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,...
# homework
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 \...
Topo Homework - Week 6
Topo Homework Week 6 T1 <div class="cbox" 7. (ERH) 设 为度量空间 的紧致子集. (1) 的直径定义为 , 证明存在 , 使得 ; (2) 设 , 证明存在 , 使得 ; (3) 设 是 的闭集, . 证明 . </div <div class="pbox" (1): 设 .因为 是紧集, 固定 后是关于...
Topo Homework - Week 5
Topo Homework Week 5 T1 <div class="cbox" 15. (ERH) 证明 空间 是豪斯多夫空间当且仅当 中任意序列至多有一个极限点. </div <div class='pbox' :因为 ,故假设序列 ,则 , 是开集.但 , ,矛盾.故任意 至多有一个极限点. :由于 ,故 , 是 的一组嵌套的( )邻域基, 是 的...
Math Analysis Homework - Sem 2 Week 5
Math Analysis Homework Sem 2 Week 5 T1 <div class="cbox" 2. 设 为 的聚点, 证明存在 中的点列 ( ), 使得 . </div <div class='pbox' 是 的聚点,即对 的任意开球邻域 , . 则取 ,取 ,则 . </div T2 <div class="cbox" 3. 对于下列...
Topo Homework - Week 4
Topo Homework Week 4 T1 <div class="cbox" 12. (ERH) 证明下面的平面区域是同胚的: 1. 整个 平面; 2. 开象限 ; 3. 开角 ; 4. 平面去掉一条射线 , 其中 . </div <div class="pbox" ,则 ,故 . ,则 ,且 ,故 . $ f 3,f 3^{ 1} f 3(C)=D...
Math Analysis Homework - Sem 2 Week 4
Math Analysis Homework Sem 2 Week 4 T1 <div class="cbox" 1. 将下列函数在指定点展开成泰勒级数： (3) ; </div <div class="pbox" $ |x 1|<1 x=2,0 0 \ln (1 + x + x^2 + x^3), x = 0 \ln ( 1,1) ( 1,1) f(x)...
Topo Homework - Week 3
Topo Homework Week 3 T1 <div class="cbox" T1. (ER) 确定下面每种情况中的 和 : (1) 在下极限拓扑空间 中, ; (2) ; (3) 在欧氏直线 上, ; (4) 在下极限拓扑空间 中, ; (5) 在欧氏平面 上, . </div <div class="pbox" (1): (2): (3): (4...
Math Analysis Homework - Sem 2 Week 3
Math Analysis Homework Sem 2 Week 3 T1 <div class="cbox" 1. 求下列幂级数的收敛域: (2) ; </div <div class="pbox" $ \sum {n=1}^\infty \left(\frac{a^n}{n} + \frac{b^n}{n^2}\right) x^n, a 0, b...
Topo Homework - Week 2
Topo Homework Week 2 T1 <div class="cbox" 1. (ER) 尝试给出 的两个非豪斯多夫的拓扑、一个豪斯多夫的拓扑, 并证明它们对应的拓扑空间不同胚. </div <div class="pbox" Not T2: T2: 离散拓扑 T2是拓扑性质:若存在连续双射 是同胚,则 中的开集/闭集与 中的一一对应,所以 均为...
Math Analysis Homework - Sem 2 Week 2
Math Analysis Homework Sem 2 Week 2 Class 1 T1 <div class="cbox" 2. 设级数 收敛, 证明 $ u n=\dfrac{a n}{n^x} a n \dfrac1{n^x} \sum {n=1}^{\infty} ( 1)^n x^n (1 x) [0, 1] \forall x,\sum {...
Topo Homework - Week 1
Topo Homework Week 1 T1 <div class="cbox" 5. (ER) 在三点集 上, 平凡拓扑有两个开集, 离散拓扑有八个开集. 对于 中的每一个数, 找出 上由 个开集组成的拓扑, 或证明这样的拓扑不存在. </div <div class="pbox" 3: 4: 5: 6: 对于 ,容易验证删掉任何一个开集都不构成拓扑....
Math Analysis Homework - Sem 2 Week 1
Math Analysis Homework Sem 2 Week 1 Class 1 T1 <div class="cbox" 2. 讨论下列函数序列在指定区间上的一致收敛性: (2) , (i) , (ii) ; </div <div class='pbox' $ f n(x) = \frac{x^n}{1+x^n} x \in (0, 1) x \i...
Math Analysis Homework - Week 14
Math Analysis Homework Week 14 Class 1 T1 <div class="cbox" 2. 讨论下列无穷乘积的敛散性: (2) </div <div class="pbox" 敛散性等价于 $ \prod {n=2}^{\infty} \left(\frac{n^2 1}{n^2 + 1}\right)^p \ (p \i...
Math Analysis Homework - Week 13
Math Analysis Homework Week 13 Class 1 T1 <div class="cbox" 1. 判断下列级数的敛散性, 绝对收敛还是条件收敛? (2) ; </div <div class="pbox" 递减,莱布尼茨判别法知收敛 $ \sum {n=1}^{\infty} \sin(\pi \sqrt{n^2+1}) \su...
Math Analysis Homework - Week 12
Math Analysis Homework Week 12 Class 1 T1 <div class="cbox" $ p\in (0,4) p\in ( \infty,0]\cup [4,+\infty) f(x) [a, b] \lim {x\to+\infty}f(x)=A, \lim {x\to \infty}f(x)=B a 0 \int {...
Math Analysis Homework - Week 11
Math Analysis Homework Week 11 Class 1 T1 计算曲线围成的面积 <div class='cbox' $ P(1,0) y=\sqrt{x 2} x x y y=\dfrac{1}{2} x \dfrac{1}{2} \iff x=2y+1 r=a(1+\cos\theta) \left(0 \leqslant \th...
Math Analysis Homework - Week 10
Math Analysis Homework Week 10 Class 1 T1 <div class='cbox' $ a<1 a=1 a\ge 1 \vert f(x) \vert <\sqrt 2+1 $ </div
Math Analysis Homework - Week 9
Math Analysis Homework Week 9 Class 1 T1 <div class='cbox' $ f(x 0)\ne 0 f(x 0) 0 (x 0 \delta,x 0+\delta) \forall x\in (x 0 \delta,x 0+\delta),f(x) \dfrac{f(x 0)}2 0 \int x^kf(x)...
Math Analysis Homework - Week 8
Math Analysis Homework Week 8 Class 1 T1 <div class='cbox' $ $ </div
Math Analysis Homework - Week 7
Math Analysis Homework Week 7 Class 1 T1 <div class='cbox' $ a,b 0 4 5 o(x^5) $ </div
Math Analysis Homework - Week 6
Math Analysis Homework Week 6 Class 1 T1 <div class='cbox' analysis the function's convexity and inflection point $ ab=0 a,b\ne 0 a $ </div
Math Analysis Homework - Week 5
Math Analysis Homework Week 5 Class 1 T1 <div class='cbox' $ f(b) f(a) \vert k \vert f'(x 1),f'(x 2) a=0 f(x) 0 (c,d) f(c)=0 f(x) 0,\forall x\in (c,d) g(x)=\ln \vert f(x) \vert \v...
Math Analysis Homework - Week 4
Math Analysis Homework Week 4 Class 1 T1 <div class='cbox' $ $ </div
Math Analysis Homework - Week 3
Math Analysis Homework Week 3 Class 1 T1 <div class='cbox' $ f(0) $ </div
Math Analysis Homework - Week 2
Math Analysis Homework Week 2 Class 4 T1 <div class="cbox" 设 </div <div class='pbox' $ 0 < a 1 < b 1 a {n+1} = \sqrt{a n b n}, b {n+1} = \frac{a n + b n}{2}, n \in \mathbb{N} + \{...
Linea Algebra Homework - Class 1
Linear Algebra Class1 Homework T2 <div class='cbox' Going around a triangle from to to to , what are those three vectors ? What is ? What are their lengths , , and ? The length sq...
Math Analysis Homework - Week 1
Math Analysis Homework Week 1 Class 1 Homework T1 <div class='cbox' </div <div class='pbox' Obviously wrong. $ a n=2^{ n} A 0 \epsilon<A n N a n 0 a n 0 ,\epsilon 1<1,\epsilon 2<A...