物理代写|广义相对论代写General relativity代考|ASP3051

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|Tensors, Component View

We continue in this section with the classic component view of vectors and tensors as indexed arrays. This section consists largely of a set of theorems which are proved by a relatively simple algebraic process often called index juggling. It should become clear that after some practice the balancing of the indices does much of the work for us.

To define a tensor we generalize the idea of a vector as defined as an $n$-tuple with a well-defined transformation between coordinate systems: a tensor is defined as a set of quantities with any number of indices, which transforms according to
$$\bar{T}{m \ldots}^{l \ldots}=\frac{\partial \bar{x}^{l}}{\partial x^{q}} \ldots \frac{\partial x^{n}}{\partial \bar{x}^{m}} \ldots T^{q \ldots . . .}, \quad \text { tensor components. }$$ The total number of indices is referred to as the rank; some of the indices may be upper, or contravariant, and others may be lower, or covariant. The number of such indices is written as $(M, N)$. Thus for example a vector is a first rank tensor and (1, 0 ). Another example is $V^{j} W{q}$, which is a second rank tensor and $(1,1)$.

From this tensor definition many simple but powerful theorems follow. We have already introduced and proved two of them in Sect. 4.2: Theorem 1 concerned the Jacobian matrices and Theorem 2 the invariance of the inner product of vectors. Let us continue to more such theorems.

Theorem 3 To contract a tensor we set an upper index equal to a lower index and sum, which gives another tensor; for example one contraction of $T^{\alpha \beta}{ }{\lambda y}$ is $T{\beta \gamma}^{\alpha \beta}=S^{\alpha}{ }{\gamma}$. Contraction of a rank $r$ tensor produces a rank $r-2$ tensor. Consider the above 4th rank tensor as an example. Then the contracted object transforms as $\bar{T}{\beta \gamma}^{\alpha \beta}=\frac{\partial \bar{x}^{\alpha}}{\partial x^{\omega}} \frac{\partial \bar{x}^{\beta}}{\partial x^{\sigma}} \frac{\partial x^{\lambda}}{\partial \bar{x}^{\beta}} \frac{\partial x^{\eta}}{\partial \bar{x}^{\gamma}} T^{\omega \sigma}{ }{\lambda \eta}=\frac{\partial \bar{x}^{\alpha}}{\partial x^{\omega}} \delta{\sigma}^{\lambda} \frac{\partial x^{\eta}}{\partial \bar{x}^{\gamma}} T^{\omega \omega \sigma}{ }{\lambda \eta}^{\partial x^{\eta}}$ $=\frac{\partial \bar{x}^{\alpha}}{\partial x^{\omega \omega}} \frac{\partial x^{\eta}}{\partial \bar{x}^{\gamma}} T{\sigma \eta}^{\omega \omega} .$

物理代写|广义相对论代写General relativity代考|Tensors, Abstract View

As with vectors we may view tensors as abstract objects instead of from the classic component point of view discussed in the previous section. In this abstract approach an $(M, N)$ tensor is defined to linearly map $M$ vectors and $N$-forms to the reals. For example a $(0,2)$ tensor $T$ operates as a linear map on vectors $\vec{V}, \vec{W}$ as follows
$$T(\vec{V}, \vec{W})=T\left(V^{\beta} \vec{e}{\beta}, W^{\mu} \vec{e}{\mu}\right)=V^{\beta} W^{\mu} T\left(\vec{e}{\beta}, \vec{e}{\mu}\right) \equiv V^{\beta} W^{\mu} T_{\beta \mu}$$
The components $T_{\beta \mu}$ defined here are the same as we discussed in the previous section. Thus a vector is also a $(1,0)$ tensor and a 1 -form is also a $(0,1)$ tensor. The metric is the most important special case of a $(0,2)$ tensor, so we explicitly note its operation in terms of components
$$g(\vec{V}, \vec{W})=V^{\beta} W^{\mu} g_{\beta \mu}$$
Let’s look at another example of a $(0,2)$ tensor. Define the direct product of two 1 -forms as something that operates linearly on two vectors to give a real in the following natural way,
$$\tilde{p} \otimes \tilde{q}=\text { direct product of 1-forms, } \tilde{p} \otimes \tilde{q}(\vec{V}, \vec{W})=\tilde{p}(\vec{V}) \tilde{q}(\vec{W})$$
That is, the first factor in the direct product operates on the first vector and the second factor in the direct product operates on the second vector. The direct product in (4.56) is thus a $(0,2)$ tensor. It should be clear that we can extend the definition to the direct product of any number $M$ of 1 -form factors to produce a $(0, M)$ tensor and so forth.
Recall that we discussed in Sect. $4.3$ a basis for 1-forms which we denoted as $\tilde{\omega}^{\alpha}$. We can similarly show that there exists a basis for the product of two 1 -forms or ( 0 , 2) tensors. Indeed the basis is a linear combination of the direct product of the $\tilde{\omega}^{\alpha}$. We write that linear combination as
$$f=f_{\alpha \beta} \tilde{\omega}^{\alpha} \otimes \tilde{\omega}^{\beta} .$$

In general relativity we often find it useful to use tetrads, a set of four basis vectors that forms an orthonormal basis as in special relativity. This sets up a reference frame at a point that is analogous to the reference frame of special relativity. The tetrad differs from the set of coordinate basis vectors in that it is normalized and need not align with the coordinate axes. More generally, in $n$ dimensions we define an $n$-trad, a set of $n$ basis vectors $\vec{e}{a}$ oriented and normalized so that $$\vec{e}{a} \cdot \vec{e}{b}=\eta{a b}$$
where the $\eta_{a b}$ matrix is chosen for convenience. It is usually taken to be the constant Lorentz metric in relativity theory but may be any constant matrix such as the Kronecker delta as needed in other situations; we refer to it as the $n$-trad metric. In this section the $n$-trads will be labeled with lower Latin indices early in the alphabet like $b$, and the space indices will usually be Greek.

Notice that the local Lorentz frame we previously discussed is essentially the same as the frame provided by the tetrads. Indeed it is possible to develop the theory of tetrads based on the transformation to the local Lorentz frame, although we will not do that here (Lawrie 1990).

In this section we will denote the coordinate basis as $\vec{g}{\beta}$ to distinguish it from the $n$-trad basis $\vec{e}{a}$, and it will be labeled with Greek indices. The $n$-trad may be expanded in terms of the coordinate basis as
$$\vec{e}{a}=e{a}^{\beta} \vec{g}{\beta}, \quad e{a}^{\beta}=n \text {-trad components in coordinate basis. }$$
This gives a beautiful relation for the $n$-trad metric in terms of the metric,
\begin{aligned} &\eta_{a b}=\vec{e}{a} \cdot \vec{e}{b}=\left(e_{a}^{\beta} \vec{g}{\beta}\right) \cdot\left(e{b}^{\mu} \vec{g}{\mu}\right)=e{a}^{\beta} e_{b}^{\mu}\left(\vec{g}{\beta} \cdot \vec{g}{\mu}\right)=e_{a}^{\beta} e_{b}^{\mu} g_{\beta \mu}, \ &\eta_{a b}=e_{a}^{\beta} e_{b}^{\mu} g_{\beta \mu} \end{aligned}

物理代写|广义相对论代写General relativity代考|Tensors, Abstract View

G(在→,在→)=在b在μGbμ

p~⊗q~= 1-形式的直接乘积， p~⊗q~(在→,在→)=p~(在→)q~(在→)

F=F一个bω~一个⊗ω~b.

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。