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Math Analysis Homework - Sem 2 Week 10
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Topo Homework - Week 10
Topo Homework Week 10 T1 <div class="cbox" 1. (ERH) 判断以下多边形表示的曲面实际上是闭曲面的拓扑分类定理中的哪一个？ (1) ; (2) ; (3) ; (4) . </div <div class='pbox' 我们已经知道公式是 或再除 了.下面用数字代表多边形顶点. (1): 首先有同向边,是不可定...
Topo Homework - Week 11
Topo Homework Week 11 T1 <div class="cbox" 1. (ER) 设 为单连通空间, 是 中有相同起点和终点的道路. 证明 . </div <div class='pbox' 为单连通空间,故 , 这一个回路等价类.故 . </div T2 <div class="cbox" 3. (ER) 设 为平凡或离散拓扑空间,...
Math Analysis Homework - Sem 2 Week 9
Math Analysis Homework Sem 2 Week 9 T1 <div class="cbox" 2. 求函数 在点 的泰勒展开式. </div <div class='pbox' $ f(x, y) = \sin(x^2 + y^2) (0, 0) f(x, y) = \ln(1 + x + y) (0, 0) n f(x, y) = 3...
Math Analysis Homework - Sem 2 Week 8
Math Analysis Homework Term 2 Week 8 T1 <div class="cbox" 6. 设函数 由方程 所确定. 证明 $ z xy=xz x+yz y z = z(x,y) \frac{x}{z} = \varphi\left(\frac{y}{z}\right) \varphi F(x+y+z, x^2+y^2+z^2...
Topo Homework - Week 9
Topo Homework Week 9 T1 <div class="cbox" 8. (ER) 找到一个不是商映射的连续满射 . </div <div class="pbox" $ S = S^1 \times I, \beta S \sim x \in \text{Int}(S) x \sim x; \forall \theta \in [0, 2\...
Topo Homework - Week 8
Topo Homework Week 8 T1 <div class="cbox" 11. (MRH) 用极坐标 表示平面上的点 , 其中 表示点 到原点的距离, 表示从原点出发过点 的射线与 轴正向的夹角. 设 $ W \mathbb{R}^2 W W S^1 (1,\theta) A \{ (\dfrac{\theta+2\pi n}{\theta+2...
Padic Homework
Padic Homework T1 <div class="cbox" 1. 试证明, 若范数 满足对任意的整数 都有 , 则 一定是非阿的. (也就是说, 这个范数同构于某个 adic 范数.) </div 首先看起来这个范数同构于p adic不对.比如 不满足() <div class="pbox" $ x \in \mathbb{Q} p |x| p...
Math Analysis Homework - Sem2 Week 7
Math Analysis Homework Sem2 Week 7 Class 1 T1 <div class="cbox" 2. 设 , 求 . </div <div class='pbox' $ f(x, y) = \begin{cases} y \sin \frac{1}{x^2 + y^2}, & x^2 + y^2 \neq 0, \\ 0,...
Topo Homework - Week 7
Topo Homework Week 7 T1 <div class="cbox" 14. (ER) 设映射 , 如果 的每个点都有一个邻域 , 使得 限制在 上为常数, 则称 是局部常值映射. 证明: (1) 任何局部常值映射都是连续的; (2) 连通空间上的局部常值映射是常值映射. </div <div class='pbox' (1): $ x\in...
Math Analysis Homework - Sem 2 Week 6
Math Analysis Homework Sem 2 Week 6 T1 <div class="cbox" 2. (2) </div <div class='pbox' 存在 令 ,显然 时 ,则原式极限为: $ \lim {(x,y)\to(0,0)} xy \frac{x^2 y^2}{x^2+y^2} 0 \lim {\substack{x \...