generates the delta function in (1.105) by virtue of (1.97).

## 物理代写|电动力学代写electromagnetism代考|The Step Function and Sign Function

The Heaviside step function $\Theta(x)$ is defined by
$$\Theta(x)= \begin{cases}0 & x<0, \ 1 & x>0 .\end{cases}$$
The delta function is the derivative of the theta function,
$$\frac{d \Theta(x)}{d x}=\delta(x) .$$

A useful representation is
$$\Theta(x)=\lim {\epsilon \rightarrow 0} \frac{i}{2 \pi} \int{-\infty}^{\infty} d k \frac{1}{k+i \epsilon} e^{-i k x} .$$
The sign function $\operatorname{sgn}(x)$ is defined by
$$\operatorname{sgn}(x)=\frac{d}{d x}|x|= \begin{cases}-1 & x<0, \ 1 & x>0 .\end{cases}$$
A convenient representation is
$$\operatorname{sgn}(x)=-1+2 \int_{-\infty}^x d y \delta(y) .$$

The definition (1.93) leads us to define a three-dimensional delta function using an integral over a volume $V$ and a smooth but otherwise arbitrary “test” function $f(\mathbf{r})$ :
$$\int_V d^3 r f(\mathbf{r}) \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)= \begin{cases}f\left(\mathbf{r}^{\prime}\right) & \mathbf{r}^{\prime} \in V, \ 0 & \mathbf{r}^{\prime} \notin V .\end{cases}$$
A less formal definition consistent with (1.112) is
$$\delta(\mathbf{r})=0 \text { for } \mathbf{r} \neq 0 \quad \text { but } \quad \int_V d^3 r \delta(\mathbf{r})= \begin{cases}1 & \mathbf{r}=0 \in V, \ 0 & \mathbf{r}=0 \notin V .\end{cases}$$
These definitions tell us that $\delta(\mathbf{r})$ has dimensions of inverse volume. In Cartesian coordinates,
$$\delta(\mathbf{r})=\delta(x) \delta(y) \delta(z)$$
In curvilinear coordinates, the constraint on the right side of (1.113) and the form of the volume elements for cylindrical and spherical coordinates imply that
$$\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)=\frac{\delta\left(\rho-\rho^{\prime}\right) \delta\left(\phi-\phi^{\prime}\right) \delta\left(z-z^{\prime}\right)}{\rho}=\frac{\delta\left(r-r^{\prime}\right) \delta\left(\theta-\theta^{\prime}\right) \delta\left(\phi-\phi^{\prime}\right)}{r^2 \sin \theta} .$$
The special case $\mathbf{r}^{\prime}=0$ requires that we define the one-dimensional radial delta function so
$$\int_0^{\infty} d r \delta(r)=1$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|The Principal Value Integral and Plemelj Formula

$$\mathcal{P} \int_{-\infty}^{\infty} d x \frac{f(x)}{x-x_0}=\lim {\epsilon \rightarrow 0}\left[\int{-\infty}^{x_0-\epsilon} d x \frac{f(x)}{x-x_0}+\int_{x_0+\epsilon}^{\infty} d x \frac{f(x)}{x-x_0}\right] .$$

$$\lim _{\epsilon \rightarrow 0} \frac{1}{x-x_0 \pm i \epsilon}=\mathcal{P} \frac{1}{x-x_0} \mp i \pi \delta\left(x-x_0\right) .$$

(1.105)的正确性可以从图1.4和恒等式中看出
$$\frac{1}{x-x_0 \pm i \epsilon}=\frac{x-x_0}{\left(x-x_0\right)^2+\epsilon^2} \mp i \frac{\epsilon}{\left(x-x_0\right)^2+\epsilon^2} .$$
(1.106)的实部生成(1.105)中的主值，因为它是$1 /\left(x-x_0\right)$的对称截止版本。(1.106)的虚部通过(1.97)生成(1.105)中的函数。

## 物理代写|电动力学代写electromagnetism代考|The Step Function and Sign Function

Heaviside阶跃函数$\Theta(x)$定义为
$$\Theta(x)= \begin{cases}0 & x<0, \ 1 & x>0 .\end{cases}$$

$$\frac{d \Theta(x)}{d x}=\delta(x) .$$

$$\Theta(x)=\lim {\epsilon \rightarrow 0} \frac{i}{2 \pi} \int{-\infty}^{\infty} d k \frac{1}{k+i \epsilon} e^{-i k x} .$$

$$\operatorname{sgn}(x)=\frac{d}{d x}|x|= \begin{cases}-1 & x<0, \ 1 & x>0 .\end{cases}$$

$$\operatorname{sgn}(x)=-1+2 \int_{-\infty}^x d y \delta(y) .$$

$$\int_V d^3 r f(\mathbf{r}) \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)= \begin{cases}f\left(\mathbf{r}^{\prime}\right) & \mathbf{r}^{\prime} \in V, \ 0 & \mathbf{r}^{\prime} \notin V .\end{cases}$$

$$\delta(\mathbf{r})=0 \text { for } \mathbf{r} \neq 0 \quad \text { but } \quad \int_V d^3 r \delta(\mathbf{r})= \begin{cases}1 & \mathbf{r}=0 \in V, \ 0 & \mathbf{r}=0 \notin V .\end{cases}$$

$$\delta(\mathbf{r})=\delta(x) \delta(y) \delta(z)$$

$$\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)=\frac{\delta\left(\rho-\rho^{\prime}\right) \delta\left(\phi-\phi^{\prime}\right) \delta\left(z-z^{\prime}\right)}{\rho}=\frac{\delta\left(r-r^{\prime}\right) \delta\left(\theta-\theta^{\prime}\right) \delta\left(\phi-\phi^{\prime}\right)}{r^2 \sin \theta} .$$

$$\int_0^{\infty} d r \delta(r)=1$$

## 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。

## 物理代写|电动力学代写electromagnetism代考|NUC-303

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

## 物理代写|电动力学代写electromagnetism代考|The Principal Value Integral and Plemelj Formula

The Cauchy principal value is a generalized function defined by its action under an integral with an arbitrary function $f(x)$, namely,
$$\mathcal{P} \int_{-\infty}^{\infty} d x \frac{f(x)}{x-x_0}=\lim {\epsilon \rightarrow 0}\left[\int{-\infty}^{x_0-\epsilon} d x \frac{f(x)}{x-x_0}+\int_{x_0+\epsilon}^{\infty} d x \frac{f(x)}{x-x_0}\right] .$$
An important application where the principal value plays a role is the Plemelj formula:
$$\lim _{\epsilon \rightarrow 0} \frac{1}{x-x_0 \pm i \epsilon}=\mathcal{P} \frac{1}{x-x_0} \mp i \pi \delta\left(x-x_0\right) .$$
This expression is symbolic in the sense that it gains meaning when we multiply every term by an arbitrary function $f(x)$ and integrate over $x$ from $-\infty$ to $\infty$.
The correctness of (1.105) can be appreciated from Figure 1.4 and the identity
$$\frac{1}{x-x_0 \pm i \epsilon}=\frac{x-x_0}{\left(x-x_0\right)^2+\epsilon^2} \mp i \frac{\epsilon}{\left(x-x_0\right)^2+\epsilon^2} .$$
The real part of (1.106) generates the principal value in (1.105) because it is a symmetrically cut-off version of $1 /\left(x-x_0\right)$. The imaginary part of (1.106) generates the delta function in (1.105) by virtue of (1.97).

## 物理代写|电动力学代写electromagnetism代考|The Step Function and Sign Function

The Heaviside step function $\Theta(x)$ is defined by
$$\Theta(x)= \begin{cases}0 & x<0, \ 1 & x>0 .\end{cases}$$
The delta function is the derivative of the theta function,
$$\frac{d \Theta(x)}{d x}=\delta(x) .$$

A useful representation is
$$\Theta(x)=\lim {\epsilon \rightarrow 0} \frac{i}{2 \pi} \int{-\infty}^{\infty} d k \frac{1}{k+i \epsilon} e^{-i k x} .$$
The sign function $\operatorname{sgn}(x)$ is defined by
$$\operatorname{sgn}(x)=\frac{d}{d x}|x|= \begin{cases}-1 & x<0, \ 1 & x>0 .\end{cases}$$
A convenient representation is
$$\operatorname{sgn}(x)=-1+2 \int_{-\infty}^x d y \delta(y) .$$

The definition (1.93) leads us to define a three-dimensional delta function using an integral over a volume $V$ and a smooth but otherwise arbitrary “test” function $f(\mathbf{r})$ :
$$\int_V d^3 r f(\mathbf{r}) \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)= \begin{cases}f\left(\mathbf{r}^{\prime}\right) & \mathbf{r}^{\prime} \in V, \ 0 & \mathbf{r}^{\prime} \notin V .\end{cases}$$
A less formal definition consistent with (1.112) is
$$\delta(\mathbf{r})=0 \text { for } \mathbf{r} \neq 0 \quad \text { but } \quad \int_V d^3 r \delta(\mathbf{r})= \begin{cases}1 & \mathbf{r}=0 \in V, \ 0 & \mathbf{r}=0 \notin V .\end{cases}$$
These definitions tell us that $\delta(\mathbf{r})$ has dimensions of inverse volume. In Cartesian coordinates,
$$\delta(\mathbf{r})=\delta(x) \delta(y) \delta(z)$$
In curvilinear coordinates, the constraint on the right side of (1.113) and the form of the volume elements for cylindrical and spherical coordinates imply that
$$\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)=\frac{\delta\left(\rho-\rho^{\prime}\right) \delta\left(\phi-\phi^{\prime}\right) \delta\left(z-z^{\prime}\right)}{\rho}=\frac{\delta\left(r-r^{\prime}\right) \delta\left(\theta-\theta^{\prime}\right) \delta\left(\phi-\phi^{\prime}\right)}{r^2 \sin \theta} .$$
The special case $\mathbf{r}^{\prime}=0$ requires that we define the one-dimensional radial delta function so
$$\int_0^{\infty} d r \delta(r)=1$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|The Principal Value Integral and Plemelj Formula

$$\mathcal{P} \int_{-\infty}^{\infty} d x \frac{f(x)}{x-x_0}=\lim {\epsilon \rightarrow 0}\left[\int{-\infty}^{x_0-\epsilon} d x \frac{f(x)}{x-x_0}+\int_{x_0+\epsilon}^{\infty} d x \frac{f(x)}{x-x_0}\right] .$$

$$\lim _{\epsilon \rightarrow 0} \frac{1}{x-x_0 \pm i \epsilon}=\mathcal{P} \frac{1}{x-x_0} \mp i \pi \delta\left(x-x_0\right) .$$

(1.105)的正确性可以从图1.4和恒等式中看出
$$\frac{1}{x-x_0 \pm i \epsilon}=\frac{x-x_0}{\left(x-x_0\right)^2+\epsilon^2} \mp i \frac{\epsilon}{\left(x-x_0\right)^2+\epsilon^2} .$$
(1.106)的实部生成(1.105)中的主值，因为它是$1 /\left(x-x_0\right)$的对称截止版本。(1.106)的虚部通过(1.97)生成(1.105)中的函数。

## 物理代写|电动力学代写electromagnetism代考|The Step Function and Sign Function

Heaviside阶跃函数$\Theta(x)$定义为
$$\Theta(x)= \begin{cases}0 & x<0, \ 1 & x>0 .\end{cases}$$

$$\frac{d \Theta(x)}{d x}=\delta(x) .$$

$$\Theta(x)=\lim {\epsilon \rightarrow 0} \frac{i}{2 \pi} \int{-\infty}^{\infty} d k \frac{1}{k+i \epsilon} e^{-i k x} .$$

$$\operatorname{sgn}(x)=\frac{d}{d x}|x|= \begin{cases}-1 & x<0, \ 1 & x>0 .\end{cases}$$

$$\operatorname{sgn}(x)=-1+2 \int_{-\infty}^x d y \delta(y) .$$

$$\int_V d^3 r f(\mathbf{r}) \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)= \begin{cases}f\left(\mathbf{r}^{\prime}\right) & \mathbf{r}^{\prime} \in V, \ 0 & \mathbf{r}^{\prime} \notin V .\end{cases}$$

$$\delta(\mathbf{r})=0 \text { for } \mathbf{r} \neq 0 \quad \text { but } \quad \int_V d^3 r \delta(\mathbf{r})= \begin{cases}1 & \mathbf{r}=0 \in V, \ 0 & \mathbf{r}=0 \notin V .\end{cases}$$

$$\delta(\mathbf{r})=\delta(x) \delta(y) \delta(z)$$

$$\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)=\frac{\delta\left(\rho-\rho^{\prime}\right) \delta\left(\phi-\phi^{\prime}\right) \delta\left(z-z^{\prime}\right)}{\rho}=\frac{\delta\left(r-r^{\prime}\right) \delta\left(\theta-\theta^{\prime}\right) \delta\left(\phi-\phi^{\prime}\right)}{r^2 \sin \theta} .$$

$$\int_0^{\infty} d r \delta(r)=1$$

## 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。

## 物理代写|电动力学代写electromagnetism代考|PHYS102

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

## 物理代写|电动力学代写electromagnetism代考|Vector Identities in Curvilinear Components

Care is needed to interpret the vector identities in Examples 1.2 and 1.3 when the vectors in question are decomposed into spherical or cylindrical components such as $\mathbf{A}=A_r \hat{\mathbf{r}}+A_\theta \hat{\boldsymbol{\theta}}+A_\phi \hat{\boldsymbol{\phi}}$. This can be seen from Example 1.3 where the final step is no longer valid because $\hat{\mathbf{r}}, \hat{\boldsymbol{\theta}}$, and $\hat{\boldsymbol{\phi}}$ are not constant vectors. In other words,
$$\nabla^2 \mathbf{A}=\nabla \cdot \nabla\left(A_r \hat{\mathbf{r}}+A_\theta \hat{\boldsymbol{\theta}}+A_\phi \hat{\boldsymbol{\phi}}\right) \neq \hat{\mathbf{r}} \nabla^2 A_r+\hat{\boldsymbol{\theta}} \nabla^2 A_\theta+\hat{\boldsymbol{\phi}} \nabla^2 A_\phi$$
One way to proceed is to work out the components of $\nabla\left(A_r \hat{\mathbf{r}}\right), \nabla\left(A_\theta \hat{\boldsymbol{\theta}}\right)$, and $\nabla\left(A_\phi \hat{\boldsymbol{\phi}}\right)$. Alternatively, we may simply define the meaning of the operation $\nabla^2 \mathbf{A}$ when $\mathbf{A}$ is expressed using curvilinear components. For example,
$$\left[\nabla^2 \mathbf{A}\right]\phi \equiv \partial\phi(\nabla \cdot \mathbf{A})-[\nabla \times(\nabla \times \mathbf{A})]\phi,$$ and similarly for $\left(\nabla^2 \mathbf{A}\right)_r$ and $\left(\nabla^2 \mathbf{A}\right)\theta$.
Exactly the same issue arises when we examine the last step in Example 1.2, namely
$$[\nabla \times(\mathbf{A} \times \mathbf{B})]i=A_i \nabla \cdot \mathbf{B}-(\mathbf{A} \cdot \nabla) B_i+(\mathbf{B} \cdot \nabla) A_i-B_i \nabla \cdot \mathbf{A} .$$ By construction, this equation makes sense when $i$ stands for $x, y$, or $z$. It does not make sense if $i$ stands for, say, $r, \theta$, or $\phi$. On the other hand, the full vector version of the identity is correct as long as we retain the $r, \theta$, and $\phi$ variations of $\hat{\mathbf{r}}, \hat{\boldsymbol{\theta}}$, and $\hat{\boldsymbol{\phi}}$. For example, $$(\mathbf{A} \cdot \nabla) \mathbf{B}=\left[A_r \frac{\partial}{\partial r}+\frac{A\theta}{r} \frac{\partial}{\partial \theta}+\frac{A_\phi}{r \sin \theta} \frac{\partial}{\partial \phi}\right]\left(B_r \hat{\mathbf{r}}+B_\theta \hat{\boldsymbol{\theta}}+B_\phi \hat{\boldsymbol{\phi}}\right) .$$

## 物理代写|电动力学代写electromagnetism代考|Functions of r and |r|

The position vector is $\mathbf{r}=r \hat{\mathbf{r}}$ with $r=\sqrt{x^2+y^2+z^2}$. If $f(r)$ is a scalar function and $f^{\prime}(r)=d f / d r$,
$$\begin{array}{cc} \nabla r=\hat{\mathbf{r}} & \nabla \times \mathbf{r}=0 \ \nabla f=f^{\prime} \hat{\mathbf{r}} & \nabla^2 f=\frac{\left(r^2 f^{\prime}\right)^{\prime}}{r^2} \ \nabla \cdot(f \mathbf{r})=\frac{\left(r^3 f\right)^{\prime}}{r^2} & \nabla \times(f \mathbf{r})=0 . \end{array}$$
Similarly, if $\mathbf{g}(r)$ is a vector function and $\mathbf{c}$ is a constant vector,
$$\begin{array}{lr} \nabla \cdot \mathbf{g}=\mathbf{g}^{\prime} \cdot \hat{\mathbf{r}} & \nabla \times \mathbf{g}=\hat{\mathbf{r}} \times \mathbf{g}^{\prime} \ (\mathbf{g} \cdot \nabla) \mathbf{r}=\mathbf{g} & (\mathbf{r} \cdot \nabla) \mathbf{g}=r \mathbf{g}^{\prime} \end{array}$$

$$\begin{array}{cr} \nabla(\mathbf{r} \cdot \mathbf{g})=\mathbf{g}+\frac{\left(\mathbf{r} \cdot \mathbf{g}^{\prime}\right) \mathbf{r}}{r} & \nabla \cdot(\mathbf{g} \times \mathbf{r})=0 \ \nabla \times(\mathbf{g} \times \mathbf{r})=2 \mathbf{g}+r \mathbf{g}^{\prime}-\frac{\left(\mathbf{r} \cdot \mathbf{g}^{\prime}\right) \mathbf{r}}{r} & \nabla(\mathbf{c} \cdot \mathbf{r})=\mathbf{c} . \end{array}$$
Functions of $\mathbf{r}-\mathbf{r}^{\prime}$
Let $\mathbf{R}=\mathbf{r}-\mathbf{r}^{\prime}=\left(x-x^{\prime}\right) \hat{\mathbf{x}}+\left(y-y^{\prime}\right) \hat{\mathbf{y}}+\left(z-z^{\prime}\right) \hat{\mathbf{z}}$. Then,
$$\nabla f(R)=f^{\prime}(R) \hat{\mathbf{R}} \quad \nabla \cdot \mathbf{g}(R)=\mathbf{g}^{\prime}(R) \cdot \hat{\mathbf{R}} \quad \nabla \times \mathbf{g}(R)=\hat{\mathbf{R}} \times \mathbf{g}^{\prime}(R) .$$
Moreover, because
$$\nabla=\hat{\mathbf{x}} \frac{\partial}{\partial x}+\hat{\mathbf{y}} \frac{\partial}{\partial y}+\hat{\mathbf{z}} \frac{\partial}{\partial z} \quad \text { and } \quad \nabla^{\prime}=\hat{\mathbf{x}} \frac{\partial}{\partial x^{\prime}}+\hat{\mathbf{y}} \frac{\partial}{\partial y^{\prime}}+\hat{\mathbf{z}} \frac{\partial}{\partial z^{\prime}},$$
it it straightforward to confirm that
$$\nabla^{\prime} f(R)=-\nabla f(R)$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Vector Identities in Curvilinear Components

$$\nabla^2 \mathbf{A}=\nabla \cdot \nabla\left(A_r \hat{\mathbf{r}}+A_\theta \hat{\boldsymbol{\theta}}+A_\phi \hat{\boldsymbol{\phi}}\right) \neq \hat{\mathbf{r}} \nabla^2 A_r+\hat{\boldsymbol{\theta}} \nabla^2 A_\theta+\hat{\boldsymbol{\phi}} \nabla^2 A_\phi$$

$$\left[\nabla^2 \mathbf{A}\right]\phi \equiv \partial\phi(\nabla \cdot \mathbf{A})-[\nabla \times(\nabla \times \mathbf{A})]\phi,$$ 类似地 $\left(\nabla^2 \mathbf{A}\right)r$ 和 $\left(\nabla^2 \mathbf{A}\right)\theta$． 当我们检查例1.2中的最后一步时，也会出现完全相同的问题，即 $$[\nabla \times(\mathbf{A} \times \mathbf{B})]i=A_i \nabla \cdot \mathbf{B}-(\mathbf{A} \cdot \nabla) B_i+(\mathbf{B} \cdot \nabla) A_i-B_i \nabla \cdot \mathbf{A} .$$ 通过构造，当 $i$ 代表 $x, y$，或 $z$． 这没有意义，如果 $i$ 代表，比如说， $r, \theta$，或 $\phi$． 另一方面，恒等式的完整向量版本是正确的，只要我们保留 $r, \theta$，和 $\phi$ 的变体 $\hat{\mathbf{r}}, \hat{\boldsymbol{\theta}}$，和 $\hat{\boldsymbol{\phi}}$． 例如， $$(\mathbf{A} \cdot \nabla) \mathbf{B}=\left[A_r \frac{\partial}{\partial r}+\frac{A\theta}{r} \frac{\partial}{\partial \theta}+\frac{A\phi}{r \sin \theta} \frac{\partial}{\partial \phi}\right]\left(B_r \hat{\mathbf{r}}+B_\theta \hat{\boldsymbol{\theta}}+B_\phi \hat{\boldsymbol{\phi}}\right) .$$

## 物理代写|电动力学代写electromagnetism代考|Functions of r and |r|

$$\begin{array}{cc} \nabla r=\hat{\mathbf{r}} & \nabla \times \mathbf{r}=0 \ \nabla f=f^{\prime} \hat{\mathbf{r}} & \nabla^2 f=\frac{\left(r^2 f^{\prime}\right)^{\prime}}{r^2} \ \nabla \cdot(f \mathbf{r})=\frac{\left(r^3 f\right)^{\prime}}{r^2} & \nabla \times(f \mathbf{r})=0 . \end{array}$$

$$\begin{array}{lr} \nabla \cdot \mathbf{g}=\mathbf{g}^{\prime} \cdot \hat{\mathbf{r}} & \nabla \times \mathbf{g}=\hat{\mathbf{r}} \times \mathbf{g}^{\prime} \ (\mathbf{g} \cdot \nabla) \mathbf{r}=\mathbf{g} & (\mathbf{r} \cdot \nabla) \mathbf{g}=r \mathbf{g}^{\prime} \end{array}$$

$$\begin{array}{cr} \nabla(\mathbf{r} \cdot \mathbf{g})=\mathbf{g}+\frac{\left(\mathbf{r} \cdot \mathbf{g}^{\prime}\right) \mathbf{r}}{r} & \nabla \cdot(\mathbf{g} \times \mathbf{r})=0 \ \nabla \times(\mathbf{g} \times \mathbf{r})=2 \mathbf{g}+r \mathbf{g}^{\prime}-\frac{\left(\mathbf{r} \cdot \mathbf{g}^{\prime}\right) \mathbf{r}}{r} & \nabla(\mathbf{c} \cdot \mathbf{r})=\mathbf{c} . \end{array}$$
$\mathbf{r}-\mathbf{r}^{\prime}$的功能

$$\nabla f(R)=f^{\prime}(R) \hat{\mathbf{R}} \quad \nabla \cdot \mathbf{g}(R)=\mathbf{g}^{\prime}(R) \cdot \hat{\mathbf{R}} \quad \nabla \times \mathbf{g}(R)=\hat{\mathbf{R}} \times \mathbf{g}^{\prime}(R) .$$

$$\nabla=\hat{\mathbf{x}} \frac{\partial}{\partial x}+\hat{\mathbf{y}} \frac{\partial}{\partial y}+\hat{\mathbf{z}} \frac{\partial}{\partial z} \quad \text { and } \quad \nabla^{\prime}=\hat{\mathbf{x}} \frac{\partial}{\partial x^{\prime}}+\hat{\mathbf{y}} \frac{\partial}{\partial y^{\prime}}+\hat{\mathbf{z}} \frac{\partial}{\partial z^{\prime}},$$

$$\nabla^{\prime} f(R)=-\nabla f(R)$$

## 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。

## 物理代写|电动力学代写electromagnetism代考|General Discussion

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

## 物理代写|电动力学代写electromagnetism代考|General Discussion

We now turn to the consideration of the distribution of energy and momentum of electromagnetic fields within material media, following closely the development of Chapter 3 (however, here we will assume no magnetic charge is present). We will base our discussion on the macroscopic form of Maxwell’s equations, (4.60). Accordingly, the rate at which the electric field does work on the free charges is
$$\mathbf{J} \cdot \mathbf{E}=\mathbf{E} \cdot\left[\frac{c}{4 \pi} \nabla \times \mathbf{H}-\frac{1}{4 \pi} \frac{\partial}{\partial t} \mathbf{D}\right]$$
If we add to this the parallel equation, appropriate to the absence of magnetic charge,
$$0=\mathbf{H} \cdot\left[-\frac{c}{4 \pi} \boldsymbol{\nabla} \times \mathbf{E}-\frac{1}{4 \pi} \frac{\partial}{\partial t} \mathbf{B}\right]$$
we obtain the suggestive form
$$\mathbf{J} \cdot \mathbf{E}=-\boldsymbol{\nabla} \cdot\left(\frac{c}{4 \pi} \mathbf{E} \times \mathbf{H}\right)-\frac{1}{4 \pi}\left(\mathbf{E} \cdot \frac{\partial}{\partial t} \mathbf{D}+\mathbf{H} \cdot \frac{\partial}{\partial t} \mathbf{B}\right) .$$
[Recall that if there were free magnetic currents, (7.2) would represent the work done on the magnetic charges.] Our aim is to write this result as a local energy conservation law. We immediately identify, from the divergence term, the energy flux or Poynting’s vector, $\mathbf{S}$, to be
$$\mathbf{S}=\frac{c}{4 \pi} \mathbf{E} \times \mathbf{H}$$
which has the same form as that of the microscopic flux, (3.5), except that here B is replaced by $\mathbf{H}$. More intractable is the identification of the last term in (7.3). To what extent is it the negative time derivative of an energy density, $-\partial U / \partial t$ ? If there does exist some quantity $U$ such that
$$\frac{\partial}{\partial t} U \stackrel{?}{=} \frac{1}{4 \pi}\left(\mathbf{E} \cdot \frac{\partial}{\partial t} \mathbf{D}+\mathbf{H} \cdot \frac{\partial}{\partial t} \mathbf{B}\right),$$
we would have a local statement of energy conservation,
$$\frac{\partial}{\partial t} U+\nabla \cdot \mathbf{S}+\mathbf{J} \cdot \mathbf{E}=0 .$$
Similarly, we consider the rate at which momentum is transferred to the charges, or equivalently, the force density, $\mathbf{f}$,
$$\mathbf{f}=\rho \mathbf{E}+\frac{1}{c} \mathbf{J} \times \mathbf{B} .$$

## 物理代写|电动力学代写electromagnetism代考|Nondispersive Medium

We cannot proceed further without specific assumptions about the properties of the material medium. The simplest hypothesis is that of a homogeneous, isotropic, nondispersive medium,
$$\mathbf{D}=\epsilon \mathbf{E}, \quad \mathbf{B}=\mu \mathbf{H},$$
where $\epsilon$ and $\mu$ are constants. This is not an unrealistic situation for many substances over a sufficiently limited frequency range. For this case, the energy density and the stress tensor exist, and have the following forms:
$$U=\frac{\epsilon E^2+\mu H^2}{8 \pi}, \quad \mathbf{T}=1 \frac{\epsilon E^2+\mu H^2}{8 \pi}-\frac{\epsilon \mathbf{E E}+\mu \mathbf{H H}}{4 \pi},$$
while we recall that the energy flux and the momentum density are given by
$$\mathbf{S}=\frac{c}{4 \pi} \mathbf{E} \times \mathbf{H}, \quad \mathbf{G}=\frac{\epsilon \mu}{4 \pi c} \mathbf{E} \times \mathbf{H}=\frac{\epsilon \mu}{c^2} \mathbf{S} .$$
It is interesting that we can transform these expressions, as well as Maxwell’s equations, to look like those in vacuum, by redefining the fields, the charges, and the speed of light as follows:
$$E^{\prime}=\sqrt{\epsilon} E, \quad H^{\prime}=\sqrt{\mu} H, \quad c^{\prime}=\frac{c}{\sqrt{\epsilon \mu}}, \quad \rho^{\prime}=\frac{1}{\sqrt{\epsilon}} \rho, \quad J^{\prime}=\frac{1}{\sqrt{\epsilon}} J .$$
(See Problem 7.1.) The ratio of $c$ to $c^{\prime}$ is the index of refraction for the medium,
$$\frac{c}{c^{\prime}}=n=\sqrt{\epsilon \mu} \text {. }$$
By this transformation we see that the speed of propagation of electromagnetic waves in the medium is $c^{\prime}$; for propagation in a definite direction, the transcription from the vacuum statement (see Section 3.4 ) that $\mathbf{E}^{\prime}$ and $\mathbf{B}^{\prime}$ are mutually perpendicular and equal in magnitude is
$$\epsilon E^2=\mu H^2, \quad \mathbf{E} \cdot \mathbf{H}=0 .$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|General Discussion

$$\mathbf{J} \cdot \mathbf{E}=\mathbf{E} \cdot\left[\frac{c}{4 \pi} \nabla \times \mathbf{H}-\frac{1}{4 \pi} \frac{\partial}{\partial t} \mathbf{D}\right]$$

$$0=\mathbf{H} \cdot\left[-\frac{c}{4 \pi} \boldsymbol{\nabla} \times \mathbf{E}-\frac{1}{4 \pi} \frac{\partial}{\partial t} \mathbf{B}\right]$$

$$\mathbf{J} \cdot \mathbf{E}=-\boldsymbol{\nabla} \cdot\left(\frac{c}{4 \pi} \mathbf{E} \times \mathbf{H}\right)-\frac{1}{4 \pi}\left(\mathbf{E} \cdot \frac{\partial}{\partial t} \mathbf{D}+\mathbf{H} \cdot \frac{\partial}{\partial t} \mathbf{B}\right) .$$

$$\mathbf{S}=\frac{c}{4 \pi} \mathbf{E} \times \mathbf{H}$$

$$\frac{\partial}{\partial t} U \stackrel{?}{=} \frac{1}{4 \pi}\left(\mathbf{E} \cdot \frac{\partial}{\partial t} \mathbf{D}+\mathbf{H} \cdot \frac{\partial}{\partial t} \mathbf{B}\right),$$

$$\frac{\partial}{\partial t} U+\nabla \cdot \mathbf{S}+\mathbf{J} \cdot \mathbf{E}=0 .$$

$$\mathbf{f}=\rho \mathbf{E}+\frac{1}{c} \mathbf{J} \times \mathbf{B} .$$

## 物理代写|电动力学代写electromagnetism代考|Nondispersive Medium

$$\mathbf{D}=\epsilon \mathbf{E}, \quad \mathbf{B}=\mu \mathbf{H},$$

$$U=\frac{\epsilon E^2+\mu H^2}{8 \pi}, \quad \mathbf{T}=1 \frac{\epsilon E^2+\mu H^2}{8 \pi}-\frac{\epsilon \mathbf{E E}+\mu \mathbf{H H}}{4 \pi},$$

$$\mathbf{S}=\frac{c}{4 \pi} \mathbf{E} \times \mathbf{H}, \quad \mathbf{G}=\frac{\epsilon \mu}{4 \pi c} \mathbf{E} \times \mathbf{H}=\frac{\epsilon \mu}{c^2} \mathbf{S} .$$

$$E^{\prime}=\sqrt{\epsilon} E, \quad H^{\prime}=\sqrt{\mu} H, \quad c^{\prime}=\frac{c}{\sqrt{\epsilon \mu}}, \quad \rho^{\prime}=\frac{1}{\sqrt{\epsilon}} \rho, \quad J^{\prime}=\frac{1}{\sqrt{\epsilon}} J .$$
(见问题7.1)$c$与$c^{\prime}$的比值是介质的折射率，
$$\frac{c}{c^{\prime}}=n=\sqrt{\epsilon \mu} \text {. }$$

$$\epsilon E^2=\mu H^2, \quad \mathbf{E} \cdot \mathbf{H}=0 .$$

## 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。

## 物理代写|电动力学代写electromagnetism代考|Plasma

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

## 物理代写|电动力学代写electromagnetism代考|Plasma

Let us combine the results of the preceding two sections by considering the motion of free charge in a conducting dielectric material, for which the conduction current is
$$\mathbf{J}=\sigma \mathbf{E}=\frac{\sigma}{\epsilon} \mathbf{D}$$
First suppose that both $\sigma$ and $\epsilon$ are taken to be independent of frequency, an approximation which is valid for low frequencies. Then the local charge conservation equation,
$$\frac{\partial}{\partial t} \rho+\nabla \cdot \mathbf{J}=0,$$
becomes
$$\frac{\partial}{\partial t} \rho+\nabla \cdot \frac{\sigma}{\epsilon} \mathbf{D}=0 .$$

In the interior of a homogeneous substance, the use of $(4.60), \nabla \cdot \mathbf{D}=4 \pi \rho$, produces the differential equation
$$\frac{\partial}{\partial t} \rho+\frac{\sigma}{\epsilon} 4 \pi \rho=0$$
The solution to this equation, corresponding to an initial charge density $\rho(\mathbf{r}, 0)$, is
$$\rho(\mathbf{r}, t)=\rho(\mathbf{r}, 0) e^{-4 \pi \sigma t / \epsilon},$$
implying that the charge disappears from the interior of the conducting body at a rate measured by
$$\gamma^{\prime}=\frac{4 \pi \sigma}{\epsilon}$$

## 物理代写|电动力学代写electromagnetism代考|Polar Molecules

In the above model for the electric susceptibility, leading to (5.21), the dipole moments were induced by the applied electric field. But what about permanent electric dipole moments? Do individual atoms possess such static properties?

With the exception of atomic hydrogen where the orbital motion respects a preferred direction in space (as in the classical elliptical orbits), atomic electric dipole moments change direction in space so rapidly in response to the fast electronic motion that no average effect survives. But things are different with molecules, specifically those of a polar nature. In the example of $\mathrm{H}^{+} \mathrm{Cl}^{-}$, the hydrogenic electron is transferred to form the chlorine ion, and a dipole moment is associated with the relative motion of the heavy ions. Other examples of polar molecules associated with familiar substances are $\mathrm{H}_2 \mathrm{O}, \mathrm{SO}_2, \mathrm{NH}_3$, and $\mathrm{CH}_3 \mathrm{Cl}$. For such molecules, in isolation, it is not misleading to think of a permanent electric dipole moment that changes its spatial orientation only in response to the slow rotation of the molecule.

But molecules are not ordinarily isolated; they exist in an environment in which other molecules collide with them at a rate determined by the temperature of the substance. The effect of these collisions is to remove any particular spatial orientation of the dipole moments; it still requires an electric field to provide a preferred direction. But now there is a competition between the organizing effect of the electric field, with its preference for lower values of the energy,
$$E=-\mathbf{d} \cdot \mathbf{E}=-|\mathbf{d}||\mathbf{E}| \cos \theta$$
and the disorganizing effect of the ambient temperature $T$. For a static field, the net balance of that competition is expressed by the Boltzmann factor, which gives the probability of finding a configuration of energy $E$,
$$e^{-E / k T}$$
where $k$, the constant of Ludwig Boltzmann (1844-1906) has the value
$$k=1.381 \times 10^{-16} \mathrm{erg} / \mathrm{K}$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Plasma

$$\mathbf{J}=\sigma \mathbf{E}=\frac{\sigma}{\epsilon} \mathbf{D}$$

$$\frac{\partial}{\partial t} \rho+\nabla \cdot \mathbf{J}=0,$$

$$\frac{\partial}{\partial t} \rho+\nabla \cdot \frac{\sigma}{\epsilon} \mathbf{D}=0 .$$

$$\frac{\partial}{\partial t} \rho+\frac{\sigma}{\epsilon} 4 \pi \rho=0$$

$$\rho(\mathbf{r}, t)=\rho(\mathbf{r}, 0) e^{-4 \pi \sigma t / \epsilon},$$

$$\gamma^{\prime}=\frac{4 \pi \sigma}{\epsilon}$$

## 物理代写|电动力学代写electromagnetism代考|Polar Molecules

$$E=-\mathbf{d} \cdot \mathbf{E}=-|\mathbf{d}||\mathbf{E}| \cos \theta$$

$$e^{-E / k T}$$

$$k=1.381 \times 10^{-16} \mathrm{erg} / \mathrm{K}$$

## 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。

## 物理代写|电动力学代写electromagnetism代考|Discussion

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

## 物理代写|电动力学代写electromagnetism代考|Discussion

We have arrived at the Maxwell-Lorentz electrodynamics by combining three ingredients: the laws of electrostatics; the Galileo-Newton principle of relativity (charges at rest, and charges with a common velocity viewed by a co-moving observer, are physically indistinguishable); and the existence of electromagnetic waves that travel in a vacuum at the speed $c$. The historical line of development was otherwise. Until the beginning of the nineteenth century, electricity and magnetism were unrelated phenomena. The discovery in 1820 by Hans Christian Oersted (1777-1851) that an electric current influences a magnet-creates a magnetic field-is formulated, for stationary currents, in the field equation
$$\nabla \times B=\frac{4 \pi}{c} \mathbf{j}$$
The symbol $c$ that appears in this equation is the ratio of electromagnetic and electrostatic units of electricity (see Appendix A). Then, in 1831, Michael Faraday (1791-1867) discovered that relative motion of a wire and a magnet induces a voltage in the wire-creates an electric field. Such is the content of
$$-\nabla \times \mathbf{E}=\frac{1}{c} \frac{\partial}{\partial t} \mathbf{B}$$
which extends the magnetostatic relation
$$\boldsymbol{\nabla} \cdot \mathbf{B}=0$$
that expresses the empirical absence of single magnetic poles. Finally, in 1864, James Clerk Maxwell (1831-1879) recognized that the restriction to stationary currents in (1.69), as expressed by $\boldsymbol{\nabla} \cdot \mathbf{j}=0$, was removed in
$$\boldsymbol{\nabla} \times \mathbf{B}=\frac{4 \pi}{c} \mathbf{j}+\frac{1}{c} \frac{\partial}{\partial t} \mathbf{E}$$
when joined to the electrostatic equation
$$\boldsymbol{\nabla} \cdot \mathbf{E}=4 \pi \rho .$$

## 物理代写|电动力学代写electromagnetism代考|A Very Brief History of Magnetic Charge

It is said that Peregrinus in 1269 observed that magnets (lodestones) always have two poles, which he called north and south. This was elevated to a “hypothesis” by Ampère in the early 19th Century. The first theoretical calculation of the motion of a charged particle in the presence of a single magnetic pole was performed by Poincaré in 1896 to explain recent observations. A few years later, Thomson showed that a static system consisting of a magnetic pole and an electric charge possessed an angular momentum-see Problem 3.8. It was Dirac in 1931 who showed that magnetic charge was consistent with quantum mechanics only if electric and magnetic charges were quantized: For a system consisting of a pure magnetic charge $g$ and a pure electric charge $e, e g$ had to be an integral (or half-integral) multiple of $\hbar c$. Many people have contributed to the theory of magnetic charge subsequently; notable is the work of Schwinger in the 1960s and 1970s, especially his concept of dyons, particles which carry both electric and magnetic charge.

Many searches, both terrestrial and cosmic, have been carried out to find magnetic monopoles in nature, but, so far, to no avail. Worth mentioning is the induction technique of Luis Alvarez, et al. Positive reports were given by Price in 1975 [cited in the Reader’s Guide] and by Blas Cabrera in 1982. These, however, were never confirmed, and are no longer believed to offer any evidence for magnetic charge, even by their authors.

However, modern unified theories of fundamental interactions typically imply the existence of magnetic monopoles, or of dyons, often at extremely high mass scales $\left(\sim 10^{16} \mathrm{GeV}\right)$, but perhaps at nearly accessible energies $(\sim 10 \mathrm{TeV})$. Moreover, there appears to be no reason why an elementary monopole or dyon of the Dirac-Schwinger type could not exist. So experimental searches continue.

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Discussion

$$\nabla \times B=\frac{4 \pi}{c} \mathbf{j}$$

$$-\nabla \times \mathbf{E}=\frac{1}{c} \frac{\partial}{\partial t} \mathbf{B}$$

$$\boldsymbol{\nabla} \cdot \mathbf{B}=0$$

$$\boldsymbol{\nabla} \times \mathbf{B}=\frac{4 \pi}{c} \mathbf{j}+\frac{1}{c} \frac{\partial}{\partial t} \mathbf{E}$$

$$\boldsymbol{\nabla} \cdot \mathbf{E}=4 \pi \rho .$$

## 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。

## 物理代写|电动力学代写electromagnetism代考|A Time-Averaging Theorem

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## 物理代写|电动力学代写electromagnetism代考|A Time-Averaging Theorem

Let $A(\mathbf{r}, t)=a(\mathbf{r}) \exp (-i \omega t)$ and $B(\mathbf{r}, t)=b(\mathbf{r}) \exp (-i \omega t)$, where $a(\mathbf{r})$ and $b(\mathbf{r})$ are complex-valued functions. If $T=2 \pi / \omega$, it is useful to know that
$$\langle\operatorname{Re}[A(\mathbf{r}, t)] \operatorname{Re}[B(\mathbf{r}, t)]\rangle=\frac{1}{T} \int_0^T d t \operatorname{Re}[A(\mathbf{r}, t)] \operatorname{Re}[B(\mathbf{r}, t)]=\frac{1}{2} \operatorname{Re}\left[a(\mathbf{r}) b^(\mathbf{r})\right]$$ We prove (1.137) by writing $\operatorname{Re}[A]=\frac{1}{2}\left(A+A^\right)$ and $\operatorname{Re}[B]=\frac{1}{2}\left(B+B^\right)$ so $$\langle\operatorname{Re}[A] \operatorname{Re}[B]\rangle=\frac{1}{4 T} \int_0^T d t\left{a b e^{-2 i \omega t}+a^ b^* e^{2 i \omega t}+a b^+b a^\right}$$
The time-dependent terms in the integrand of (1.138) integrate to zero over one full period. Therefore,
$$\langle\operatorname{Re}[A] \operatorname{Re}[B]\rangle=\frac{1}{4}\left[a b^+a^ b\right]=\frac{1}{2} \operatorname{Re}\left[a b^\right]=\frac{1}{2} \operatorname{Re}\left[a^ b\right]$$

## 物理代写|电动力学代写electromagnetism代考|Orthogonal Transformations

Let $\left(\hat{\mathbf{e}}1, \hat{\mathbf{e}}_2, \hat{\mathbf{e}}_3\right)$ and $\left(\hat{\mathbf{e}}_1^{\prime}, \hat{\mathbf{e}}_2^{\prime}, \hat{\mathbf{e}}_3^{\prime}\right)$ be two sets of orthogonal Cartesian unit vectors. Each is a complete basis for vectors in three dimensions, so $$\hat{\mathbf{e}}_i^{\prime}=A{i j} \hat{\mathbf{e}}j$$ The set of scalars $A{i j}$ are called direction cosines. Using the unit vector properties from Section 1.2,
$$\delta_{i j}=\hat{\mathbf{e}}i^{\prime} \cdot \hat{\mathbf{e}}_j^{\prime}=A{i k} A_{j k}$$
Equation (1.141) says that the transpose of the matrix $\mathbf{A}$, called $\mathbf{A}^{\top}$, is identical to the inverse of the matrix $\mathbf{A}$, called $\mathbf{A}^{-1}$. This is the definition of a matrix that describes an orthogonal transformation,
$$\mathbf{A} \mathbf{A}^{\top}=\mathbf{A A}^{-1}=1$$
There are two classes of orthogonal coordinate transformations. These follow from the determinant of $(1.142)$ :
$$\operatorname{det}\left[\mathbf{A A}^{\top}\right]=\operatorname{det} \mathbf{A} \operatorname{det} \mathbf{A}^{\top}=(\operatorname{det} \mathbf{A})^2=1 .$$
A rotation has det $\mathbf{A}=1$. Figure 1.5(a) shows an example where
$$\mathbf{A}=\left[\begin{array}{ccc} \cos \theta & \sin \theta & 0 \ -\sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{array}\right]$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|A Time-Averaging Theorem

$$\langle\operatorname{Re}[A(\mathbf{r}, t)] \operatorname{Re}[B(\mathbf{r}, t)]\rangle=\frac{1}{T} \int_0^T d t \operatorname{Re}[A(\mathbf{r}, t)] \operatorname{Re}[B(\mathbf{r}, t)]=\frac{1}{2} \operatorname{Re}\left[a(\mathbf{r}) b^(\mathbf{r})\right]$$我们通过写$\operatorname{Re}[A]=\frac{1}{2}\left(A+A^\right)$和$\operatorname{Re}[B]=\frac{1}{2}\left(B+B^\right)$来证明(1.137)所以$$\langle\operatorname{Re}[A] \operatorname{Re}[B]\rangle=\frac{1}{4 T} \int_0^T d t\left{a b e^{-2 i \omega t}+a^ b^* e^{2 i \omega t}+a b^+b a^\right}$$
(1.138)的被积函数中与时间相关的项在一个完整周期内积分为零。因此，
$$\langle\operatorname{Re}[A] \operatorname{Re}[B]\rangle=\frac{1}{4}\left[a b^+a^ b\right]=\frac{1}{2} \operatorname{Re}\left[a b^\right]=\frac{1}{2} \operatorname{Re}\left[a^ b\right]$$

## 物理代写|电动力学代写electromagnetism代考|Orthogonal Transformations

$$\mathbf{A} \mathbf{A}^{\top}=\mathbf{A A}^{-1}=1$$

$$\operatorname{det}\left[\mathbf{A A}^{\top}\right]=\operatorname{det} \mathbf{A} \operatorname{det} \mathbf{A}^{\top}=(\operatorname{det} \mathbf{A})^2=1 .$$

$$\mathbf{A}=\left[\begin{array}{ccc} \cos \theta & \sin \theta & 0 \ -\sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{array}\right]$$

## 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。

## 物理代写|电动力学代写electromagnetism代考|Generalized Functions

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

## 物理代写|电动力学代写electromagnetism代考|The Delta Function in One Dimension

The one-dimensional generalized function $\delta(x)$ is defined by its “filtering” action on a smooth but otherwise arbitrary test function $f(x)$ :
$$\int_{-\infty}^{\infty} d x f(x) \delta\left(x-x^{\prime}\right)=f\left(x^{\prime}\right) .$$

An informal definition consistent with (1.93) is
$$\delta(x)=0 \text { for } x \neq 0 \text { but } \int_{-\infty}^{\infty} d x \delta(x)=1 .$$
If the variable $x$ has dimensions of length, the integrals in these equations make sense only if $\delta(x)$ has dimensions of inverse length. Note also that the integration ranges in (1.93) and (1.94) need only be large enough to include the point where the argument of the delta function vanishes.
The delta function can be understood as the limit of a sequence of functions which become more and more highly peaked at the point where its argument vanishes. Some examples are
$$\begin{gathered} \delta(x)=\lim {m \rightarrow \infty} \frac{\sin m x}{\pi x} \ \delta(x)=\lim {m \rightarrow \infty} \frac{m}{\sqrt{\pi}} \exp \left(-m^2 x^2\right) \ \delta(x)=\lim _{\epsilon \rightarrow 0} \frac{\epsilon / \pi}{x^2+\epsilon^2} . \end{gathered}$$
We prove the correctness of any of these proposed representations by showing that it possesses the filtering property (1.93). The same method is used to prove delta function identities like
$$\begin{gathered} \delta(a x)=\frac{1}{|a|} \delta(x), \quad a \neq 0 \ \int_{-\infty}^{\infty} d x f(x) \frac{d}{d x} \delta\left(x-x^{\prime}\right)=-\left.\frac{d f}{d x}\right|{x=x^{\prime}} \ \delta[g(x)]=\sum_n \frac{1}{\left|g^{\prime}\left(x_n\right)\right|} \delta\left(x-x_n\right) \quad \text { where } \quad g\left(x_n\right)=0, \quad g^{\prime}\left(x_n\right) \neq 0 \ \delta\left(x-x^{\prime}\right)=\frac{1}{2 \pi} \int{-\infty}^{\infty} d k e^{i k\left(x-x^{\prime}\right)} . \end{gathered}$$
Formula (1.101) may be read as a statement of the completeness of plane waves labeled with the continuous index $k$ :
$$\psi_k(x)=\frac{1}{\sqrt{2 \pi}} e^{-i k x}$$
The general result for a complete set of normalized basis functions $\psi_n(x)$ labeled with the discrete index $n$ is $^2$
$$\delta\left(x-x^{\prime}\right)=\sum_{n=1}^{\infty} \psi_n^*(x) \psi_n\left(x^{\prime}\right)$$

## 物理代写|电动力学代写electromagnetism代考|The Principal Value Integral and Plemelj Formula

The Cauchy principal value is a generalized function defined by its action under an integral with an arbitrary function $f(x)$, namely,
$$\mathcal{P} \int_{-\infty}^{\infty} d x \frac{f(x)}{x-x_0}=\lim {\epsilon \rightarrow 0}\left[\int{-\infty}^{x_0-\epsilon} d x \frac{f(x)}{x-x_0}+\int_{x_0+\epsilon}^{\infty} d x \frac{f(x)}{x-x_0}\right] .$$
An important application where the principal value plays a role is the Plemelj formula:
$$\lim _{\epsilon \rightarrow 0} \frac{1}{x-x_0 \pm i \epsilon}=\mathcal{P} \frac{1}{x-x_0} \mp i \pi \delta\left(x-x_0\right)$$
This expression is symbolic in the sense that it gains meaning when we multiply every term by an arbitrary function $f(x)$ and integrate over $x$ from $-\infty$ to $\infty$.
The correctness of (1.105) can be appreciated from Figure 1.4 and the identity
$$\frac{1}{x-x_0 \pm i \epsilon}=\frac{x-x_0}{\left(x-x_0\right)^2+\epsilon^2} \mp i \frac{\epsilon}{\left(x-x_0\right)^2+\epsilon^2} .$$
The real part of (1.106) generates the principal value in (1.105) because it is a symmetrically cut-off version of $1 /\left(x-x_0\right)$. The imaginary part of (1.106) generates the delta function in (1.105) by virtue of (1.97).

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|The Delta Function in One Dimension

$$\int_{-\infty}^{\infty} d x f(x) \delta\left(x-x^{\prime}\right)=f\left(x^{\prime}\right) .$$

$$\delta(x)=0 \text { for } x \neq 0 \text { but } \int_{-\infty}^{\infty} d x \delta(x)=1 .$$

$$\begin{gathered} \delta(x)=\lim {m \rightarrow \infty} \frac{\sin m x}{\pi x} \ \delta(x)=\lim {m \rightarrow \infty} \frac{m}{\sqrt{\pi}} \exp \left(-m^2 x^2\right) \ \delta(x)=\lim {\epsilon \rightarrow 0} \frac{\epsilon / \pi}{x^2+\epsilon^2} . \end{gathered}$$ 我们通过证明这些表述具有过滤性质(1.93)来证明其正确性。同样的方法被用来证明函数恒等式，比如 $$\begin{gathered} \delta(a x)=\frac{1}{|a|} \delta(x), \quad a \neq 0 \ \int{-\infty}^{\infty} d x f(x) \frac{d}{d x} \delta\left(x-x^{\prime}\right)=-\left.\frac{d f}{d x}\right|{x=x^{\prime}} \ \delta[g(x)]=\sum_n \frac{1}{\left|g^{\prime}\left(x_n\right)\right|} \delta\left(x-x_n\right) \quad \text { where } \quad g\left(x_n\right)=0, \quad g^{\prime}\left(x_n\right) \neq 0 \ \delta\left(x-x^{\prime}\right)=\frac{1}{2 \pi} \int{-\infty}^{\infty} d k e^{i k\left(x-x^{\prime}\right)} . \end{gathered}$$

$$\psi_k(x)=\frac{1}{\sqrt{2 \pi}} e^{-i k x}$$

$$\delta\left(x-x^{\prime}\right)=\sum_{n=1}^{\infty} \psi_n^*(x) \psi_n\left(x^{\prime}\right)$$

## 物理代写|电动力学代写electromagnetism代考|The Principal Value Integral and Plemelj Formula

$$\mathcal{P} \int_{-\infty}^{\infty} d x \frac{f(x)}{x-x_0}=\lim {\epsilon \rightarrow 0}\left[\int{-\infty}^{x_0-\epsilon} d x \frac{f(x)}{x-x_0}+\int_{x_0+\epsilon}^{\infty} d x \frac{f(x)}{x-x_0}\right] .$$

$$\lim _{\epsilon \rightarrow 0} \frac{1}{x-x_0 \pm i \epsilon}=\mathcal{P} \frac{1}{x-x_0} \mp i \pi \delta\left(x-x_0\right)$$

(1.105)的正确性可以从图1.4和恒等式中看出
$$\frac{1}{x-x_0 \pm i \epsilon}=\frac{x-x_0}{\left(x-x_0\right)^2+\epsilon^2} \mp i \frac{\epsilon}{\left(x-x_0\right)^2+\epsilon^2} .$$
(1.106)的实部生成(1.105)中的主值，因为它是$1 /\left(x-x_0\right)$的对称截止版本。(1.106)的虚部通过(1.97)生成(1.105)中的函数。

## 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。

## 物理代写|电动力学代写electromagnetism代考|The Kronecker and Levi-Civita Symbols

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

## 物理代写|电动力学代写electromagnetism代考|The Kronecker and Levi-Civita Symbols

The Kronecker delta symbol $\delta_{i j}$ and Levi-Cività permutation symbol $\epsilon_{i j k}$ have roman indices $i, j$, and $k$ which take on the Cartesian coordinate values $x, y$, and $z$. They are defined by
$$\delta_{i j}= \begin{cases}1 & i=j, \ 0 & i \neq j\end{cases}$$
and
$$\epsilon_{i j k}= \begin{cases}1 & i j k=x y z \quad y z x \quad z x y, \ -1 & i j k=x z y \quad y x z \quad z y x, \ 0 & \text { otherwise. }\end{cases}$$
Some useful Kronecker delta and Levi-Cività symbol identities are
$$\begin{array}{cc} \hat{\mathbf{e}}i \cdot \hat{\mathbf{e}}_j=\delta{i j} & \delta_{k k}=3 \ \partial_k r_j=\delta_{j k} & V_k \delta_{k j}=V_j \ {[\mathbf{V} \times \mathbf{F}]i=\epsilon{i j k} V_j F_k} & {[\nabla \times \mathbf{A}]i=\epsilon{i j k} \partial_j A_k} \ \delta_{i j} \epsilon_{i j k}=0 & \epsilon_{i j k} \epsilon_{i j k}=6 . \end{array}$$
A particularly useful identity involves a single sum over the repeated index $i$ :
$$\epsilon_{i j k} \epsilon_{i s t}=\delta_{j s} \delta_{k t}-\delta_{j t} \delta_{k s} \text {. }$$
A generalization of (1.39) when there are no repeated indices to sum over is the determinant
$$\epsilon_{k i \ell} \epsilon_{m p q}=\left|\begin{array}{lll} \delta_{k m} & \delta_{i m} & \delta_{\ell m} \ \delta_{k p} & \delta_{i p} & \delta_{\ell p} \ \delta_{k q} & \delta_{i q} & \delta_{\ell q} \end{array}\right| .$$

## 物理代写|电动力学代写electromagnetism代考|Vector Identities in Cartesian Components

The Kronecker and Levi-Cività symbols simplify the proof of vector identities. An example is
$$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\mathbf{b}(\mathbf{a} \cdot \mathbf{c})-\mathbf{c}(\mathbf{a} \cdot \mathbf{b})$$
Using the left side of (1.37), the $i$ th component of $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ is
$$[\mathbf{a} \times(\mathbf{b} \times \mathbf{c})]i=\epsilon{i j k} a_j(\mathbf{b} \times \mathbf{c})k=\epsilon{i j k} a_j \epsilon_{k \ell m} b_l c_m .$$
The definition (1.34) tells us that $\epsilon_{i j k}=\epsilon_{k i j}$. Therefore, the identity (1.39) gives
$$[\mathbf{a} \times(\mathbf{b} \times \mathbf{c})]i=\epsilon{k i j} \epsilon_{k \ell m} a_j b_{\ell} c_m=\left(\delta_{i \ell} \delta_{j m}-\delta_{i m} \delta_{j \ell}\right) a_j b_{\ell} c_m=a_j b_i c_j-a_j b_j c_i .$$
The final result, $b_i(\mathbf{a} \cdot \mathbf{c})-c_i(\mathbf{a} \cdot \mathbf{b})$, is indeed the $i$ th component of the right side of (1.44). The same method of proof applies to gradient-, divergence-, and curl-type vector identities because the components of the $\nabla$ operator transform like the components of a vector [see above (1.8)]. The next three examples illustrate this point.

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|The Kronecker and Levi-Civita Symbols

$$\delta_{i j}= \begin{cases}1 & i=j, \ 0 & i \neq j\end{cases}$$

$$\epsilon_{i j k}= \begin{cases}1 & i j k=x y z \quad y z x \quad z x y, \ -1 & i j k=x z y \quad y x z \quad z y x, \ 0 & \text { otherwise. }\end{cases}$$

$$\begin{array}{cc} \hat{\mathbf{e}}i \cdot \hat{\mathbf{e}}j=\delta{i j} & \delta{k k}=3 \ \partial_k r_j=\delta_{j k} & V_k \delta_{k j}=V_j \ {[\mathbf{V} \times \mathbf{F}]i=\epsilon{i j k} V_j F_k} & {[\nabla \times \mathbf{A}]i=\epsilon{i j k} \partial_j A_k} \ \delta_{i j} \epsilon_{i j k}=0 & \epsilon_{i j k} \epsilon_{i j k}=6 . \end{array}$$

$$\epsilon_{i j k} \epsilon_{i s t}=\delta_{j s} \delta_{k t}-\delta_{j t} \delta_{k s} \text {. }$$

$$\epsilon_{k i \ell} \epsilon_{m p q}=\left|\begin{array}{lll} \delta_{k m} & \delta_{i m} & \delta_{\ell m} \ \delta_{k p} & \delta_{i p} & \delta_{\ell p} \ \delta_{k q} & \delta_{i q} & \delta_{\ell q} \end{array}\right| .$$

## 物理代写|电动力学代写electromagnetism代考|Vector Identities in Cartesian Components

Kronecker和levi – civit符号简化了向量恒等式的证明。一个例子是
$$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\mathbf{b}(\mathbf{a} \cdot \mathbf{c})-\mathbf{c}(\mathbf{a} \cdot \mathbf{b})$$

$$[\mathbf{a} \times(\mathbf{b} \times \mathbf{c})]i=\epsilon{i j k} a_j(\mathbf{b} \times \mathbf{c})k=\epsilon{i j k} a_j \epsilon_{k \ell m} b_l c_m .$$

$$[\mathbf{a} \times(\mathbf{b} \times \mathbf{c})]i=\epsilon{k i j} \epsilon_{k \ell m} a_j b_{\ell} c_m=\left(\delta_{i \ell} \delta_{j m}-\delta_{i m} \delta_{j \ell}\right) a_j b_{\ell} c_m=a_j b_i c_j-a_j b_j c_i .$$

## 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。

## 物理代写|电动力学代写electromagnetism代考|QED and Symmetry

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|电动力学代写electromagnetism代考|QED and Symmetry

The gauge invariance of the QED Hamiltonian $\mathrm{H}$ is an important dynamical symmetry that we have described in detail in this chapter. It leads to the fundamental principle that only gauge-invariant quantities can be candidates for physical observables. The other symmetries of the QED Hamiltonian are easily summarised. A simple additive construction of particle and field quantities suffices to construct the total linear momentum ${ }^5(\mathbf{P})$ and total angular momentum $(\mathbf{J})$ operators of the combined system of charges and field, for example,
$$\mathbf{P}=\sum_i^n \mathbf{p}i+\sum{\mathbf{k}, \lambda} \hbar \mathbf{k} \mathrm{n}_{\mathbf{k}, \lambda},$$
in terms of the individual particle momenta and the photon number operator. $\mathbf{J}$ can be constructed in a similar fashion. The operators $\mathbf{P}$ and $\mathbf{J}$ so formed, together with $\mathrm{H}$, satisfy the Lie bracket relations for the relativity groups described in Chapters 3 and 5. This is true classically as well as in the quantum mechanical account. Thus the total linear momentum and the total angular momentum are conserved quantities. However, it is not possible to define a boost operator $\mathbf{K}$ with the requisite properties to complete the algebras of either the Galilean or Poincaré groups. The theory is conventionally described as ‘non-relativistic’.

An evident limitation of a presentation of quantum electrodynamics based on canonical quantisation of classical electrodynamics is that the charges are necessarily spin 0 ; there are familiar examples in particle physics ( $\pi$ – and $K$-mesons), but they are not of interest here. It is well known that the overall gross properties of atomic matter can be described in terms of non-relativistic quantum mechanics without an explicit reference to the concept of particle spin, provided the Pauli exclusion principle (or more generally, the permutation group symmetry of the particle Hamiltonian) is recognised. Electrons are fermions with spin $\frac{1}{2}$, while nuclei can have either integer or half-integer spin $\geq 0$. Interactions involving the particle spin operators occur naturally in relativistic (that is, Lorentz-invariant) formulations of quantum mechanics. The standard Lorentz-invariant formulation of quantum electrodynamics is a quantum theory of interacting electron and electromagnetic fields (‘electrons and photons’) in which particle number is not a conserved quantity. Nuclei sit rather uncomfortably in this framework other than as fixed classical sources of external fields [47], not least because their anomalous magnetic moments may make such a theory unrenormalisable. As discussed in Chapter 5 , there is no known Lorentz-invariant quantum theory of an $N$-particle system involving electromagnetic interactions with fixed $N$, and so there is no such theory of atoms and molecules.

A widely used approach to the theoretical description of the interactions between atoms or molecules and electromagnetic radiation is based on the notion that the field can be treated as a classical electromagnetic field described by Maxwell’s theory (cf. Chapter 2) and that the quantum properties of the atomic system are given by an appropriate time-dependent Schrödinger equation. This is the ‘semiclassical radiation model’. A static electric or magnetic field is always classical, and its interaction with charged particles can be described by the inclusion of additional terms in the timeindependent Schrödinger equation that modify the spectrum of the atomic system; such perturbations may lead to shifts in eigenvalues (Zeeman effect) or the conversion of eigenvalues into (metastable) resonances (Stark effect). These topics are discussed thoroughly in numerous standard quantum mechanics texts.

On the other hand, the treatment of the interaction of atomic/molecular matter with an optical field using classical electromagnetism is not a trivial matter; its relationship to quantum electrodynamics does not seem to be well described in the literature. Given the extensive evidence that the electromagnetic field is a quantum mechanical system, one may enquire how an approach that eschews that information can possibly succeed, in an admittedly limited set of experimental situations. In the following we attempt to answer that question. The main limitation of such an approach is that the atom can only respond to a non-zero classical field; thus stimulated absorption and emission, and light scattering can be considered, but phenomena that derive from spontaneous emission, for example fluorescence, luminescence and, phosphorescence, or involve virtual photons, for example intermolecular interactions, resonant energy transfer processes and the problems of self-interaction are outside the scope of the semiclassical model.
Given a quantum Hamiltonian constructed by canonical quantisation of the corresponding classical theory in Hamiltonian form (P.B.s $\rightarrow$ quantum commutators, $x \rightarrow \mathrm{x}$ etc.), we know that the classical equations of motion for the classical variables are replaced by operator equations of motion for the corresponding quantum mechanical operators. Furthermore, linear equations of motion such as the Maxwell equations for the electromagnetic field have the same form in both cases with a suitable operator interpretation of the particle and field variables in the quantum case. The classical Hamiltonian equations of motion yield the wave equation for the vector potential ${ }^8$
$$\square \mathbf{a}(\mathbf{x}, t)=\mu_0 \mathbf{j}(\mathbf{x}, t)$$
which may be solved in the usual way by the Green’s function technique,
$$\mathbf{a}(\mathbf{x}, t)=\mu_0 \int \mathbf{G}\left(\mathbf{x}, t ; \mathbf{x}^{\prime}, t^{\prime}\right) \cdot \mathbf{j}\left(\mathbf{x}^{\prime}, t^{\prime}\right) \mathrm{d}^3 \mathbf{x}^{\prime} \mathrm{d} t^{\prime},$$
where $\mathbf{G}$ satisfies the equation ${ }^9$
$$\mathbf{G}\left(\mathbf{x}, t ; \mathbf{x}^{\prime}, t^{\prime}\right)=\delta^3\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \delta\left(t-t^{\prime}\right)$$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|QED and Symmetry

QED 哈密顿量的规范不变性 $\mathrm{H}$ 是我们在本章中详细描述的一个重要的动力学对称性。它引出了一个基本 原理，即只有规范不变的量才能成为物理可观察量的候选者。QED 哈密顿量的其他对称性很容易总结。 粒子和场量的简单加法构造足以构造总线性动量 ${ }^5(\mathbf{P})$ 和总角动量 $(\mathbf{J})$ 收费和现场联合系统的运营商，例 如，
$$\mathbf{P}=\sum_i^n \mathbf{p} i+\sum \mathbf{k}, \lambda \hbar \mathbf{k} \mathbf{n}_{\mathbf{k}, \lambda}$$

$$\square \mathbf{a}(\mathbf{x}, t)=\mu_0 \mathbf{j}(\mathbf{x}, t)$$

$$\mathbf{a}(\mathbf{x}, t)=\mu_0 \int \mathbf{G}\left(\mathbf{x}, t ; \mathbf{x}^{\prime}, t^{\prime}\right) \cdot \mathbf{j}\left(\mathbf{x}^{\prime}, t^{\prime}\right) \mathrm{d}^3 \mathbf{x}^{\prime} \mathrm{d} t^{\prime},$$

$$\mathbf{G}\left(\mathbf{x}, t ; \mathbf{x}^{\prime}, t^{\prime}\right)=\delta^3\left(\mathbf{x}-\mathbf{x}^{\prime}\right) \delta\left(t-t^{\prime}\right)$$

## 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。