Posted on Categories:Differential Manifold, 微分流形, 数学代写

# 数学代考|微分流形代考Differential Manifold代写|MATH881 Tensor fields

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## 数学代考|微分流形代考Differential Manifold代写|Tensor fields

We now specialize the previous considerations to the special situation when $E$ is the tangent bundle of $M, E \cong T M$. The cotangent bundle is then
$$T^* M:=(T M)^* .$$
We define the tensor bundles of $M$
$$\mathcal{T}_s^r(M):=\mathcal{T}_s^r(T M)=(T M)^{\otimes r} \otimes\left(T^* M\right)^{\otimes s} .$$
Definition 2.3.4. (a) A tensor field of type $(r, s)$ over the open set $U \subset M$ is a section of $\mathcal{T}_s^r(M)$ over $U$.

(b) A degree $r$ differential form (r-form for brevity) is a section of $\Lambda^r\left(T^* M\right)$. The space of (smooth) $r$-forms over $M$ is denoted by $\Omega^r(M)$. We set
$$\Omega^{\bullet}(M):=\bigoplus_{r \geq 0} \Omega^r(M) .$$
(c) A Riemannian metric on a manifold $M$ is a metric on the tangent bundle. More precisely, it is a symmetric $(0,2)$-tensor $g$, such that for every $x \in M$, the bilinear map
$$g_x: T_x M \times T_x M \rightarrow \mathbb{R}$$
defines a Euclidean metric on $T_x M$.

## 数学代考|微分流形代考Differential Manifold代写|Fiber bundles

We consider useful at this point to bring up the notion of fiber bundle. There are several reasons to do this.

On one hand, they arise naturally in geometry, and they impose themselves as worth studying. On the other hand, they provide a very elegant and concise language to describe many phenomena in geometry.

We have already met examples of fiber bundles when we discussed vector bundles. These were “smooth families of vector spaces”. A fiber bundle wants to be a smooth family of copies of the same manifold. This is a very loose description, but it offers a first glimpse at the notion about to be discussed.

The model situation is that of direct product $X=F \times B$, where $B$ and $F$ are smooth manifolds. It is convenient to regard this as a family of manifolds $\left(F_b\right)_{b \in B}$. The manifold $B$ is called the base, $F$ is called the standard (model) fiber, and $X$ is called the total space. This is an example of trivial fiber bundle.

In general, a fiber bundle is obtained by gluing a bunch of trivial ones according to a prescribed rule. The gluing may encode a symmetry of the fiber, and we would like to spend some time explaining what do we mean by symmetry.

## 数学代考|微分流形代考Differential Manifold代写|Tensor fields

$$T^* M:=(T M)^* .$$

$$\mathcal{T}s^r(M):=\mathcal{T}_s^r(T M)=(T M)^{\otimes r} \otimes\left(T^* M\right)^{\otimes s} .$$ 定义 2.3.4。 (a) 尖型的张荲场 $(r, s)$ 在开集上 $U \subset M$ 是一部分 $\mathcal{T}_s{ }^r(M)$ 超过 $U$. (b) 学位 $r$ 微分形式 (为简洁起见, $r$ 形式) 是 $\Lambda^r\left(T^* M\right)$. (平滑) 的空间 $r$-表格结束 $M$ 表示为 $\Omega^r(M)$. 我们设置 $$\Omega^{\bullet}(M):=\bigoplus{r \geq 0} \Omega^r(M) .$$
(c) 流形上的黎曼度量 $M$ 是切从上的度量。更准确地说，它是一个对称的 $(0,2)$-张量 $g$, 这样对于每个 $x \in M$ ，双线性映射
$$g_x: T_x M \times T_x M \rightarrow \mathbb{R}$$

## 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。