assignmentutor™您的专属作业导师

assignmentutor-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|Differentiation of Invariants and Vectors

In the previous chapters, the importance of tensors in spacetime was stressed. Any time a new quantity is encountered, it will have to be checked to see if it is a tensor. If it isn’t, its transformation properties are not obvious. Construction of new tensors has, so far, taken the form of products of known tensors or total differentiation with respect to $\tau$. For example, $(d \tau)^{2}=d r_{\mu} d r^{\mu}, g^{\mu \nu} g_{\xi \nu}=\delta_{\xi}^{\mu}=\delta_{\xi}^{\mu}$, or $U^{\mu}=\frac{d r^{\mu}}{d \tau}$. From studies of the calculus of 3 -vectors, one recalls that partial differentiation with respect to the coordinates produces new 3-vectors and scalars through the gradient and divergence operations. In spacetime, such partial differentiation also leads to important new tensors.

Consider an invariant that is a function of position, $\Phi=\Phi\left(x^{\mu}\right)=$ $\Phi\left(x^{\mu^{\prime}}\right)$, e.g., $d \tau$. It has no index associated with it. Taking the partial derivative with respect to a coordinate yields
$$\Phi_{, \nu}=x^{\xi^{\prime}},{ }_{\nu} \Phi, \xi^{\prime} .$$
However, this is the rule for the transformation of a covariant vector and so another vector is added to our arsenal.

The gradient of a scalar $\Phi$ is given by $g^{\mu \nu} \Phi, \nu$ because in an inertial frame the expected results for the spatial components are obtained
\begin{aligned} \nabla^{\bar{\mu}} \Phi & \equiv g^{\bar{\mu} \bar{\nu}} \Phi_{, \bar{\nu}}=\eta^{\mu \nu} \Phi_{, \bar{\nu}} \ \vec{\nabla} \Phi &=\Phi,{x} \hat{e}{x}+\Phi,{y} \hat{e}{y}+\Phi_{, z} \hat{e}_{z} \end{aligned}

## 物理代写|广义相对论代写General relativity代考|Differentiation of Tensors

Given two vectors $V$ and $W$, the product, $V^{\mu} W_{\nu}$, transforms like a mixed tensor of rank 2 , and its covariant derivative yields
\begin{aligned} T_{\nu}^{\mu} i_{\alpha} &=\left(V^{\mu} W_{\nu}\right) ;{\alpha}=V^{\mu}{ }{\alpha} W_{\nu}+V^{\mu} W_{\nu} ;{\alpha} \ &=\left(V^{\mu}{ }{\alpha \alpha}+\Gamma_{\beta \alpha}^{\mu} V^{\beta}\right) W_{\nu}+V^{\mu}\left(W_{\nu} ;{\alpha}-\Gamma{\nu \alpha}^{\beta} W_{\beta}\right) \ &=\left(V^{\mu} W_{\nu}\right),{\alpha}+\Gamma{\beta \alpha}^{\mu} V^{\beta} W_{\nu}-\Gamma_{\nu \alpha}^{\beta} V^{\mu} W_{\beta} \ &=T_{\nu}^{\mu}{ }{\nu \alpha}+\Gamma{\beta \alpha}^{\mu} T_{\nu}^{\beta}-\Gamma_{\nu \alpha}^{\beta} T_{\beta}^{\mu}{ }{\beta} \end{aligned} yielding a mixed tensor of rank 3 . The contravariant index requires a positive sign, while the covariant index requires a negative sign for the $\mathrm{C}$ symbol. In a similar manner, one obtains the covariant derivatives of a covariant or contravariant tensor of rank 2 . If the rank is higher, say $n$, then $n \mathrm{C}$ symbols with appropriate signs are needed. In the case of the metric tensor, \begin{aligned} &g^{\mu \nu} ;{\alpha}=g^{\mu \nu}{ }{\alpha}+\Gamma{\beta \alpha}^{\mu} g^{\beta \nu}+\Gamma_{\alpha \beta}^{\nu} g^{\mu \beta}=0, \ &g_{\mu \nu} i_{\alpha}=g_{\mu \nu},{\alpha}-\Gamma{\mu \alpha}^{\beta} g_{\beta \nu}-\Gamma_{\alpha \nu}^{\beta} g_{\mu \beta}=0 . \end{aligned}
The reason the above tensors are zero is that in an inertial frame $g_{\bar{\mu} \bar{\nu}} \bar{\alpha}{\bar{\alpha}}=$ $\eta{\mu \nu} i_{\bar{\alpha}}=\eta_{\mu \nu}, \bar{\alpha}{\alpha}=0$. As this is a tensor equation, it holds in all frames, and leads to the more useful form for $\Gamma{\mu \nu}^{\lambda}$,
\begin{aligned} 0=& g_{\mu \nu} ;{\alpha}+g{\mu \alpha} ; \nu-g_{\alpha \nu} ; \mu \ =& g_{\mu \nu},{ }{\alpha}+g{\mu \alpha},{ }{\mu}-g{\alpha \nu, \mu}-\Gamma_{\mu \alpha}^{\beta} g_{\beta \nu}-\Gamma_{\alpha \nu}^{\beta} g_{\mu \beta} \ &-\Gamma_{\mu \nu}^{\beta} g_{\beta \alpha}-\Gamma_{\alpha \nu}^{\beta} g_{\mu \beta}+\Gamma_{\mu \alpha}^{\beta} g_{\beta \nu}+\Gamma_{\mu \nu}^{\beta} g_{\alpha \beta} \ 2 g_{\mu \beta} \Gamma_{\alpha \nu}^{\beta}=&\left(g_{\mu \nu},{\alpha}+g{\mu \alpha, \nu}-g_{\alpha \nu}, \mu\right.\ 2 g^{\mu \lambda} g_{\mu \beta} \Gamma_{\alpha \nu}^{\beta}=& 2 \delta_{\beta}^{\lambda} \Gamma_{\alpha \nu}^{\beta}=g^{\mu \lambda}\left(g_{\mu \nu},{ }{\alpha}+g{\mu \alpha},{ }{\nu}-g{\alpha \nu}, \mu\right.\ \Gamma_{\alpha \nu}^{\lambda}=& g^{\mu \lambda}\left(g_{\mu \nu},{ }{\alpha}+g{\mu \alpha, \nu}-g_{\alpha \nu}, \mu\right) / 2 \end{aligned}

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|Differentiation of Invariants and Vectors

$$\Phi_{, \nu}=x^{\xi^{\prime}},{ }{\nu} \Phi, \xi^{\prime} .$$ 然而，这是协变向量变换的规则，因此另一个向量被添加到我们的武器库中。 标量的梯度 $\Phi$ 是 (准) 给的 $g^{\mu \nu} \Phi, \nu$ 因为在惯性框架中，获得了空间分量的预期结果 $$\nabla^{\bar{\mu}} \Phi \equiv g^{\bar{\mu} \bar{\nu}} \Phi{, \bar{\nu}}=\eta^{\mu \nu} \Phi_{, \bar{\nu}} \vec{\nabla} \Phi \quad=\Phi, x \hat{e} x+\Phi, y \hat{e} y+\Phi_{, z} \hat{e}_{z}$$

## 物理代写|广义相对论代写General relativity代考|Differentiation of Tensors

$$T_{\nu}^{\mu} i_{\alpha}=\left(V^{\mu} W_{\nu}\right) ; \alpha=V^{\mu} \alpha W_{\nu}+V^{\mu} W_{\nu} ; \alpha \quad=\left(V^{\mu} \alpha \alpha+\Gamma_{\beta \alpha}^{\mu} V^{\beta}\right) W_{\nu}+V^{\mu}\left(W_{\nu} ; \alpha-\Gamma \nu \alpha^{\beta} W_{\beta}\right)=\left(V^{\mu} W_{\nu}\right), \alpha+\Gamma \beta \alpha^{\mu} V^{\beta} W_{\nu}-\Gamma_{\nu \alpha}^{\beta} V^{\mu} W_{\beta}$$

$$g^{\mu \nu} ; \alpha=g^{\mu \nu} \alpha+\Gamma \beta \alpha^{\mu} g^{\beta \nu}+\Gamma_{\alpha \beta}^{\nu} g^{\mu \beta}=0, \quad g_{\mu \nu} i_{\alpha}=g_{\mu \nu}, \alpha-\Gamma \mu \alpha^{\beta} g_{\beta \nu}-\Gamma_{\alpha \nu}^{\beta} g_{\mu \beta}=0 .$$

$$0=g_{\mu \nu} ; \alpha+g \mu \alpha ; \nu-g_{\alpha \nu} ; \mu=\quad g_{\mu \nu}, \alpha+g \mu \alpha, \mu-g \alpha \nu, \mu-\Gamma_{\mu \alpha}^{\beta} g_{\beta \nu}-\Gamma_{\alpha \nu}^{\beta} g_{\mu \beta}-\Gamma_{\mu \nu}^{\beta} g_{\beta \alpha}-\Gamma_{\alpha \nu}^{\beta} g_{\mu \beta}+\Gamma_{\mu \alpha}^{\beta} g_{\beta \nu}+\Gamma_{\mu \nu}^{\beta} g_{\alpha \beta} 2 g_{\mu \beta} \Gamma_{\alpha \nu}^{\beta}=\quad\left(g_{\mu \nu}, \alpha+g \mu \nu\right.$$

## 有限元方法代写

assignmentutor™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。

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