## 物理代写|广义相对论代写General relativity代考|MATH4105

statistics-lab™ 为您的留学生涯保驾护航 在代写广义相对论General relativity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。我们的专家在代写广义相对论General relativity代写方面经验极为丰富，各种代写广义相对论General relativity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|广义相对论代写General relativity代考|Affine Connection

At first, we are providing some basic concepts:
A set $M$, which is locally Euclidean of dimension $n$ is called a manifold of dimension $n$. Locally Euclidean means that every $x$ that belongs to $M$ possesses a neighborhood, which is homeomorphic to an open subset of $R^n$ (see Fig. 3).

A mapping $f: X \rightarrow Y$ is homeomorphic, if $f$ is bijective mapping, continuous and $f^{-1}$ exists. A manifold $\mathrm{M}$ is said to be Hausdorff, if for any two distinct points $\mathrm{x}$ and $\mathrm{y}$ in $\mathrm{M}$, there exist disjoint neighborhoods of $\mathrm{x}$ and $\mathrm{y}$.
A manifold $\mathrm{M}$ is said to be compact if each open cover of $\mathrm{M}$ has a finite subcover.
A manifold $\mathrm{M}$ is said to be paracompact if every open cover of $\mathrm{M}$ has an open refinement that is locally finite.

In the absence of paracompact, a manifold does not admit a real analytic differentiable structure. A manifold M is said to be connected space if it cannot be represented as the union of two or more disjoint nonempty open subsets.

A manifold is said to be differentiable manifold if it is continuous and differentiable. Usually in physics, one can describe the spacetime by a differentiable manifold.

Suppose $M$ is a differential manifold of dimension $m$. The tangent space $T_p M$ is a collection of tangent vectors $v_p$ to $M$ at the point $p \in M$. The tangent bundle $T M$ of a manifold $M$ is defined by $T M=\cup_{p \in M} T_p M=(p, v) \mid p \in M, v \in T_p M$.

Let a contravariant vector $A^i$ at point $\mathrm{P}\left(x^i\right)$. $\mathrm{F}\left(x^i+\delta x^i\right)$ be a neighborhood point of $\mathrm{P}$. Now we shift the vector $A^i$ from $\mathrm{P}$ to $\mathrm{F}$ such that its magnitude and direction do not change. This scheme is known as parallel transport (see Fig. 4). In this scheme, the tangent vector is propagated parallel to itself. The changes $\delta A^i$ of the components of $A^i$ in going from $\mathrm{P}$ to $\mathrm{F}$ under such a parallel transport will be proportional to the original components of $A^i$ and also to the displacement $\delta x^l$, i.e., the changes $\delta A^i$ will be linear functions of $\delta x^I$ and $A^k$. Thus
$$\delta A^i=-{ }^a \Gamma_{k l}^i A^k \delta x^l .$$
The notations ${ }^a \Gamma_{k l}^j$ are known as the affine connection of the spacetime region, which contains $4^3=64$ components entity. This connection is torsion-free. Here, the notion of local parallelism, i.e., parallelism over infinitesimal distances or the parallel transport of connecting two nearby vectors is affine connection of the spacetime. These symbols are mentioned as Christoffel symbols, i.e., ${ }^a \Gamma_{k l}^i=\Gamma_{k l}^i$

## 物理代写|广义相对论代写General relativity代考|Covariant Derivative

As the partial derivative of a tensor is not, in general, a tensor, therefore, it is demanded to introduce a new kind of differentiation, which gives rise to a tensor when applied to a tensor. This new type of derivative is actually covariant derivative. It is independent of the choice of coordinates.

Consider a contravariant vector $A^i$ at the point $\mathrm{P}\left(x^i\right)$ and then displace the vector to a point $\mathrm{F}$ $\left(x^i+\delta x^i\right)$. The actual physical change in $A^i$ from $\mathrm{P}$ to $\mathrm{F}$ is given by $d A^i-\delta A^i$, where $d A^i$ is due to point differences and $\delta A^i$ due to parallel transport.
We know
\begin{aligned} d A^i & =A^i\left(x^i+\delta x^i\right)-A^i\left(x^i\right) \ & \cong A^i\left(x^i\right)+\delta x^I \frac{\partial A^i}{\partial x^l}-A^i\left(x^i\right)=\frac{\partial A^i}{\partial x^l} \delta x^I \end{aligned}
The rate of change with respect to $x^i$ is
$$\frac{d A^i-\delta A^i}{\delta x^l}=A^i, l$$
This rate of change is called covariant derivative of $A^i$ for $\delta x^I \longrightarrow 0$.
Now putting the values of $d A^i$ and $\delta A^i$, we get the covariant derivative of a contravariant vector as
$$A_{; l}^i=\frac{\partial A^i}{\partial x^l}+\Gamma_{k l}^i A^k .$$
Remember that $A_{\mu ; \lambda}=g_{\mu v} A_{; \lambda^*}^v$
Hint: Differentiating both side of $A_\mu=g_{\mu v} A^v$ with respect to $x^\lambda$, we obtain
$$\frac{\partial A_\mu}{\partial x^\lambda}=\frac{\partial g_{\mu v}}{\partial x^\lambda} A^v+g_{\mu v} \frac{\partial A^v}{\partial x^\lambda}=\left(\Gamma_{\mu \lambda}^\delta g_{v \delta}+\Gamma_{\lambda v}^\delta g_{\mu \delta}\right) A^v+g_{\mu v} \frac{\partial A^v}{\partial x^\lambda}$$
or
$$\frac{\partial A_\mu}{\partial x^\lambda}-\Gamma_{\lambda \mu}^\delta g_{v \delta} A^v=g_{\mu v} \frac{\partial A^v}{\partial x^\lambda}+\Gamma_{\lambda \delta}^v g_{\mu v} A^\delta$$
(replacing the dummy index $v$ by $\delta$ )

## 物理代写|广义相对论代写General relativity代考|Affine Connection

$: M$, 这是局部的欧几里德维数 $n$ 称为维度流形 $n$. 局部欧几里德意味着每个 $x$ 那属于 $M$ 拥有一 个邻域，该邻域同胚于 $R^n$ (见图 3)。

$$\delta A^i=-{ }^a \Gamma_{k l}^i A^k \delta x^l .$$

## 物理代写|广义相对论代写General relativity代考|Covariant Derivative

$$d A^i=A^i\left(x^i+\delta x^i\right)-A^i\left(x^i\right) \quad \cong A^i\left(x^i\right)+\delta x^I \frac{\partial A^i}{\partial x^l}-A^i\left(x^i\right)=\frac{\partial A^i}{\partial x^l} \delta x^I$$

$$\frac{d A^i-\delta A^i}{\delta x^l}=A^i, l$$

$$A_{; l}^i=\frac{\partial A^i}{\partial x^l}+\Gamma_{k l}^i A^k .$$

$$\frac{\partial A_\mu}{\partial x^\lambda}=\frac{\partial g_{\mu v}}{\partial x^\lambda} A^v+g_{\mu v} \frac{\partial A^v}{\partial x^\lambda}=\left(\Gamma_{\mu \lambda}^\delta g_{v \delta}+\Gamma_{\lambda v}^\delta g_{\mu \delta}\right) A^v+g_{\mu v} \frac{\partial A^v}{\partial x^\lambda}$$

$$\frac{\partial A_\mu}{\partial x^\lambda}-\Gamma_{\lambda \mu}^\delta g_{v \delta} A^v=g_{\mu v} \frac{\partial A^v}{\partial x^\lambda}+\Gamma_{\lambda \delta}^v g_{\mu v} A^\delta$$
(替换虚拟索引 $v$ 经过 $\delta$ )

## 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|广义相对论代写General relativity代考|PHYC90012

statistics-lab™ 为您的留学生涯保驾护航 在代写广义相对论General relativity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。我们的专家在代写广义相对论General relativity代写方面经验极为丰富，各种代写广义相对论General relativity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|广义相对论代写General relativity代考|The Line Element

The distance between two neighboring points $P\left(\vec{r}\left(x^i\right)\right)$ and $F\left(\vec{r}\left(x^i\right)+d \vec{r}\left(x^i\right)\right)\left(x^i\right.$ are the coordinates of the space) in an $n$-dimensional space is given by (see Fig. 2)
$$d s^2=d \vec{r} \cdot d \vec{r}=g_{a b} d x^a d x^b$$

Here,
$$d \vec{r}\left(x^1\right)=\frac{\partial \vec{r}}{\partial x^1} d x^1+\frac{\partial \vec{r}}{\partial x^2} d x^2+\ldots \ldots \ldots+\frac{\partial \vec{r}}{\partial x^n} d x^n=\alpha_1 d x^1+\alpha_2 d x^2+\ldots+\alpha_n d x^n$$
with
$$\alpha_i=\frac{\partial \vec{r}}{\partial x^i} \text { and } g_{a b}=\alpha_a \cdot \alpha_b .$$
The distance between two neighboring points is referred as line element and is given by Eq. (1.10).

Here, $g_{a b}$ are known as metric tensor, which are functions of $x^a$. If $g=\left|g_{a b}\right| \neq 0$ and $d s$ is adopted to be invariant, then the space is called Riemannian space.

In mathematics, Riemannian space is used for a positive-definite metric tensor, whereas in theoretical physics, spacetime is modeled by a pseudo-Riemannian space in which the metric tensor is indefinite.
The metric tensor $g_{a b}$ is also called fundamental tensor (covariant tensor of order two).
In Euclidean space:
$$d s^2=d x^2+d y^2+d z^2 .$$
In Minkowski flat spacetime, the line element
$$d s^2=d x^{0^2}-d x^{1^2}-d x^{2^2}-d x^{3^2} .$$
Since the distance $d s$ between two neighboring points is real, the Eq. (1.10) will be amended to
$$d s^2=e g_{i j} d x^i d x^j,$$
where $e$ is known as the indicator and assumes the value $+1$ or $-1$ in order that $d s^2$ be always positive.

## 物理代写|广义相对论代写General relativity代考|Levi-Civita Tensor or Alternating Tensor

Levi-Civita tensor is a tensor of order three in three dimensions and is denoted by $\epsilon_{a b c}$ and defined as
$$\epsilon_{a b c}=+1,$$
if a,b,c is an even permutation of $1,2,3$, i.e., in cyclic order.
$$=-1,$$
if a,b,c is odd permutation of $1,2,3$, i.e., not in cyclic order.
$$=0$$
if any two indices are equal.

Levi-Civita tensor is a tensor of order four in four dimensions and denoted by $\epsilon^{a b c d}$.
$$\epsilon^{a b c d}=+1,$$
if a,b,c,d is an even permutation of $0,1,2,3$, i.e., in cyclic order.
$$=-1,$$
if a,b,c,d is odd permutation of $0,1,2,3$, i.e., not in cyclic order.
$$=0$$
if any two indices are equal.
The components of $\epsilon_{a b c d}$ can be found from $\epsilon^{a b c d}$ by lowering the indices in a typical way, just multiplying it by $(-g)^{-1}$ :
$$\epsilon_{a b c d}=g_{a \mu} g_{b v} g_{c \gamma} g_{d \sigma}(-g)^{-1} \epsilon^{\mu v \gamma \sigma} .$$
For example,
\begin{aligned} \epsilon_{0123} & =g_{0 \mu} g_{1 v} g_{2 \gamma} g_{3 \sigma}(-g)^{-1} \epsilon^{\mu \gamma \gamma \sigma} \ & =(-g)^{-1} \operatorname{det} g_{\mu v}=-1 \end{aligned}

## 物理代写|广义相对论代写General relativity代考|The Line Element

$$d s^2=d \vec{r} \cdot d \vec{r}=g_{a b} d x^a d x^b$$

$$d \vec{r}\left(x^1\right)=\frac{\partial \vec{r}}{\partial x^1} d x^1+\frac{\partial \vec{r}}{\partial x^2} d x^2+\ldots \ldots \ldots+\frac{\partial \vec{r}}{\partial x^n} d x^n=\alpha_1 d x^1+\alpha_2 d x^2+\ldots+\alpha_n$$

$$\alpha_i=\frac{\partial \vec{r}}{\partial x^i} \text { and } g_{a b}=\alpha_a \cdot \alpha_b .$$

$$d s^2=d x^2+d y^2+d z^2 .$$

$$d s^2=d x^{0^2}-d x^{1^2}-d x^{2^2}-d x^{3^2} .$$

$$d s^2=e g_{i j} d x^i d x^j,$$

## 物理代写|广义相对论代写General relativity代考|Levi-Civita Tensor or Alternating Tensor

Levi-Civita 张量是三维空间中的三阶张量，表示为 $\epsilon_{a b c}$ 并定义为
$$\epsilon_{a b c}=+1,$$

$$=-1,$$

$$=0$$

Levi-Civita 张量是四维四阶张量，表示为 $\epsilon^{a b c d}$.
$$\epsilon^{a b c d}=+1,$$

$$=-1,$$

$$=0$$

$$\epsilon_{a b c d}=g_{a \mu} g_{b v} g_{c \gamma} g_{d \sigma}(-g)^{-1} \epsilon^{\mu v \gamma \sigma} .$$

$$\epsilon_{0123}=g_{0 \mu} g_{1 v} g_{2 \gamma} g_{3 \sigma}(-g)^{-1} \epsilon^{\mu \gamma \gamma \sigma} \quad=(-g)^{-1} \operatorname{det} g_{\mu v}=-1$$

## 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|广义相对论代写General relativity代考|MATH7105

statistics-lab™ 为您的留学生涯保驾护航 在代写广义相对论General relativity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。我们的专家在代写广义相对论General relativity代写方面经验极为丰富，各种代写广义相对论General relativity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|广义相对论代写General relativity代考|Covariant and Contravariant Vector and Tensor

Usually one can describe the tensors by means of their properties of transformation under coordinate transformation. There are two possible ways of transformations from one coordinate system $\left(x^i\right)$ to the other coordinate system $\left(\bar{x}^i\right)$.

Let us consider a set of $n$ functions $A_i$ of the coordinates $x^i$. The functions $A_i$ are said to be the components of covariant vector if these components transform according to the following rule
$$\bar{A}_i=\frac{\partial x^j}{\partial \bar{x}^i} A_j .$$
Also, one can find by multiplying $\frac{\partial x^i}{\partial x^k}$ and using $\frac{\partial x^i}{\partial x^k} \frac{\partial x^j}{\partial x^i}=\delta_k^j$ and $\delta_k^j A_j=A_k$
$$A_k=\frac{\partial \bar{x}^i}{\partial x^k} \overline{A_i} .$$

Here, $A_i$ is known as the covariant tensor of first order or of the type $(0,1)$.
The functions $A^i$ are said to be the components of the contravariant vector if these components transform according to the following rule
$$\bar{A}^i=\frac{\partial \bar{x}^i}{\partial x^j} A^j$$
Also, one can find by multiplying both sides with $\frac{\partial x^l}{\partial x^i}$ and using $\delta_j^k A^j=A^k$
$$A^k=\frac{\partial x^k}{\partial \bar{x}^i} \bar{A}^i$$
Here, $A^i$ is known as the contravariant tensor of first order or of the type $(1,0)$.

## 物理代写|广义相对论代写General relativity代考|Contravariant and covariant tensors of rank two

Let $C^j$ and $B^j$ be two contravariant vectors with $n$ components, then $C^j B^j=A^{i j}$ has $n^2$ quantities, i.e., $A^{i j}$ are the set of $n^2$ functions of the coordinates $x^i$. If the transformation of $A^{i j}$ is like
$$A^{i j}=\frac{\partial \bar{x}^i}{\partial x^k} \frac{\partial \bar{x}^j}{\partial x^l} A^{k l},$$
then $A^{i j}$ is known as contravariant tensor of rank two. Here, $A^{i j}$ is also known as the contravariant tensor of order two or of the type $(2,0)$.
If we multiply both sides of (1.8) by $\frac{\partial x^{\prime}}{\partial x^{\frac{1}{r}}} \frac{\partial x^2}{\partial \bar{x}}$, then
$$A^{r s}=\frac{\partial x^r}{\partial \bar{x}^i} \frac{\partial x^s}{\partial \bar{x}^j} \bar{A}^{i j} .$$
Again, if $C_i$ and $B_j$ are two covariant vectors with $n$ components, then $C_i B_j=A_{i j}$ form $n^2$ quantities, i.e., $A_{i j}$ are the set of $n^2$ functions of the coordinates $x^i$.
If the transformation of $A_{i j}$ is like
$$\bar{A}{i j}=\frac{\partial x^k}{\partial \bar{x}^i} \frac{\partial x^I}{\partial \bar{x}^j} A{k l},$$
then $A_{i j}$ is known as covariant tensor of rank two.
Here, $A_{i j}$ is also known as the covariant tensor of order two or of the type $(0,2)$.

If we multiply both sides of (1.9) by $\frac{\partial x^i}{\partial x^2} \frac{\partial j}{\partial x^x}$, then
$$A_{r s}=\frac{\partial \bar{x}^i}{\partial x^r} \frac{\partial \bar{x}^j}{\partial x^s} \bar{A}_{i j} .$$

## 物理代写|广义相对论代写General relativity代考|Covariant and Contravariant Vector and Tensor

$$\bar{A}_i=\frac{\partial x^j}{\partial \bar{x}^i} A_j .$$

$$A_k=\frac{\partial \bar{x}^i}{\partial x^k} \overline{A_i} .$$

$$\bar{A}^i=\frac{\partial \bar{x}^i}{\partial x^j} A^j$$

$$A^k=\frac{\partial x^k}{\partial \bar{x}^i} \bar{A}^i$$

## 物理代写|广义相对论代写General relativity代考|Contravariant and covariant tensors of rank two

$$A^{i j}=\frac{\partial \bar{x}^i}{\partial x^k} \frac{\partial \bar{x}^j}{\partial x^l} A^{k l},$$

$$A^{r s}=\frac{\partial x^r}{\partial \bar{x}^i} \frac{\partial x^s}{\partial \bar{x}^j} \bar{A}^{i j} .$$

$$\bar{A} i j=\frac{\partial x^k}{\partial \bar{x}^i} \frac{\partial x^I}{\partial \bar{x}^j} A k l,$$

$$A_{r s}=\frac{\partial \bar{x}^i}{\partial x^r} \frac{\partial \bar{x}^j}{\partial x^s} \bar{A}_{i j} .$$

## 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|光学代写Optics代考|CSCI031

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics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。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics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|光学代写Optics代考|Quantum Harmonic Oscillator

As described in Chap. 1, the Hamiltonian for the quantum harmonic oscillator (QHO) is obtained by canonnical quantization where the coordinătès $(x, \not p)$ aree replaced by their quantum operators:

\begin{aligned} & \text { canonical } \ & H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2 \stackrel{\text { quantization }}{\longrightarrow} \widehat{H}=\frac{\widehat{p}^2}{2 m}+\frac{1}{2} m \omega^2 \widehat{x}^2 \ & \end{aligned}
where $\widehat{x}$ and $\widehat{p}$ obey the commutation relation:
$$[\widehat{x}, \widehat{p}]=i \hbar$$
Equation (2.12) can be used to find the momentum operator $\hat{p}$ in terms of the $x$ coordinate. Starting from Eq. (2.12) and according to the definition of the commutation relation:
$$(\widehat{x} \hat{p}-\widehat{p} \hat{x})|\psi\rangle=i \hbar|\psi\rangle$$
Expanding the left side of Eq. (2.13) gives
$$\widehat{x} \hat{p}|\psi\rangle-\widehat{p} \hat{x}|\psi\rangle=i \hbar|\psi\rangle$$
If $|\psi\rangle$ is in the position representation (i.e., $|\psi\rangle$ represents the familiar wavefunction, $\psi(\mathrm{x})$ ), then the operator $\hat{x}$ is simply the position, $x$; that is, $\hat{x}|\psi\rangle=x|\psi\rangle$. Thus, Eq. (2.14) becomes
$$x \hat{p}|\psi\rangle-\widehat{p} x|\psi\rangle=i \hbar|\psi\rangle$$
In the second term on the left, $\widehat{p} x|\psi\rangle$, we apply the rules of partial differentiation, that is, the operator $\widehat{p}$ operates on $x$ while keeping $|\psi\rangle$ constant, and then $\widehat{p}$ operates on $|\psi\rangle$ while keeping $x$ constant. This gives
$$x \widehat{p}|\psi\rangle-(\widehat{p} x)|\psi\rangle-x(\widehat{p}|\psi\rangle)=i \hbar|\psi\rangle$$
In the second term on the left, $\widehat{p}$ operates on $x$ only.

## 物理代写|光学代写Optics代考|Dirac Formalism

Paul Dirac formulated an alternative approach to solve the QHO. Suppose $\hat{H}$ can be factorized as follows:
$$\widehat{H}=\widehat{O}^{\dagger} \widehat{O}+E_0$$
where $\widehat{O}$ is some operator and $\widehat{O}^{\dagger}$ is the Hermitian conjugate. If $\left|\psi_n\right\rangle$ is an eigenstate of $\widehat{H}$, then the eigenenergies are
$$E_n=\left\langle\psi_n|\widehat{H}| \psi_n\right\rangle$$
Substituting Eq. (2.30) for $\widehat{H}$ gives
\begin{aligned} E_n & =\left\langle\psi_n\left|\left(\widehat{O}^{\dagger} \widehat{O}+E_0\right)\right| \psi_n\right\rangle \ & =\left\langle\psi_n\left|\widehat{O}^{\dagger} \widehat{O}\right| \psi_n\right\rangle+E_0 \end{aligned}
This means:
$$E_n \geq E_0$$
If $\widehat{O}\left|\psi_0\right\rangle=0$, then the minimum energy (ground state energy, $E_0$ ) is found. At this point, it is helpful to define dimensionless operators, $\widehat{Q}$ and $\widehat{P}$ :
$$\widehat{Q}=\sqrt{\frac{m \omega}{\hbar}} \widehat{x}$$ $$\widehat{P}=\sqrt{\frac{1}{m \hbar \omega}} \widehat{p}$$
It is easily shown that Eqs. (2.11) and (2.12) become
$$\begin{gathered} \widehat{H}=\frac{\hbar \omega}{2}\left(\widehat{Q}^2+\widehat{P}^2\right) \ {[\widehat{Q}, \widehat{P}]=i} \end{gathered}$$

# 光学代考

## 物理代写|光学代写Optics代考|Quantum Harmonic Oscillator

$$\text { canonical } \quad H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2 \stackrel{\text { quantization }}{\longrightarrow} \widehat{H}=\frac{\hat{p}^2}{2 m}+\frac{1}{2} m \omega^2 \widehat{x}^2$$

$$[\widehat{x}, \hat{p}]=i \hbar$$

$$(\widehat{x} \hat{p}-\hat{p} \hat{x})|\psi\rangle=i \hbar|\psi\rangle$$

$$\widehat{x} \hat{p}|\psi\rangle-\hat{p} \hat{x}|\psi\rangle=i \hbar|\psi\rangle$$

$$x \hat{p}|\psi\rangle-\hat{p} x|\psi\rangle=i \hbar|\psi\rangle$$

$$x \hat{p}|\psi\rangle-(\hat{p} x)|\psi\rangle-x(\hat{p}|\psi\rangle)=i \hbar|\psi\rangle$$

## 物理代写|光学代写Optics代考|Dirac Formalism

Paul Dirac 制定了另一种解决 $\mathrm{QHO}$ 的方法。认为 $\hat{H}$ 可以分解如下:
$$\widehat{H}=\widehat{O}^{\dagger} \widehat{O}+E_0$$

$$E_n=\left\langle\psi_n|\widehat{H}| \psi_n\right\rangle$$

$$E_n=\left\langle\psi_n\left|\left(\widehat{O}^{\dagger} \widehat{O}+E_0\right)\right| \psi_n\right\rangle=\left\langle\psi_n\left|\widehat{O}^{\dagger} \widehat{O}\right| \psi_n\right\rangle+E_0$$

$$E_n \geq E_0$$

\begin{aligned} \widehat{Q} & =\sqrt{\frac{m \omega}{\hbar}} \widehat{x} \ \widehat{P} & =\sqrt{\frac{1}{m \hbar \omega}} \hat{p} \end{aligned}

$$\widehat{H}=\frac{\hbar \omega}{2}\left(\widehat{Q}^2+\widehat{P}^2\right)[\widehat{Q}, \widehat{P}]=i$$

## 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|光学代写Optics代考|UNITS24

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics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。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics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|光学代写Optics代考|Commutation Relations

Dirac showed that the canonically conjugate variables $\left(\widehat{q}i, \widehat{p}_j\right)$ of the quantum system satisfy the commutation relation: $$\left[\widehat{q}_i, \widehat{p}_j\right]=i \hbar \delta{i j}$$
where $\delta_{i j}$ is the Kronecker function and, by definition,
$$\left[\widehat{q}_i, \widehat{p}_j\right]=\widehat{q}_i \widehat{p}_j-\widehat{p}_j \widehat{q}_i$$
Thus, when $i=j$, we say that $\widehat{q}_i$ and $\widehat{p}_i$ “do not commute”; that is, $\left[\widehat{q}_i, \widehat{p}_i\right]=$ $\widehat{q}_i \widehat{p}_i-\widehat{p}_i \widehat{q}_i=i \hbar$. Otherwise, the operators commute. For example, when the generalized coordinates $\left(\widehat{q}_i, \widehat{p}_i\right)$ are the position and momentum, we have
\begin{aligned} & {\left[\widehat{x}, \widehat{p}_x\right]=i \hbar} \ & {\left[\widehat{y}, \widehat{p}_y\right]=i \hbar} \ & {\left[\widehat{z}, \widehat{p}_z\right]=i \hbar} \end{aligned}
Thus, position and momentum along the same direction do not commute (e.g., $\hat{x}$ and $\widehat{p}_x$ do not commute), while position and momentum along different directions do commute (e.g., $\widehat{x}$ and $\widehat{p}_y$ commute). Eq. (1.20) leads to the well-known Heisenberg uncertainty relation:
$$\Delta x \Delta p_x \geq \frac{\hbar}{2}$$
with the same relation for the $y$ and $z$ directions arising from Eq. (1.21) and (1.22), respectively. In Eq. (1.23), $\Delta x$ is the uncertainty in position $x$ and $\Delta p_x$ is the uncertainty in momentum along $x$. Uncertainty is defined as the standard deviation or root mean square (rms) error:
$$\begin{gathered} \Delta x=\sqrt{\left\langle(x-\langle x\rangle)^2\right\rangle} \ =\sqrt{\left\langle x^2+\langle x\rangle^2-2 x\langle x\rangle\right\rangle} \ =\sqrt{\left\langle x^2\right\rangle-\langle x\rangle^2} \end{gathered}$$
where the brackets \langle\rangle denote an average (in quantum mechanics, this is called the “expectation value” of $x$ ).

## 物理代写|光学代写Optics代考|Classical Harmonic Oscillator

Consider a classical system comprised of a particle of mass, $m$, and position, $x$, moving in a one-dimensional parabolic potential:
$$U(x)=\frac{1}{2} k x^2=\frac{1}{2} m \omega^2 x^2$$
where $k$ is a force constant and $\omega=\sqrt{k / m}$ is the angular frequency. The harmonic oscillator arises in a wide variety of classical systems, but most often as a mass on a spring described by Hooke’s law $(F=-k x)$. The generalized coordinates for this system are simply the position and momentum:
$$\begin{gathered} q \rightarrow x \ p \rightarrow m \frac{d x}{d t} \end{gathered}$$
and the Hamiltonian becomes

$$H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2$$
The Hamilton equations become
$$\begin{gathered} \frac{d x}{d t}=\frac{\partial H}{\partial p}=\frac{p}{m}=v \ \frac{d p}{d t}=-\frac{\partial H}{\partial x}=-m \omega^2 x=-\frac{\partial U}{\partial x}=F \end{gathered}$$
The first equation is the definition of momentum $(p=m v)$, while the second equation reproduces Newton’s equation $\left(F=\frac{d p}{d t}\right)$. Thus, $x$ and $p$ satisfy the Hamilton equations (they give the correct dynamical behavior) and are therefore canonically conjugate variables.

Equations (2.5) and (2.6) are easily solved. Combining the two equations gives
$$\frac{d^2 x}{d t^2}=-\omega^2 x$$
with the solution
$$x=a \cos (\omega t+\varphi)$$
where the amplitude, $a$, and phase, $\varphi$, are determined by initial conditions. Equivalently, the solution may be written as
$$x=A e^{-i \omega t}+\text { c.c. }$$

# 光学代考

## 物理代写|光学代写Optics代考|Commutation Relations

$$\left[\hat{q}_i, \hat{p}_j\right]=\hat{q}_i \hat{p}_j-\hat{p}_j \hat{q}_i$$

$$\left[\widehat{x}, \hat{p}_x\right]=i \hbar \quad\left[\hat{y}, \hat{p}_y\right]=i \hbar\left[\hat{z}, \hat{p}_z\right]=i \hbar$$

$$\Delta x \Delta p_x \geq \frac{\hbar}{2}$$

$$\Delta x=\sqrt{\left\langle(x-\langle x\rangle)^2\right\rangle}=\sqrt{\left\langle x^2+\langle x\rangle^2-2 x\langle x\rangle\right\rangle}=\sqrt{\left\langle x^2\right\rangle-\langle x\rangle^2}$$

## 物理代写|光学代写Optics代考|Classical Harmonic Oscillator

$$U(x)=\frac{1}{2} k x^2=\frac{1}{2} m \omega^2 x^2$$

$$q \rightarrow x p \rightarrow m \frac{d x}{d t}$$

$$H=\frac{p^2}{2 m}+\frac{1}{2} m \omega^2 x^2$$

$$\frac{d x}{d t}=\frac{\partial H}{\partial p}=\frac{p}{m}=v \frac{d p}{d t}=-\frac{\partial H}{\partial x}=-m \omega^2 x=-\frac{\partial U}{\partial x}=F$$

$$\frac{d^2 x}{d t^2}=-\omega^2 x$$

$$x=a \cos (\omega t+\varphi)$$

$$x=A e^{-i \omega t}+\text { c.c. }$$

## 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|光学代写Optics代考|PHS2062

statistics-lab™ 为您的留学生涯保驾护航 在代写光学Optics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。我们的专家在代写光学Optics代写方面经验极为丰富，各种代写光学Optics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|光学代写Optics代考|Hamiltonian Mechanic

Hamiltonian mechanics was formulated by William Rowan Hamilton in 1833. Hamiltonian mechanics is equivalent to Newton’s laws of motion but provides a simplification of the analysis for many dynamical systems. Another approach is Lagrangian mechanics, which we leave to the reader as a topic for independent study. In Hamiltonian mechanics, a system is described by canonically conjugate variables denoted by $q_i$ and $p_i$ :
$$q_1, q_2, \ldots, q_i, \ldots ; p_1, p_2, \ldots, p_i, \ldots$$
$q_i$ and $p_i$ are also called the generalized position and momentum coordinates, respectively. For example, $q_1, q_2$ and $q_3$ may refer to the actual position coordinates $(x, y, z)$ of a particle and $p_1, p_2$ and $p_3$ correspond to its linear momentum $\left(p_x, p_y\right.$ and $\left.p_2\right)$. If there is more than one particle, then $q_4, q_5, q_6, p_4, p_5$ and $p_6$ are the corresponding variables for the second particle, and so on. In general, $q_i$ and $p_i$ may represent dynamic variables other than position and momentum, depending on the system. For example, to describe a pendulum (Exercise 1.1), it is easier to assign $q_i$ as the angle of the pendulum and $p_i$ as the angular momentum. The $q_i$ and $p_i$ variables, if they are canonically conjugate variables, satisfy the Hamilton equations:

$$\begin{gathered} \frac{d q_i}{d t}=\frac{\partial H}{\partial p_i} \ \frac{d p_i}{d t}=-\frac{\partial H}{\partial q_i} \end{gathered}$$
where $H$ is the Hamiltonian and $t$ is the time. The Hamiltonian is the total energy of the system (kinetic energy plus potential energy) expressed in terms of the generalized coordinates.

To illustrate Hamilton’s approach, let us find the equations of motion for a particle of mass $m$ in a one-dimensional potential, $U(x)$, shown in Fig. 1.1. Although $U(x)$ is actually the potential energy, physicists often abbreviate this simply as “the potential”. In this example, suppose the generalized coordinates $\left(q_i, p_i\right)$ are the position $(x)$ and momentum $(p)$ of the particle:
$$\begin{gathered} q \rightarrow x \ p \rightarrow m \frac{d x}{d t} \end{gathered}$$
The Hamiltonian is the total energy (kinetic energy plus potential energy) expressed in terms of the generalized coordinates from Eqs. (1.4) and (1.5):
$$H-\frac{p^2}{2 m}+U(x)$$

## 物理代写|光学代写Optics代考|Canonical Quantization

Canonical quantization is a prescribed method of finding the Hamiltonian of a quantum system. The procedure was developed by Paul Dirac in 1925 (Fig. 1.2). Dirac proposed that any system for which we have a classical description can be quantized according to the procedure of canonical quantization. In canonical quantization, the generalized coordinates of the classical description, found by Hamilton’s approach (described in the previous section), are replaced by the corresponding quantum operators (denoted by a “hat”, )):
$$H\left(q_1, \ldots, q_i, \ldots ; p_1, \ldots, p_i, \ldots\right) \stackrel{\substack{\text { canonical } \ \text { quantization }}}{\longrightarrow} \widehat{H}\left(\widehat{q}_1, \ldots, \widehat{q}_i, \ldots ; \widehat{p}_1, \ldots, \widehat{p}_i, \ldots\right)$$ where the classical description is on the left and the quantum description is on the right. The Hamiltonian of the quantum system, $\widehat{H}$, is expressed in terms of the generalized coordinates (now operators) on the right-hand side of Eq. (1.14). For example, according to Sect. 1.1, the generalized coordinates for a particle of mass $m$ in a potential, $U(x)$, are $x$ and $p$. The Hamiltonian for the corresponding quantum system becomes
\begin{aligned} & \text { canonical } \ & H=\frac{p^2}{2 m}+U(x) \stackrel{\text { quantization }}{\longrightarrow} \widehat{H}=\frac{\widehat{p}^2}{2 m}+U(\hat{x}) \ & \end{aligned}
Once you know $\widehat{H}$ of the quantum system, you can determine its quantum properties from the time-dependent Schrodinger equation:
$$i \hbar \frac{\partial|\psi\rangle}{\partial t}=\widehat{H}|\psi\rangle$$
where $|\psi\rangle$ is the state of the system and $\hbar$ is the reduced Planck constant $(\hbar=h / 2 \pi)$. You may remember from introductory quantum mechanics that Eq. (1.16) reduces to the time-independent Schrodinger equation for stationary states:
$$\widehat{H}\left|\psi_n\right\rangle=E_n\left|\psi_n\right\rangle$$
where $E_n$ are the eigenenergies and $\left|\psi_n\right\rangle$ are the eigenstates (basis states) of the system.

# 光学代考

## 物理代写|光学代写Optics代考|Hamiltonian Mechanic

$$q_1, q_2, \ldots, q_i, \ldots ; p_1, p_2, \ldots, p_i, \ldots$$
$q_i$ 和 $p_i$ 也分别称为广义位置和动量坐标。例如， $q_1, q_2$ 和 $q_3$ 可参考实际位置坐标 $(x, y, z)$ 一个 粒子和 $p_1, p_2$ 和 $p_3$ 对应于它的线性动量 $\left(p_x, p_y\right.$ 和 $\left.p_2\right)$. 如果有一个以上的粒子，则 $q_4, q_5, q_6, p_4, p_5$ 和 $p_6$ 是第二个粒子的相应变量，依此类推。一般来说， $q_i$ 和 $p_i$ 可能表示位置 和动量以外的动态变量，具体取决于系统。例如，要描述一个钟摆（练习 1.1），更容易分配 $q_i$ 作为摆的角度和 $p_i$ 作为角动量。这 $q_i$ 和 $p_i$ 变量，如果它们是典型共轭变量，则满足 Hamilton 方程:
$$\frac{d q_i}{d t}=\frac{\partial H}{\partial p_i} \frac{d p_i}{d t}=-\frac{\partial H}{\partial q_i}$$

$$q \rightarrow x p \rightarrow m \frac{d x}{d t}$$

$$H-\frac{p^2}{2 m}+U(x)$$

## 物理代写|光学代写Optics代考|Canonical Quantization

1.2）。狄拉克提出，任何具有经典描述的系统都可以根据规范量化过程进行量化。在规范量化 中，通过哈密顿方法 (在上一节中描述) 找到的经典描述的广义坐标被相应的量子算子 (用“帽 子”表示) 代替:
$$H\left(q_1, \ldots, q_i, \ldots ; p_1, \ldots, p_i, \ldots\right) \stackrel{\text { canonical quantization }}{\longrightarrow} \widehat{H}\left(\hat{q}_1, \ldots, \hat{q}_i, \ldots ; \hat{p}_1, \ldots, \hat{p}_i\right.$$

$$\text { canonical } \quad H=\frac{p^2}{2 m}+U(x) \stackrel{\text { quantization }}{\longrightarrow} \widehat{H}=\frac{\hat{p}^2}{2 m}+U(\hat{x})$$

$$i \hbar \frac{\partial|\psi\rangle}{\partial t}=\widehat{H}|\psi\rangle$$

$$\widehat{H}\left|\psi_n\right\rangle=E_n\left|\psi_n\right\rangle$$

## 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|几何光学代写Geometrical Optics代考|OPTI502

statistics-lab™ 为您的留学生涯保驾护航 在代写几何光学Geometrical Optics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。我们的专家在代写几何光学Geometrical Optics代写方面经验极为丰富，各种代写几何光学Geometrical Optics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|几何光学代写Geometrical Optics代考|Set of Maxwell Equations for Electrostatic Field

First, we introduce the set of Maxwell equations for the electrostatic field in free space. Using Gauss’s Law (see Chap. 2), we can write the electric flux of electric field created by continuous charge distribution in a volume $V$ enclosed by the surface $A$ as
$$\oint_A \mathbf{E} \cdot d \mathbf{A}=\frac{Q_{i n}}{\epsilon_0}$$
Note that in Eq. (4.65) $\mathbf{E}$ is the electrostatic field created by all charges in space, and $Q_{i n}$ is the electric charge inside the volume $V$ enclosed by the surface $A$. The left-hand side of Eq. (4.65) can be written in the following form using Gauss formula: $$\oint_A \mathbf{E} \cdot d \mathbf{A}=\int_V \nabla \cdot \mathbf{E} d V$$
where $V$ is the volume enclosed by the surface $A$. In addition, the right-hand side of Eq. (4.65) can be written as
$$\frac{Q_{i n}}{\epsilon_0}=\int_V \frac{\rho(\mathbf{r})}{\epsilon_0} d V$$
Combining Eqs. (4.65), (4.66) and (4.67), we get
$$\int_V \nabla \cdot \mathbf{E} d V=\int_V \frac{\rho(\mathbf{r})}{\epsilon_0} d V$$
where $\nabla \cdot \mathbf{E}$ is the divergence of the vector $\mathbf{E}$, which produces a scalar.
Comparing both sides of Eq. (4.68), we obtain the first Maxwell equation in free space:
$$\nabla \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})}{\epsilon_0}$$
where both $\mathbf{E}$ and $\rho$ can be functions of the position $\mathbf{r}$.
Using the expression of the electrostatic potential difference in free space, Eq. (4.10) (Chap.3), we have
$$\Delta \phi=-\int_A^B \mathbf{E} \cdot d \mathbf{s}$$
where $A$ and $B$ are two points in free space, and $d \mathbf{s}$ is an infinitesimal displacement along the curve joining points $A$ and $B$. If we consider a closed path, that is, $A=B$, then $\Delta \phi=\phi_B-\phi_A=\phi_A-\phi_A=0$, and hence
$$\oint_{\mathcal{L}} \mathbf{E} \cdot d \mathbf{s}=0$$

## 物理代写|几何光学代写Geometrical Optics代考|Maxwell Equations for Dielectric Media Electrostatic Field

We mentioned that in the dielectric medium, an average over macroscopically small volumes, which are microscopically large, is necessary to obtain the Maxwell equations of the macroscopic phenomena.
The first observation is that Eq. (4.74) holds microscopically, that is
$$\nabla \times \mathbf{E}_{\text {micro }}=0$$
When averaging is made of the homogeneous Eq. (4.75), we obtain
$$\nabla \times \mathbf{E}=0$$
Equation (4.76) indicates that Eq. (4.74) holds for the averaged macroscopic electric field $\mathbf{E}$.

Using Eq. (4.57) for the effective charge density in the medium, Eq. (4.69) becomes
$$\nabla \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})-\nabla \cdot \mathbf{P}(\mathbf{r})}{\epsilon_0}$$
Rearranging Eq. (4.77), we get
$$\nabla \cdot\left(\epsilon_0 \mathbf{E}(\mathbf{r})+\mathbf{P}(\mathbf{r})\right)=\rho(\mathbf{r})$$
Using the definition of the electric displacement vector given by Eq. (4.58), we write Eq. (4.78) as
$$\nabla \cdot \mathbf{D}(\mathbf{r})=\rho(\mathbf{r})$$
Note that Eqs. (4.76) and (4.79) are the macroscopic Maxwell equations in the dielectric medium, which are the counterparts of Eqs. (4.69) and (4.74).

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Set of Maxwell Equations for Electrostatic Field

$$\oint_A \mathbf{E} \cdot d \mathbf{A}=\frac{Q_{i n}}{\epsilon_0}$$

$$\oint_A \mathbf{E} \cdot d \mathbf{A}=\int_V \nabla \cdot \mathbf{E} d V$$

$$\frac{Q_{i n}}{\epsilon_0}=\int_V \frac{\rho(\mathbf{r})}{\epsilon_0} d V$$

$$\int_V \nabla \cdot \mathbf{E} d V=\int_V \frac{\rho(\mathbf{r})}{\epsilon_0} d V$$

$$\nabla \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})}{\epsilon_0}$$

$$\Delta \phi=-\int_A^B \mathbf{E} \cdot d \mathbf{s}$$

$$\oint_{\mathcal{L}} \mathbf{E} \cdot d \mathbf{s}=0$$

## 物理代写|几何光学代写Geometrical Optics代考|Maxwell Equations for Dielectric Media Electrostatic Field

$$\nabla \times \mathbf{E}_{\text {micro }}=0$$

$$\nabla \times \mathbf{E}=0$$

$$\nabla \cdot \mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})-\nabla \cdot \mathbf{P}(\mathbf{r})}{\epsilon_0}$$

$$\nabla \cdot\left(\epsilon_0 \mathbf{E}(\mathbf{r})+\mathbf{P}(\mathbf{r})\right)=\rho(\mathbf{r})$$

$$\nabla \cdot \mathbf{D}(\mathbf{r})=\rho(\mathbf{r})$$

## 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|几何光学代写Geometrical Optics代考|PHYSICS134A

statistics-lab™ 为您的留学生涯保驾护航 在代写几何光学Geometrical Optics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。我们的专家在代写几何光学Geometrical Optics代写方面经验极为丰富，各种代写几何光学Geometrical Optics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|几何光学代写Geometrical Optics代考|Energy Stored in Capacitor

The energy stored in the absence of the dielectric is
$$U_0=\frac{Q_0^2}{2 C_0}$$
After the battery is removed and the dielectric inserted, the charge on the capacitor remains the same. Hence, the energy stored in the presence of the dielectric is
$$U=\frac{Q_0^2}{2 C}$$
Using the relation $C=\varepsilon C_0$, then
$$U=\frac{Q_0^2}{2 \varepsilon C_0}$$
or
$$U=\frac{U_0}{\varepsilon}$$
Because $\varepsilon>1$, the final energy is less than the initial energy (see also Eq. (4.48)) $\Delta U=U-U_0<0$. We can account for the “missing” energy by noting that the dielectric, when inserted, gets pulled into the device. An external agent must do negative work to keep the dielectric from accelerating.
This work is simply the difference
$$W_a=U-U_0$$
Alternatively, the positive work done on the external agent by the system is
$$W=-W_a=U_0-U$$

## 物理代写|几何光学代写Geometrical Optics代考|Electric Polarization

Consider an electric field applied to a medium made up of a large number of particles, such as atoms or molecules. The charges bound in molecules will then respond to the external electric field, and they will follow the perturbed motion to align with the external field. Thus, the charge density within the molecules will be distorted. The dipole moments ${ }^3$ of each molecule will be different in comparison to the dipole moments in the absence of the applied electric field. That is, in the absence of the external field, the average dipole moments over all molecules of the substance are zero because the dipole vectors are oriented randomly. In contrast, in the presence of the applied electric field, the net dipole moment of the substance is different from zero. Therefore, in the medium, there is an average dipole moment per unit volume, which is called electric polarization $\mathbf{P}$, given as
$$\mathbf{P}(\mathbf{r})=\sum_i n_i\left\langle\mathbf{p}_i\right\rangle$$
In Eq. (4.51), $\mathbf{p}_i$ is the dipole moment of the molecule type $i$ in the medium, $\langle\cdots\rangle$ denotes the average over a small volume around $\mathbf{r}$, and $n_i$ is the average number per unit volume of the molecule type $i$ at the position $\mathbf{r}$.

If the net charge of the molecule $i$ is $Q_i$, and there is a macroscopic excess or free charge, the charge density at the macroscopic level is
$$\rho(\mathbf{r})=\sum_i n_i\left\langle Q_i\right\rangle+\rho_{\text {free }}$$
Note that, in general, average charge of a molecule $i$ is zero, $\left\langle Q_i\right\rangle=0$, and hence, the charge density $\rho$ is equal to the macroscopic excess or free charge, $\rho_{\text {free }}$.

In the following, we will consider the case of a continuous charge distribution, as in Fig. $3.6$ (Chap. 3), and see the medium from a macroscopic viewpoint. The potential at some point $P$ at the position $\mathbf{r}$ from a macroscopic small volume element $d V$ at the position $\mathbf{r}^{\prime}$ is the sum of the potential created by the charge of $d V, d q-\rho\left(\mathbf{r}^{\prime}\right) d V$ and the dipole moment of $d V$ is $\mathbf{P}\left(\mathbf{r}^{\prime}\right) d V$, assuming that there are no higher macroscopic multipole moment densities:
$$d \phi\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=k_e\left(\frac{\rho\left(\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}+\frac{\mathbf{P}\left(\mathbf{r}^{\prime}\right) \cdot\left(\mathbf{r}-\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^3}\right)$$

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Energy Stored in Capacitor

$$U_0=\frac{Q_0^2}{2 C_0}$$

$$U=\frac{Q_0^2}{2 C}$$

$$U=\frac{Q_0^2}{2 \varepsilon C_0}$$

$$U=\frac{U_0}{\varepsilon}$$

$$W_a=U-U_0$$

$$W=-W_a=U_0-U$$

## 物理代写|几何光学代写Geometrical Optics代考|Electric Polarization

$$\mathbf{P}(\mathbf{r})=\sum_i n_i\left\langle\mathbf{p}i\right\rangle$$ 在等式中。(4.51), $\mathbf{p}_i$ 是分子类型的偶极矩 $i$ 在媒体中， $\langle\cdots\rangle$ 表示周围小体积的平均值 $\mathbf{r}$ ，和 $n_i$ 是分子类型 每单位体积的平均数 $i$ 在那个位置r. 如果分子的净电荷 $i$ 是 $Q_i$ ，并且存在宏观过剩或自由电荷，宏观层面的电荷密度为 $$\rho(\mathbf{r})=\sum_i n_i\left\langle Q_i\right\rangle+\rho{\text {free }}$$

$$d \phi\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=k_e\left(\frac{\rho\left(\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}+\frac{\mathbf{P}\left(\mathbf{r}^{\prime}\right) \cdot\left(\mathbf{r}-\mathbf{r}^{\prime}\right) d V}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^3}\right)$$

## 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|几何光学代写Geometrical Optics代考|PHYS201

statistics-lab™ 为您的留学生涯保驾护航 在代写几何光学Geometrical Optics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。我们的专家在代写几何光学Geometrical Optics代写方面经验极为丰富，各种代写几何光学Geometrical Optics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|几何光学代写Geometrical Optics代考|Energy Storage in the Electric Field

To transfer an amount of charge from one plate of a capacitor to the other during the process of charging the capacitor, an external work is done against the electric field. That work stores in the capacitor in the form of the potential energy. For that, let $q$ be the charge on the capacitor at some instant during the charging process when the potential difference across the capacitor is $\Delta V=q / C$. At that instant, one of the plates is carrying a charge $+q$ and the other $-q$. To transfer an increment of charge $d q$ from the plate with charge $-q$ (which is at a lower electric potential) to the plate carrying charge $+q$ (which is at a higher electric potential) an elementary work is done against the electric field:
$$d W=\Delta V d q=\frac{q}{C} d q$$
To calculate the total work required to charge the capacitor from $q=0$ to final charge $Q$, we integrate Eq. (4.27) as follows:
$$W=\int_0^Q \frac{q}{C} d q=\frac{1}{2} \frac{Q^2}{C}$$

This work done to charge the capacitor stores in the capacitor as an electric potential energy $U$. Therefore, $U=W$. Also, we can express the potential energy $U$ in the following forms:
\begin{aligned} U & =\frac{1}{2} \frac{Q^2}{C} \ & =\frac{1}{2} Q \Delta V \ & =\frac{1}{2} C(\Delta V)^2 \end{aligned}
Note that all expressions given by Eqs. (4.29)-(4.31) are equivalent; that is, they can all be used to calculate the potential energy stored in a capacitor depending on what is known. We can consider the energy stored in a capacitor as being stored in the electric field created between the plates as the capacitor is charged. This description is reasonable from the viewpoint that the electric field is proportional to the charge $Q$ stored on a capacitor. For a capacitor of two parallel plates, the potential difference is related to the electric field through a simple relationship $\Delta V=E d$. Furthermore, its capacitance is $C=\epsilon_0 \frac{A}{d}$. Then, we obtain
$$U=\frac{1}{2}\left(\epsilon_0 \frac{A}{d}\right)(E d)^2=\frac{1}{2} \epsilon_0(A d) E^2$$
Since the volume is $A d$, then the energy density is given
$$u_E=\frac{U}{A d}=\frac{1}{2} \epsilon_0 E^2$$
This expression is generally valid. That is, the energy density in any electric field is proportional to the square of the magnitude of the electric field at a given point.

## 物理代写|几何光学代写Geometrical Optics代考|Electrostatics of Macroscopic Media and Dielectrics

There exist many materials that do not allow electric charges to move freely within them, or may allow such motion to occur only very slowly. Those materials are used to block the flow of electrical current, and to form the insulators. For example, they can create insulating layers between the plates of a capacitor. Those materials are known as dielectric materials. As an application, the use of the dielectric material for a capacitor reduces its size for a given capacitance or increases its working voltage. Note that a dielectric material subject to a high enough electric field becomes a conductor; that is, the dielectric material experiences a dielectric breakdown. Thus, there exists a maximum voltage for dielectric capacitors to work. For example, there is a maximum power that a coaxial cable can adequately function in high-power applications such as radio transmitters; similarly, for microcircuits there are maximum voltages, which can be handled.

To know about the differences between dielectric and conducting materials, we can consider their behavior in electric fields. In particular, we have shown in Fig. 4.7 a conducting and dielectric sheet between the parallel plates in which a potential difference exists. That is, there are an equal amount of opposite charges on the two plates.

In the conducting sheet, the conducting electrons are free to move, and they establish a surface charge which exactly cancels the electric field within the conductor, as shown in Fig.4.7. That is, the surface charge density of the plates and conducting sheet is the same but with opposite sign. On the other hand, the electrons in the dielectric material are bound to atoms, and the external electric field causes only a displacement of the electronic configuration of atoms (see Fig. 4.7). However, it is sufficient to produce some surface charge with density $\sigma_{\text {ind }}$ (called an induced charge). We say that the dielectric is polarized. Note that the surface charge is not able to cancel the external electric field within the sheet; however, it does reduce. In the following, we will introduce a simplified molecular theory of dielectrics to understand the behavior of dielectric materials in the presence of an external electric field. ${ }^1$ A more complicated, but more precise theory, will be introduced in the following sections, accounting for electric polarization of the ponderable media. ${ }^2$

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Energy Storage in the Electric Field

$$d W=\Delta V d q=\frac{q}{C} d q$$

$$W=\int_0^Q \frac{q}{C} d q=\frac{1}{2} \frac{Q^2}{C}$$

$$U=\frac{1}{2} \frac{Q^2}{C} \quad=\frac{1}{2} Q \Delta V=\frac{1}{2} C(\Delta V)^2$$

$$U=\frac{1}{2}\left(\epsilon_0 \frac{A}{d}\right)(E d)^2=\frac{1}{2} \epsilon_0(A d) E^2$$

$$u_E=\frac{U}{A d}=\frac{1}{2} \epsilon_0 E^2$$

## 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|电磁学代写electromagnetism代考|PHYC20014

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism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磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism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|电磁学代写electromagnetism代考|Electric Field Lines

By definition, electric field lines are drawn to follow the same direction as the electric field vector at any point. Furthermore, the electric field vector is tangent to the line at every point along the field line.

The electric field lines are such that $\mathbf{E}$ is tangent to the electric field line at each point. The number of lines per unit surface area passing a surface perpendicular to the lines is proportional to the magnitude $|\mathbf{E}|$ in that region. Furthermore, the lines are directed radially away from the positive point charge. Moreover, the lines are directed radially toward the negative point charge.

In Fig. 1.7, we show the electric field lines of a negative and positive point charge. It can be seen that for a negative point charge, $-q$, the electric field lines are drawn toward the charge (see Fig. 1.7a). On the other hand, for a positive point charge, $+q$, electric field lines are leaving the charge, as shown in Fig. 1.7b.

The following general rules for drawing electric field lines apply:
The lines start from a positive charge and end on a negative charge. Also, the number of lines drawn, leaving a positive charge, or approaching a negative charge is proportional to the magnitude of the charge. Moreover, no two field lines can cross.

In Fig. 1.8, we show the electric field vector for a positive point charge $+q$ located at the point $(0,3,0)$ (Fig. 1.8b) and a negative point charge $-q$ located at $(0,-3,0)$ (Fig. 1.8a), colored according to the magnitude of the electric field $\mathbf{E}$ using a color scaling. as depicted in Fig. 1.8. Besides, the electric field lines of the resultant electric field are shown in Fig. 1.8c.

## 物理代写|电磁学代写electromagnetism代考|Motion in Uniform Electric Field

Suppose a charge particle of mass $m$ and charge $q$ is moving in a uniform electric field $\mathbf{E}$. Electric field $\mathbf{E}$ exerts on a particle placed in it the force
$$\mathbf{F}=q \mathbf{E}$$

If that force is equal to the resultant force exerted on the particle, it causes the particle to accelerate, based on Newton’s second law:
$$m \mathbf{a}=q \mathbf{E}$$
The acceleration gained by the charge is given as
$$\mathbf{a}=\frac{q}{m} \mathbf{E}$$
Therefore, if $\mathbf{E}$ is uniform (that is, constant in magnitude and direction), then a is constant. Furthermore, if the particle has a positive charge, then its acceleration is in the direction of the electric field. On the other hand, if the particle has a negative charge, then its acceleration is in the direction opposite the electric field.

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Motion in Uniform Electric Field

$$\mathbf{F}=q \mathbf{E}$$

$$m \mathbf{a}=q \mathbf{E}$$

$$\mathbf{a}=\frac{q}{m} \mathbf{E}$$

## 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。