(History of Natural sciences)

History

1. circle squaring: construct a square equal in area to a circle
2. cube duplication (Delian problem): give the length of an edge of a cube, that a second cube be constructed having double the volume of the first
3. angle trisection: divide an arbitrary angle into three equal angles

"Liber Abaci": introduced Indian-Arabic mathematics, numeration, calculation, etc.

Cardano–Tartaglia formula (= cubic equation): ax³ + bx² + cx + d = 0

invented Gunter rule (prototype of slide rule)

find out the original curve by detecting the characteristics of tangent = the beginning of infinitesimal calculus

Leibniz, Gottfried Wilhelm 1646-1716, Germany: competition to the priority of differential and integral calculus

1/1² + 1/2² + 1/3² + … → convergence

mathematics and physics

1735 solved Basel problem
≈1735 discovered Euler-Maclaurin formula (オイラーの和公式)
1736 The Seven Bridges of Königsberg (ケーニヒスベルクの橋)

Königsberger Brückenproblem
the foundations of graph theory

1748 proved Euler's formula  e^iz = cosz + isinz
1748 published Introductio in analysin infinitorum
1755 published Institutiones calculi differentialis
1760 developed Euler's equations of rigid body dynamics
1761 discovered Euler's totient function

Graph theory (グラフ理論)

V, a set of vertices (nodes or points, 頂点/節点)
E ⊆ {(x, y)}|x, y ∈ V and x ≠ y}, a set of edges (links or lines, 枝/辺), which are unordered pairs of vertices

The number of vertices of odd degree must be either zero or two
If not then there is no Euler Path
And if there are two vertices with odd degree, then they are the starting and ending vertices
⇒ Vertices N, E and S = 3. W = 5 → no Euler Path //

[ calculus ]

quaternion (四元数)
Hamiltonian path or traceable path (ハミルトン路): a graph that visits each vertex exactly once
Hamiltonian cycle or circuit: a cycle that visits each vertex exactly once

icosian game or Hamilton puzzle

edited Encyclopedie universelle des arts et sciences (百科全書) with Diderot, Denis (1713-1784)

left his name Laplace transformation or transform

left his name gauss (G) on the electromagnetic unit (EMU), gauss
Basic theorem of algebra: every non-constant single-variable polynomial with complex coefficients has at least one complex root
1801 Disquisitiones arithmeticae (数論研究)

proposed the analytical theory of heat or thermodynamics (contributed the development of mathematical physics)

Probability theory and static electromagnetics

Study on geometry when he was in jail because he became captive during joining an army for the conquest to Russia by Napoleon I
→ founder of projective geometry
left his name a unit of power Poncelet (= 100 kg·m/sec)

1831: complex number (複素数)

Lobachevskian geometry → hyperbolic geometry
Non-Euclidean axiom: denied Euclidean axiom 5
= for any given point there exists more than one line that can be extended through that point and run parallel to another line of which that point is not part

Research on the bacground of theory proposed by Abel → establishing group theory (群論)

Non-Euclidean axiom: no pararell lines exist

Cousin of The president Poincaré R (1860-1934)
Multidisciplinary research represented by differential equation

Prime number theorem (PNT)
D ≡ prime density = (count prime number < ^∃N)/N)
N → ∞ ⇒ D = 1/lnN

= rational function (有理関数) + irrational function (無理関数)

Transcendental function (超越関数)

Elementary transcendental function (初等超越関数)

Ex. trigonometric function (f), exponent f, logarithmic f

Exponent function (指数関数)

a^xa^y = a^x+y → a^x/a^y = a^x-y
(a^x)^y = a^xy
(ab)^x = a^xb^x → (b/a)^x = b^x/a^x

Logarithmic function (対数関数)

Def. Power function (冪関数)

x → +∞, a > 0 → y → +∞, a < 0 → y → –∞
∴ (1 + 1/y)^y = e //

Def. Hyperbolic function (双曲線関数)

⇒ graph: catenary (懸垂線)

(2) tanhx = coshx/sinhx
(3) cosh(x ± y) = coshxcoshy ± sinhxsinhy, and

sinh(x ± y) = sinhxcoshy ± coshxsinhy
(double-sign corresponds)

(4) limx→∞tanhx = 1, and limx→ -∞tanhx = -1

(2) trivial //
(3) coshxcoshy + sinhxsinhy

= (e^x + e^-x)/2·(e^y + e^-y)/2 + (e^x - e^-x)/2·(e^y - e^-y)/2
= (e^x+y + e^-x+y + e^x-y + e^-x-y)/4 + (e^x+y - e^-x+y - e^x-y + e^-x-y)/4
= (e^x+y + e^-(x+y))/2 = cosh(x + y), and so forth //

(4) limx→∞tanhx = limx→∞((1 - e^-2x)/(1 + e^-2x)) = 1
__ limx→ -∞tanhx = limx→ -∞((e^2x - 1)/(e^2x + 1)) = -1

Def. Trigonometric function (三角関数)

→ sin(π/2 – θ) = cosθ, cos(π/2 – θ) = sinθ, sinnπ = 0

lim_θ→0(sinθ/θ) = 1, |θ| ∠ 1 → sinθ ≈ θ

= 1·1 = 1 (∵ x → 0 ⇒ sinx → 0)

ΔABC, AB = a, BC = b, CA = c, ∠A, ∠B, ∠C ⇒

= 2R (R: radius of circumcircle)

a² = b² + c² + bccosA
b² = a² + c² + accosB
c² = a² + b² + abcosC

sin(x ± y) = sinxcosy ± cosxsiny
cos(x ± y) = cosxcosy –(±) sinxsiny
tan(x ± y) = (tanx ± tany)/(1 –(±) tanxtany)

sin2x = 2sinxcosx
cos2x = cos²x – sin2x = 2cos²x – 1 = 1 – 2sin²x

sin3x =3sinx – 4sin3x
cos3x = 4cos³x – 3cosx

sin²x/2 = (1 – cosx/2)
cos²x/2 = (1 + cosx)/2
tan²x/2 = (1 – cosx)/(1 + cosx)

sinαcosβ = 1/2{sin(α + β) + sin(α – β)}
cosαsinβ = 1/2{sin(α + β) – sin(α – β)}
cosαcosβ = 1/2{cos(α + β) + cos(α – β)}
sinαsinβ = 1/2{cos(α + β) – cos(α – β)}

sinα + sinβ = 2sin((α + β)/2)cos((α – β)/2)
sinα – sinβ = 2cos((α + β)/2)sin((α – β)/2)
cosα + cosβ = 2cos((α + β)/2)cos((α – β)/2)
cosα – cosβ = 2sin((α + β)/2)sin((α – β)/2)

|sinx – sina| = |2sin((x – a)/2)·cos((x + a)/2)| < x – a < δ (= ε)
∵ |cos((x + a)/2)| ≤ 1, |sin((x – a)/2)| < (x – a)/2___ditto cosine //

asinθ + bcosθ = √(a² + b²)·sin(θ + α)

Here, α is obtained by: cosα = a/√(a² + b²), sinα = b/√(a² + b²)

= √(a² + b²)·(cosαsinθ + sinαcosθ) = √(a² + b²)·sin(θ + α) // _____________________from addition theorem

cosx = (1 - t²)/(1 + t²), sinx = 2t/(1 + t²), dx = 2/(1 + t²)·dt

From double-angle formula,
cosx = 2cos²(x/2) - 1 = 2·1/(1 + t²) -1 = (1 - t²)/(1 + t²)
sinx = 2sin(x/2)cos(x/2) = 2cos²(x/2)tan(x/2)

= 2·1/(1 + t²)·t = 2t/(1 + t²)

t = tan(x/2), differentiate both sides
dt = 1/(cos²(x/2))·1/2·dx ∴ dx = 2cos²(x/2)·dt = 2/(1 + t²)·dt

Def. Inverse trigonometric function (逆三角関数)

(-π/2 ≤ θ ≤ π/2)

∴ sin^-1(sin(3/5·π)) = θ = 2/5·π

Law. Basic formulae (基本公式)

A∪A = A, A∩A = A

A∪B = B∪A, A∩B = B∩A

A∪(B∪C) = (A∪B)∪C
A∩(B∩C) = (A∩B)∩C
A∪(B∩C) = (A∪B)∩(A∪C)
A∩(B∪C) = (A∩B)∪(A∪C)

A∪(A∩B) = A
A∩(A∪B) = A

(A^C)^C = A

(A∪B)^C = A^C∩B^C
(A∩B)^C = A^C∪B^C

⇒ max(a) ≡ [x]

Fermat's Last Theorem (フェルマーの最終定理) 1670

aⁿ + bⁿ = cⁿ

Plane geometry (平面幾何学) = two-dimensional Euclidean geometry
Solid geometry (立体幾何学) = three-dimensional Euclidean geometry

Euclid's five axioms (5公理)

1. Things that are equal to the same thing are also equal to one another
2. If equals are added to equals, then the wholes are equal (Addition property of equality)
3. If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality)
4. Things that coincide with one another are equal to one another (Reflexive Property)
5. The whole is greater than the part.

Euclid's five postulates (5公準)

1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Trochoid (トロコイド)

x = aθ - bsinθ
y = a - bcosθ

L: a line (the circle center C moves parallel to L)
P (trochoid) ≡ every other point in the rotating plane attached to the circle traces
a: radius of circle
θ: the variable angle through which the circle rolls

Eepitrochoid (外トロコイド)/Hypotrochoid (内トロコイド)

Cycloid (CP = b)

Cycloid (サイクロイド)

If the cycloid has a cusp (尖点) at the origin and its humps are oriented upward, its parametric equation is:
x = a(t - sint), y = a(1 - cost)

x^2/3 + y^2/3 = a^2/3 ⊂ superellipse
⇒ x = acos³t = a/4·(3cost + cos³t), y = asin³t = a/4·(3sint + sin³t)

polar coordinates: r = a(1 + cosθ)
orthogonal coordinates: (x² + y²)(x² + y² + 2ax) - a²y² = 0
parameters: x = a(1 + cosθ)cosθ, y = a(1 + cosθ)sinθ

S = 3/2·πa², L = 8a

Non-Euclidean geometry (非ユークリッド幾何学)

Expression: det(A), det A, or |A|
det(A) = = a₁₁a₂₂ - a₁₂a₂₁

Simultaneous equation (連立方程式)

D₁² = = 15 - 4 = 11, D₂² = = -7 + 18 = 11
x = D₁²/D = 1, y = D₂²/D = 1

D₁³ = = D₂³ = = D₃³ =

= ab + bc + ca - a² - b² - c²

a) a + b + c ≠ 0, a ≠ b ≠ c →

x = D₁³/D = 1/(a + b + c),
y = D₂³/D = 1/(a + b + c),
z = D₃³/D = 1/(a + b + c)

b) a + b + c = 0 → incompatible (不能)
c) a + b + c ≠ 0, a = b = c → indeterminate (不定)

Two dimensions (二次元)