Euler’s formula with introductory group theory

During the 19th century French Revolution, a young French boy Evariste Galois self-studied Math and invented a totally strange math called “Group Theory“, in his own saying –

 “A new Math not on calculation but on reasoning“. 

With Group, Galois proved a 300-year-old problem that Quintic equations with degree 5 (or more) have no radical solution (ie formula using +, -,  ×,  ÷,  nth root). During his short tragic life (21 years) his work was not understood by the world masters like Cauchy, Fourier, Poisson, Gauss, Jacobi…

“Group Theory” is the foundation of Modern Math today.

\boxed {e^{i \pi} = -1}  

Why call “Affine” Geometry ?

The term “Affine” was coined by Euler  (1748b), motivated by the idea that images related by affine transformation  have an affinity with one another.

Note 1: The term “Affine Geometry” is never used in GCE A-level Math, but commonly taught in French Baccalaureate.

Note 2:  “Affinity” 亲和力 => 模仿
Affine transformation => 仿射 变化

In geometry, an affine transformationaffine map[1] or an affinity (from the Latin, affinis, “connected with”) is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

Examples of affine transformations include translationscalinghomothetysimilarity transformationreflectionrotationshear mapping, and compositions of them in any combination and sequence.

Differentiating under integral

Prove: (Euler Gamma Γ Function)
\displaystyle n! = \int_{0}^{\infty}{x^{n}.e^{-x}dx}

Proof:
∀ a>0
Integrate by parts:

\displaystyle\int_{0}^{\infty}{e^{-ax}dx}=-\frac{1}{a}e^{-ax}\Bigr|_{0}^{\infty}=\frac{1}{a}

∀ a>0
\displaystyle\int_{0}^{\infty}{e^{-ax}dx}=\frac{1}{a} …[1]

Feynman trick: differentiating under integral => d/da left side of [1]

\displaystyle\frac{d}{da}\displaystyle\int_{0}^{\infty}e^{-ax}dx= \int_{0}^{\infty}\frac{d}{da}(e^{-ax})dx=\int_{0}^{\infty} -xe^{-ax}dx

Differentiate the right side of [1]:
\displaystyle\frac{d}{da}(\frac{1}{a}) = -\frac{1}{a^2}
=>
a^{-2}=\int_{0}^{\infty}xe^{-ax}dx

Continue to differentiate with respect to ‘a’:
-2a^{-3} =\int_{0}^{\infty}-x^{2}e^{-ax}dx
2a^{-3} =\int_{0}^{\infty}x^{2}e^{-ax}dx
\frac{d}{da} \text{ both sides}
2.3a^{-4} =\int_{0}^{\infty}x^{3}e^{-ax}dx


2.3.4\dots n.a^{-(n+1)} =\int_{0}^{\infty}x^{n}e^{-ax}dx
Set a = 1
\boxed{n!=\int_{0}^{\infty}x^{n}e^{-x}dx} [QED]

Another Example using “Feynman Integration”:

\displaystyle \text{Evaluate }\int_{0}^{1}\frac{x^{2}-1}{ln x} dx

\displaystyle \text{Let I(b)} = \int_{0}^{1}\frac{x^{b}-1}{ln x} dx ; for b > -1

\displaystyle \text{I'(b)} = \frac{d}{db}\int_{0}^{1}\frac{x^{b}-1}{ln x} dx = \int_{0}^{1}\frac{d}{db}(\frac{x^{b}-1}{ln x}) dx

x^{b} = e^{ln x^{b}} = e^{b.ln x}

\frac{d}{db}(x^{b}) = \frac{d}{db}e^{b.ln x}=e^{b.ln x}.{ln x}= e^{ln x^{b}}.{ln x}=x^{b}.{ln x}

\text{I'(b)}=\int_{0}^{1} x^{b} dx=\frac{x^{b+1}}{b+1}\Bigr|_{0}^{1} = \frac{1}{b+1}
=>
\text {I(b)} = ln (b+1) + C

Let b=0
I(0) = 0= ln (1) + C = 0+C => C=0

\boxed{I(b)=ln(b+1)}

Let b= 2
\displaystyle\int_{0}^{1}\frac{x^{2}-1}{ln x} dx = I(2) = ln (3) [QED]

The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math

Euler on Math Education

Euler was invited by Peter I of Russia in 1727 to work in the
Petersburg Academy of Sciences. He introduced the fundamental math
disciplines in school math education:
1. Arithmetic
2. Geometry
3. Trigonometry
4. Algebra
these 4 are taught as separate and specific subjects, versus 19 duplicated disciplines in Europe.

Euler influenced not only in Russia schools, but in schools worldwide today.

Source: Russian Mathematics Education
Vol 1: History and world significance
Vol 2: Programs and practices
(Publisher: World Scientific)