金庸武术的道理 和学数学一样：
1）华山正邪二派 ： 气宗 (正) vs 剑宗 (邪)
数学：数学理论 vs 刷题技巧
2）虚竹 ：忘掉以前的少林功夫才能学 消遥派
数学： 忘掉A-level 前的思维 (concrete) ，才能学好大学抽象(Abstract ) 数学.
3) 少林寺僧好高鹜远：还没学精本派“一阳指”，就想去换 印度鸠摩罗的功夫。
数学： 先打好 大学基础数学 (Epsilon-Delta, Abstract Algebraic Structures) ，才去学其他的高深东西 (Category, Differential Geometry, Algebraic Topology,… )
金庸的武林江湖 宗师 vs 数学江湖
东邪(孤僻冷漠) ：黄药师 vs 德国. Gauss
西毒 (妒忌，狠毒) ：欧阳锋 vs 法国. Cauchy (迫害 年轻天才 Abel, Galois )
南僧（避世隐士）: 大理国王 一灯大师 vs 法国.Fermat
北丐 （流浪天涯）：洪七公 vs 匈牙利. Paul Erdos
理查德·费曼非常聪明的求导 differentiate 技巧
Feymann (Nobel Physicist) has many funny speedy Math tricks for Calculus eg. Differentiate an Integral (Applied Fundamental Theorem of Calculus) , and this one below.
This is a free video lecture series of the Princeton Math Textbook 《Calculus Lifesaver》(link Amazon.com) for university non-math majors.
He explains the “funny” but smart Calculus basics : First & 2nd Fundamental Theorems of Calculus (by Newton’s co-inventor Leibniz), some how never taught in the 1970s GCE A level (now also ? ) bcos not Newton’s Anglo-saxxon invention. Neither the French textbooks teach so (bcos Leibniz was German ?). They only appear in pure American/German Calculus books.
“Funny” bcos by “1st Theorem” you could ‘d/dx’ any intergral to get the anti-derivative . Richard Feymann self-learnt this trick in High school to solve complex integrations.
Also why the “Coset” (+ C) is explained by the 2nd Theorem in this video.
群论的原本是Galois 发明，没有现在这么抽象的Axiom-based, 是100年后 Noether 在 WW2 归纳的抽象化。Galois 只发明 左/右陪集 Left /Right Coset (l’ensemble à gauche / à droite)，当 左陪集＝右陪集，就是Normal subgroup (正规子群，l’invariant)， 破解300年的数学难题：5次方程以上(Quintic equations & above ) 无根式 (radical root) 解, 从而诞生Group Theory， 开辟抽象代数Abstract Algebra / New Math.
这位中国年轻老师 教得很棒, 证明严谨rigorous, 很好的方法：eg. Test “well-defined”, necessary & sufficient condition (Set Prove Technique : ⊂ left inclusion, ⊃ right inclusion, then equal =), Bijection 双射 (Surjective 满射 trivial + Injective proof).
他的习题答案：
｜Z : nZ｜Index is easy ?
意思：
把 Z 用coset 分类 (partitioned) 为
0Z，1Z, 2Z,… (n-1)Z 个陪集,
Total = n个
｜Z : nZ｜＝ n
Note: Coset in Calculus ( Indefinite Integration) :
Integral Solution “+ C” 就是 Right Coset ＝ solution的右陪集：意思所有 “+C” 都是solution Set 的 右陪伴。
Good education video for students on Maths applied in Big Data.
莱布尼茨微积分——
Leibniz’s original proof : Integration by Parts (分部积分公式”)
【函数概念并不难，理解“函”字是关键——函数概念如何理解】
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清. 李善兰 翻译 Function 为函数。函，信也。只能有一个收信人，所以 只有一个 f(x) 值。
…
The unique 1 single output of a function becomes very important for subsequent development in Math & IT:
functions are composable, associative, identify function，etc (distributive,… ) => it can be treated like vector => structure of a Vector Space “Vect”
Extended to..
“Vect” is a bigger structure “Category” in which “function of functions” is a
“Functor” （函子）F：F(f)
Example : F(f) = fmap (in Haskell)
fmap (+1) {2,7,6,3}
=> {3,8,7,4}
here F = fmap, f = +1
The Math branch in the study of functions is called “functional” 泛函。
IT : Functional Programming in Lisp, Haskell, Scala, ensure safety of guaranteed output by math function property. Any unexpected exception (side effects: IO, errors) is handled by a special function called “Monad” (endo-Functor).
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This American Math educator proposes teaching Primary/Sec math in new ways :
Mathematician Cédric Villani (Fields Medal) is a MP in President Macron’s young party “En Marche”.
He introduced “Singapore Modeling Math” to French primary schools Arithmetic in 2017.
吴文俊，陈省身在 WW2昆明西南联大的弟子， 留法Strasbourg University, 兼任 博士生的tutor， 教导 法国未来的 大师Grothendieck (两人都是新数Bourbaki 学派 最后一批会员) 。
吴文俊在1975 文革后才研究 中国古代数学，从中得灵感，发明 电脑 机器化 AI 证明axiom-based 几何定理。得 1st batch Run- Run Shaw (东方Nobel Prize) $1 m Prize.
数学家清心寡慾不爱斗争，一心专注 “数学之美” ，其他生活繁琐的事(文革 批斗) 都看得开。所以多长寿 (Newton, 陈省身， Hadamard, 杨振宁, 也都90＋) ，吴文俊也高寿活到98岁。
【「给我进来做题啦!」法国高考数学题难度如何？居然只有四道题？】
Baccalaureate Scientifique Math 2019 (Analyse).
Yet to see the Baccalaureate (S) Paper 1 “Algèbre” which is tougher on Abstract Algebra (Group, Ring, Field, Vector Space / Linear Algebra,…)
The integration is meaningless if
$latex displaystyle {int x^{x} dx}&fg=00aa00&s=3$
unless specifying the domain of defintion, eg. interval [0,1] :
$latex displaystyle {int_{0}^{1} x^{x} dx = frac {pi^{2}}{12} }&fg=aa0000&s=3$
Note: Gamma function proof by Feynman trick : differentiate under integral.
DEMO:
$latex boxed{a = a^{a^{a^ {{…}^{a}}}}}&fg=aa0000&s=3$ $latex text {…. [A] } $
$latex text {Let } a = sqrt {2} $
Method…: Wrong !!!
Method 1 : Rigorous!
1) Prove [A] Convergent ?
2) If converge, find Limit ?
Method 2: by RMI Homorphism
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Mathtuition88 