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物理代写|拓扑物理代写Physical topology代考| Introduction: Smooth Manifolds, Tangent Spaces,Derivatives

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物理代写|拓扑物理代写Physical topology代考|Examples of manifolds

1. $\mathbb{R}^{n}$, with atlas consisting of the single chart $\left(\mathbb{R}^{n}, i d\right)$.
2. The unit circle $S^{1} \subset \mathbb{R}^{2}$, with atlas consisting of two charts giving the usual angular coordinate,
$$\theta_{1}: S^{1}-{(-1,0)} \rightarrow(-\pi, \pi), \theta_{2}: S^{1}-{(1,0)} \rightarrow(0,2 \pi)$$
3. The unit sphere $S^{2} \subset \mathbb{R}^{3}$, with atlas consisting of two charts given by stereographic projections from the North Pole $N=(0,0,1)$ and South Pole $S=(0,0,-1)$ respectively,
$$\phi_{1}: S^{2}-{N} \rightarrow \mathbb{R}^{2}, \phi_{2}: S^{2}-{P} \rightarrow \mathbb{R}^{2} .$$
The same construction works for the $n$-sphere $S^{n} \subset \mathbb{R}^{n+1}$.
4. The two-torus $T^{2}=S^{1} \times S^{1} \subset \mathbb{R}^{4}$, with atlas consisting of 4 charts
$$\begin{array}{lll} U_{1} \times U_{1} & \rightarrow & (-\pi, \pi) \times(-\pi, \pi) \ U_{1} \times U_{2} & \rightarrow & (-\pi, \pi) \times(0,2 \pi) \ U_{2} \times U_{2} & \rightarrow & (0,2 \pi) \times(0,2 \pi) \ U_{2} \times U_{1} & \rightarrow & (0,2 \pi) \times(-\pi, \pi) \end{array}$$
giving the two angular coordinates on the torus, where
$$U_{1}=S^{1}-{(-1,0)}, U_{2}=S^{1}-{(1,0)}$$
Similarly the Cartesian product $M \times N$ of two manifolds $M, N$ is a manifold with charts $\phi \times \psi: U \times V \rightarrow \mathbb{R}^{m} \times \mathbb{R}^{n}$, where $(U, \phi),(V, \psi)$ are charts on $M, N$ respectively.

物理代写|拓扑物理代写Physical topology代考|Manifolds as configuration spaces of mechanical systems

The configuration spaces of many mechanical systems are often manifolds. We give some examples below, taken from Arnold’s book.
The configuration space of a planar pendulum is the circle $S^{1}$.
The configuration space of the “spherical” mathematical pendulum is the two-sphere $S^{2}$.
Example 3. The configuration space of a “planar double pendulum” is the two torus $T^{2}$.
The configuration space of a spherical double pendulum is the Cartesian product of two spheres $S^{2} \times S^{2}$.
The configuration space of a rigid line segment in the $\left(q_{1}, q_{2}\right)$-plane is the manifold $\mathbb{R}^{2} \times S^{1}$ with coordinates $q_{1}, q_{2}, q_{3}$.
Example 6. A rigid right-angled triangle OAB moves around the vertex $O$ in $\mathbb{R}^{3}$. The position of the triangle is completely described by an orthogonal righthanded frame $\overrightarrow{e_{1}}=O A /|O A|, \overrightarrow{e_{2}}=O B /|O B|, \overrightarrow{e_{3}}=\overrightarrow{e_{1}} \times \overrightarrow{e_{2}}$, or equivalently by the $3 \times 3$ orthogonal matrix $\left[\overrightarrow{e_{1}}\left|\overrightarrow{e_{2}}\right| \overrightarrow{e_{3}}\right]$ with determinant $+1$. The configuration space of the triangle OAB is the group $S O(3)$ of such matrices, which is a 3 -manifold.

物理代写|拓扑物理代写PHYSICAL TOPOLOGY代考|Tangent spaces and derivatives

Just as smooth curves in $\mathbb{R}$ or surfaces in $\mathbb{R}^{3}$ have at each point a tangent line or tangent plane, to each point $x$ of a $k$-manifold $M$ embedded in $\mathbb{R}^{n}$ one can associate a $k$-dimensional subspace $T_{x} M$ of $\mathbb{R}^{n}$, called the tangent space to $M$ at $p$. The vector space $T_{x} M$ can be described as the set of velocities $\dot{\gamma}(0)$ of smooth curves $\gamma$ in $M$ passing through $x$ at time 0 .

For an abstract manifold without a given embedding into Euclidean space, such as those defined as quotient spaces, it is not immediately clear however how to define tangent spaces. While the Whitney Embedding Theorem ensures that any smooth manifold may be embedded into some Euclidean space, it is useful (and aesthetically satisfying) to have a definition of the tangent space $T_{p} M$ as an abstract vector space defined intrinsically independent of the choice of an embedding into Euclidean space. We give two definitions, one geometric and the other algebraic.

物理代写|拓扑物理代写PHYSICAL TOPOLOGY代考|EXAMPLES OF MANIFOLDS

1. Rn，图集由单个图表组成(Rn,一世d).
2. 单位圆小号1⊂R2，地图集由两个图表组成，给出通常的角坐标，
θ1:小号1−(−1,0)→(−圆周率,圆周率),θ2:小号1−(1,0)→(0,2圆周率)
3. 单位球体小号2⊂R3, 地图集由北极的立体投影给出的两个图表组成ñ=(0,0,1)和南极小号=(0,0,−1)分别，
φ1:小号2−ñ→R2,φ2:小号2−磷→R2.
同样的建筑工程n-领域小号n⊂Rn+1.
4. 两环吨2=小号1×小号1⊂R4，图集由 4 个图表组成
在1×在1→(−圆周率,圆周率)×(−圆周率,圆周率) 在1×在2→(−圆周率,圆周率)×(0,2圆周率) 在2×在2→(0,2圆周率)×(0,2圆周率) 在2×在1→(0,2圆周率)×(−圆周率,圆周率)
给出环面上的两个角坐标，其中
在1=小号1−(−1,0),在2=小号1−(1,0)
同样的笛卡尔积米×ñ两个流形的米,ñ是带有图表的流形φ×ψ:在×在→R米×Rn， 在哪里(在,φ),(在,ψ)是图表米,ñ分别。

物理代写|拓扑物理代写PHYSICAL TOPOLOGY代考|MANIFOLDS AS CONFIGURATION SPACES OF MECHANICAL SYSTEMS

“球面”数学摆的构型空间是二球面小号2.

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。