Derivative Meaning

The derivative of a function can be thought of as:

(1) Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.

(2) Symbolic: The derivative of
$Latex x^{n} = nx^{n-1} $
the derivative of sin(x) is cos(x),
the derivative of f°g is f'°g*g',
etc.

(3) Logical:
$Latex \boxed{\text{f'(x) = d}} $
$Latex \Updownarrow $
$latex \forall \varepsilon, \exists \delta, \text{ such that }$
$latex \boxed{
0 < |\Delta x| < \delta,
\implies
\Bigr|\frac{f(x+\Delta x)-f(x)}{\Delta x} - d \Bigr| < \varepsilon
}$

(4) Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.

(5) Rate: the instantaneous speed of f(t), when t is time.

(6) Approximation: The derivative of a function is the best linear approximation to the function near a point.

(7) Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.

(8) The derivative of a real-valued function f in a domain D is the Lagrangian section of the cotangent bundle T*(D) that gives the connection form for the unique flat connection on the trivial R-bundle ßxR for which the graph of f is parallel.

[Source]: Extract from "On Proof and Progess in Mathematics" by William Thurston.

Math Chants

Math Chants make learning Math formulas or Math properties fun and easy for memory . Some of them we learned in secondary school stay in the brain for whole life, even after leaving schools for decades.

Math chant is particularly easy in Chinese language because of its single syllable sound with 4 musical tones (like do-rei-mi-fa) - which may explain why Chinese students are good in Math, as shown in the International Math Olympiad championships frequently won by China and Singapore school students.

1. A crude example is the quadratic formula which people may remember as a little chant:
"ex equals minus bee plus or minus the square root of bee squared minus four ay see all over two ay."

$latex \boxed{
x = \frac{-b \pm \sqrt{b^{2}-4ac}}
{2a}
}$

2. $latex \mathbb{NZQRC}$
Nine Zulu Queens Rule China

3. $latex \boxed {\cos 3A = 4\cos^{3} A - 3\cos A }$

cos = $ 1 (kö, Singapore Chinese Fujian福建話 / Taiwan 台湾闽南语dialect) [一元]

cos 3= $1.3
4cos^3 = $4.3
3cos = $3

$1.3 = $4.3 - $3

4. $latex \boxed{\pi = 3.14159\dots}$
Chant in Chinese Mandarin (Beijing):
山顶一寺一壶酒
Chinese sound means: On the mountain (3) top (.) there is one (1) temple (4) with a (1) bottle (5) of wine (9)

Note : Below is the 22 decimal memory chant:
山顶一寺一壶酒，尔乐, 我三壶把酒吃，酒杀尔，杀不死，乐而乐。

$latex \boxed{
\pi = 3.14159 \:26 \:535897 \:932 \:384 \:626 \dots
}$


5. Group Properties:
$latex \boxed{\text{CAN I ?} }$
C: Closure
A: Associative
N: Neutral (e)
I: Inverse

If only 50%, it is Semi-Group(C, A) 半群
If No Inverse, it is MoNoId (C,A,N) 幺群
(MoNoId : No I)

6. The eccentric choice of letters (a,h,g,b,f,c, skipping iand e) for the coefficients in Conic equation: $latex \boxed
{ ax^{2} + 2hxy + 2gx + by^{2}+ 2fy + c = 0
}$
Math Chant for the coefficients:
{a h g b f}
"all hairy guys big feet"

c: constant term (understood)

Proof & Progress in Math

"On Proof and Progress in Mathematics" by the late William Thurston

Great article in Mathematics Education, enjoy reading !

http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf

成语数学

中国的小学离校考试 (PSLE) 「神题」: 猜成语
1) 20 除 3
2）1 除100
3）9寸+1寸=1尺
4）12345609
5）1,3,5,7,9

答案：:
1) 20/3= 6.666 六六大顺
2）百中挑一
3）得寸進尺
4）七零八落
5）举世无双

Sequence Limit

Definition: $latex \text{Sequence } (a_n) $
has limit a

$latex \boxed{\forall \varepsilon >0, \exists N, \forall n \geq N \text { such that } |(a_n) -a| < \varepsilon}$

$latex \Updownarrow $

$latex \displaystyle \boxed{ \lim_{n\to\infty} (a_n) = a }$

What if we reverse the order of the definition like this:

∃ N such that ∀ε > 0, ∀n ≥ N,
$latex |(a_n) -a| < \varepsilon$

This means:

$latex \boxed {\forall n \geq N, (a_n) = a }$

Example:

$latex \displaystyle (a_n) = \frac{3n^{2} + 2n +1}{n^{2}-n-3}$

$latex \displaystyle\text{Prove: } (a_n) \text { convergent? If so, what is the limit ?}$

Proof:
$latex \displaystyle (a_n) = 3 + \frac{5n +10}{n^{2}-n-3}$

$latex n \to \infty, (a_n) \to 3$

Let's prove it.

$latex \text {Let } \varepsilon >0$
$latex \text{Choose N such that } \forall n \geq N, $
$latex \displaystyle |(a_n) -3| = \Bigr|\frac{5n +10}{n^{2}-n-3}\Bigr| < \varepsilon$

$latex \text{Simplify: } \displaystyle \Bigr|\frac{5n +10}{n^{2}-n-3}\Bigr|$
$latex \text{Let } n > 10 $
$latex \displaystyle \Bigr|\frac{5n +10}{n^{2}-n-3}\Bigr| < \frac{6n}{\frac{1}{2}n^{2}}= \frac{12}{n} < \varepsilon$

$Latex \text{Choose } N = \max (10, \frac{12}{\varepsilon})$
$Latex \displaystyle\forall n \geq N,
|(a_n) -3 | < \frac{12}{n} < \varepsilon$

Therefore,
$latex \displaystyle \boxed{ \lim_{n\to\infty} (a_n) = 3 }$ [QED]

[Source]: Excellent Introduction in Modern Math:
"A Concise Introduction to Pure Mathematics (3rd Edition)"
by Martin Liebeck
CRC Press @ 2011

Abel Prize 2013

Algebraic Geometry

Belgian mathematician Pierre Deligne is a 'perfect' mathematician: he won all the coveted 'Oscar' Math Prizes:
Fields Medal, Wolf Prize and in 2013 $1m Abel Prize.

http://www.nature.com/news/mathematician-wins-award-for-shaping-algebra-1.12644

Turn Sphere Inside Out

Watch this amazing video, mathematically you can turn a sphere inside out, but not a circle:

http://www.snotr.com/video/3107/How_to_turn_a_sphere_inside_out