MY-ASSIGNMENTEXPERT™可以为您提供 berkeley Math214 Manifold learning 流行学习课程的代写代考和辅导服务！
这是加州大學 流行学习课程代写成功案例。
Math214课程简介
he official textbook for the course is John Lee, Introduction to smooth manifolds, second edition. (The first edition presents the material in a different order and omits some key topics such as Sard’s theorem.) The following are some other books which you might also find useful, in order of increasing difficulty:
- Munkres, Topology, second edition. Clearly and gently explains point set topology, if you need to review this. (However we won’t be going into details of point set topology very much in the course.) Also gives a nice introduction to the fundamental group and the classification of surfaces. (Familiarity with the fundamental group is useful but we will not use this much.)
- Guillemin and Pollack, Differential topology. Explains the basic s of smooth manifolds (defining them as subsets of Euclidean space instead of giving the abstract definition). More elementary than Lee’s book, but gives nice explanations of transversality and differential forms (which we wil be covering).
- Spivak, A comprehensive introduction to differential geometry, vol. I, 3rd edition. This is a classic, which I considered using as the text for this course. Volumes 2-5 are also good (but go beyond this course). I learned a lot from this series when I was a student.
- Bott and Tu, Differential forms in algebraic topology. As the title suggests, it introduces various topics in algebraic topology using differential forms. We will not be doing much algebraic topology in this class, but you might still enjoy looking at this book while we are discussing differential forms.
Prerequisites
The basic plan is to cover most of the material in chapters 1-19 of Lee’s book (adding a few interesting things which are not in the book, and possibly some material from chapters 20-22, as time permits). My goal is for you to understand the basic concepts listed below and to be able to work with them. This material is all essential background for graduate level geometry (except for the most algebraic kind). In class I will try to introduce the main ideas, explain where they come from, and demonstrate how to use them. I will tend to leave technical lemmas for you to read in the book (or not).
Math214 Manifold learning HELP（EXAM HELP， ONLINE TUTOR）
Consider spherical coordinates on $\mathbb{R}^3$ (not including the line $z=0$ ) $\rho, \phi, \theta$ defined in terms of the Euclidean coordinates $x, y, z$ by
$$
x=\rho \sin \phi \cos \theta, \quad y=\rho \sin \phi \sin \theta, \quad z=\rho \cos \phi .
$$
(a) Express $\partial / \partial \rho, \partial / \partial \phi$, and $\partial / \partial \theta$ as linear combinations of $\partial / \partial x$, $\partial / \partial y$, and $\partial / \partial z$. (The coefficients in these linear combinations will be functions on $\mathbb{R}^3 \backslash(z=0)$.)
(b) Express $d \rho, d \phi$, and $d \theta$ as linear combinations of $d x, d y$, and $d z$.
Let $V$ and $W$ be finite dimensional vector spaces and let $A: V \rightarrow W$ be a linear map. Show that the dual map $A^: W^ \rightarrow V^$ is given in coordinates as follows. Let $\left{e_i\right}$ and $\left{f_j\right}$ be bases for $V$ and $W$, and let $\left{e^i\right}$ and $\left{f^j\right}$ be the corresponding dual bases for $V^$ and $W^$. If $A e_i=A_i^j f_j$ then $A^ f^j=A_i^j e^i$.
Let $V$ be a finite dimensional vector space and let $\langle\cdot, \cdot\rangle$ be an inner product on $V$. The inner product determines an isomorphism $\phi: V \rightarrow$ $V^$. (a) Show that the isomorphism $\phi$ is given in coordinates as follows. Let $\left{e_i\right}$ be a basis for $V$, let $\left{e^i\right}$ be the dual basis, and write $g_{i j}=\left\langle e_i, e_j\right\rangle$. Then $\phi\left(e_i\right)=g_{i j} e^j$. (b) The inner product, together with the isomorphism $\phi$, define an inner product on $V^$. Write this in coordinates as $g^{i j}=\left\langle e^i, e^j\right\rangle$. Show that the matrix $\left(g^{i j}\right)$ is the inverse of the matrix $\left(g_{i j}\right)$.
Show that if $M$ and $N$ are smooth manifolds and $p \in M, q \in N$, then there is a canonical isomorphism $T_{(p, q)}(M \times N)=T_p M \oplus T_q N$. Describe this isomorphism in terms of derivations, coordinate charts, and velocity vectors of curves.
MY-ASSIGNMENTEXPERT™可以为您提供 berkeley Math214 Manifold learning 流行学习课程的代写代考和辅导服务！
