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...
# math-analysis
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...
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,...
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 \...
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...
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)...
Your Classmates' Chaos Power
Your Classmates' Chaos Power Period 3 Means Period K 首先是一些准备工作: <div class='dbox' $ I=[a,b] f(x) x f(x) f(x) I 1=[f(a),f(b)] c c=\sup \{x|f(x)=f(a)\} f(c)=f(a) d=\inf \{x|f(x)=f(b...
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...
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 {...
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 (Class Note 3)
Math Analysis (Class Note 3) 20260302 讲了一些函数项级数和一致收敛 <div class='cbox' 证明 上 一致收敛. </div <div class='pbox' 极限是 法1:Dini引理,单调性+连续秒了. 法2: 老师写的是你先求导,发现: $ 0=((1+\dfrac xn)^n e^x)'=(1+\...
Math Analysis Huashi Xia
Math Analysis Huashi Xia 只挑了几点 函数项级数 不能乱交换 <div class='cbox' 构造:求极限不保证连续性 </div <div class='pbox' </div <div class='cbox' 求导和求极限不交换 </div <div class='pbox' 第一种是我们直接让导函数发散: $ d(x)...
I am a pig
I am a pig <div class='cbox' $ f(n)=r n,f'(n)=r {n+1} r {n}= a n f n 0 a n r n f(n)=a n,f'(n)=a {n+1} a n,f''(n)=f'(n+1) f'(n) f f f \int \dfrac{f'}{f}=\ln f x n=\ln(r {n+1}) \ln(...
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 (Class Note) 2
Math Analysis (Class Note) 2 Class 24 定积分的应用 <div class='cbox' 参数方程的面积 围成的面积 满足 $ C^1 \sqrt{(x a)(b x)} \dfrac{dx}{y}=d\theta 2\pi \dfrac{\vert \sin x \vert }{x} X x X \implies \d...
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
A Fun Problem about Function Limits
A Fun Problem about Function Limits <div class='cbox' $ \epsilon x n \lim {n \to \infty} x n=+\infty f(x n) \epsilon \delta n,\forall x\in (x i \delta i,x i+\delta i) \vert f(x) f...
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
Right Continuous Function
Can you find a function such that right continuous at every point but not left continuous on a point $ q 1,q 2\ldots 2^{ i} x \epsilon \forall \delta,\exists y\in [x \delta,x],\ve...
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} + \{...
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...
Math Analysis(Class Note)
Math analysis Class 1 Some Inequality and Def of limit Triangle Inequality <div class='cbox' $ x i=\dfrac{1}{x i} 2 2 n X=\sqrt[n]{\prod i x i} N=N(\epsilon) N \epsilon (0,a),a 0...