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# 物理代写|广义相对论代写General Relativity代考|Tensor fields

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## 物理代写|广义相对论代写General Relativity代考|Tensor fields

By definition, a ‘tensor’ is a field with some downstairs indices (that transform as in (3.62)) and some upstairs indices (that transform as in $(3.66))$
$\tilde{T}^{a \ldots} b_{\ldots}(\tilde{x})=\frac{\partial \tilde{X}^a}{\partial x^c} \ldots \frac{\partial x^d}{\partial \tilde{X}^b} \ldots \tilde{T}^{c \ldots}{ }{d \ldots}(x(\tilde{x}))$ The metric $g{a b}$ is a tensor: (3.30) is a special case of (3.64).
An equation between tensors is called a tensorial equation. If a tensorial equation holds in one coordinate system, it holds in all coordinate systems, because since the transformation of a tensor is linear, a tensor that vanishes in one coordinate system vanishes in all coordinate systems.

Careful: not all fields with indices are tensors. For instance, the Christoffel symbol $\Gamma_{b c}^a$ is not a tensor. The way it transforms under a change of coordinates can be derived from the definition [do it!] and is not given by (3.64).

A bit of cleaner maths. The definition of vector fields (and tensors) given above (quantities with indices that transform in a certain manner under change of coordinates) is customary in physics textbooks. I have always found it awkward. So, for once I prefer to mention a cleaner definition given by mathematicians.

Intuitively, the tangent space is the space of the ‘directions with length’ at the point. The mathematician’s solution to make this notion precise is to define a (controvariant) vector $v$ at a point $p$ as a derivative operator acting on scalar fields $\varphi(x)$ at $p$, measuring how they change in a certain direction. Its general form in terms of the coordinates $x^a$ is
$$\left.v \equiv v^a \frac{\partial}{\partial X^a}\right|_{X_p},$$

where $x_p$ are the coordinates of $p$. The set of these derivative operators at $p$ form a vector space (with coordinates $v^a$ ), which is by definition the tangent space $T_p$ at $p$. The quantities $v^a$ are the components of the vector $v$ in the basis of the vectors $\left.e_{|a|} \equiv \frac{\partial}{\partial X^a}\right|_{X_p}$. The components change under a change of coordinates as
$$\tilde{V}^a=\frac{\partial \tilde{x}^a}{\partial x^b} v^b$$
as follows easily from the Leibniz rule [do it!].

## 物理代写|广义相对论代写General Relativity代考|Covariant derivative

Tensors are important because an equation between tensors true in one coordinate system is true in any other coordinate system. Hence it is a relation that does not depend on a particular coordinate system. Importantly, in general the derivative of a tensor is not a tensor. For instance, the quantity $\partial_a V^b \equiv \frac{\partial V^a}{\partial x^b}$ is not a tensor. This is because the derivative of the transformed tensor includes also the derivative of the Jacobian $\frac{\partial \tilde{X}^a}{\partial X^b}$, which in general is not a constant. However, it turns out that the quantity

$$D_a v^b \equiv \partial_a v^b+\Gamma_{a c}^b v^c$$
happens to be a tensor, as an explicit calculation shows [do it!]. This quantity is called the ‘covariant derivative’ of $v^a$. The covariant derivative of a covariant tensor is defined with a minus sign:
$$D_a w_b \equiv \partial_a w_b-\Gamma_{a b}^c w_c$$
And the covariant derivative of a tensor with more indices picks one such term for each index. That is, for instance,
$$D_a w_c^b \equiv \partial_a w_c^b+\Gamma_{a d}^b w_c^d-\Gamma_{a c}^d w_d^b{ }^b$$
The covariant derivative is a coordinate-independent notion. If a tensor field $T_a^b=D_a V^b$ is the covariant derivative of the vector field $v^a$ in a coordinate system, it is so in any coordinate system. This is not true for a relation like $\partial_a v^b=T_a^b$.

If a vector has a covariant derivative null along a path $\gamma$, namely $\dot{\gamma}^a D_a v^b=0$, the angle between the vector and the tangent to the path remains constant along the path. This follows directly from the fact that this equation does not depend on the coordinates and is true in locally Cartesian coordinates. We say that such a vector is ‘parallel transported’ along the path. This result clarifies the meaning of $\Gamma_{b c}^a$ : it tells us how to parallel transport vectors.

## 物理代写|广义相对论代写General Relativity代考|Tensor fields

$$\left.v \equiv v^a \frac{\partial}{\partial X^a}\right|{X_p},$$ 在哪里 $x_p$ 是的坐标 $p$. 这些导数算子的集合在 $p$ 形成一个向量空间 (坐标 $\left.v^a\right)$ ，根据定义是切线空间 $T_p$ 在 $p$. 数量 $v^a$ 是向量的分量 $v$ 在向量的基础上 $\left.e{|a|} \equiv \frac{\partial}{\partial X^a}\right|_{X_p}$ 组件在坐标变化下变化为
$$\tilde{V}^a=\frac{\partial \tilde{x}^a}{\partial x^b} v^b$$

## 物理代写|广义相对论代写General Relativity代考|Covariant derivative

$$D_a v^b \equiv \partial_a v^b+\Gamma_{a c}^b v^c$$

$$D_a w_b \equiv \partial_a w_b-\Gamma_{a b}^c w_c$$

$$D_a w_c^b \equiv \partial_a w_c^b+\Gamma_{a d}^b w_c^d-\Gamma_{a c}^d w_d^{b b}$$

## 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。