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# 数学代写|微分拓扑作业代写differential topology代考|Regular Values

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## 数学代写|微分拓扑作业代写differential topology代考|The Rank

Remember that the rank of a linear transformation is the dimension of its image. In terms of matrices, this can be captured by saying that a matrix has rank at least $r$ if it contains an $r \times r$ invertible submatrix.

Let $\bar{f}:(M, p) \rightarrow(N, f(p))$ be a smooth germ. The rank $\mathrm{rk}{p} f$ of $f$ at $p$ is the rank of the linear map $T{p} f$. We say that a germ $\bar{f}$ has constant rank $r$ if it has a representative $f: U_{f} \rightarrow N$ whose rank rk $T_{q} f=r$ for all $q \in U_{f}$. We say that a germ $\bar{f}$ has rank $\geq r$ if it has a representative $f: U_{f} \rightarrow N$ whose rank rk $T_{q} f \geq r$ for all $q \in U_{f}$.

In view of Proposition 3.3.11, the rank of $f$ at $p$ is the same as the rank of the Jacobi matrix $D\left(y f x^{-1}\right)(x(p))$, where $(x, U)$ is a chart around $p$ and $(y, V)$ a chart around $f(p)$.

## 数学代写|微分拓扑作业代写differential topology代考|The Inverse Function Theorem

The technical foundation for the theorems to come is the inverse function theorem from multivariable calculus which we cite below. A proof can be found in Theorem 2.11 of 19, or in any other decent book on multivariable calculus.

Let $f: U_{1} \rightarrow U_{2}$ be a smooth function where $U_{1}, U_{2} \subseteq \mathbf{R}^{n}$. Let $p \in U_{1}$ and assume the Jacobi matrix $D f(p)$ is invertible in the point $p$. Then there exists a neighborhood around $p$ on which $f$ is smoothly invertible, i.e., there exists an open subset $U_{0} \subseteq U_{1}$ containing $p$ such that
$$\left.f\right|{U{0}}: U_{0} \rightarrow f\left(U_{0}\right)$$
is a diffeomorphism onto an open subset of $U_{2}$. See Figure 4.3.

## 数学代写|微分拓扑作业代写DIFFERENTIAL TOPOLOGY代考|The Rank Theorem

The rank theorem says that, if the rank of a smooth map $f: M \rightarrow N$ is constant in a neighborhood of a point, then there are charts such that $f$ looks like a composite $\mathbf{R}^{m} \rightarrow \mathbf{R}^{r} \subseteq \mathbf{R}^{n}$, where the first map is the projection onto the first $r \leq m$ coordinate directions, and the last one is the inclusion of the first $r \leq n$ coordinates. So, for instance, a map of rank 1 between 2-manifolds looks locally like
$$\mathbf{R}^{2} \rightarrow \mathbf{R}^{2}, \quad\left(q_{1}, q_{2}\right) \mapsto\left(q_{1}, 0\right) .$$
Not only does the rank theorem have an enormous impact, but also its proof carries two very neat ideas, namely
(1) if the question is “local”, we may reduce to the case where our manifolds are Euclidean spaces, and
(2) the inverse function theorem in the guise of Corollary 4.2.3 is an extremely efficient device for checking that a concrete formula actually gives rise to a chart – you simply calculate the Jacobian and observe that it is invertible.

## 数学代写|微分拓扑作业代写DIFFERENTIAL TOPOLOGY代考|THE INVERSE FUNCTION THEOREM

$$\left.f\right| {U {0}}: U_{0} \rightarrow f\leftU_{0}\右U_{0}\右$$

## 数学代写|微分拓扑作业代写DIFFERENTIAL TOPOLOGY代考|THE RANK THEOREM

R2→R2,(q1,q2)↦(q1,0).

1如果问题是“局部的”，我们可以简化为我们的流形是欧几里得空间的情况，并且
2以推论 4.2.3 为幌子的反函数定理是一种非常有效的工具，用于检查一个具体公式是否真的产生了一个图表——你只需计算雅可比并观察它是可逆的。

## 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。