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Mount Usu / Sarobetsu postmined peatland
From left: Crater basin in 1986 and 2006. Cottongrass / Daylily
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Gr. μαθημα. Engl. Mathematics
HistoryThree Greek geometric problems (ギリシア幾何学三大問題)a set of geometric problems whose solution was sought using only compass and straightedge:
Fibonacci, Leonardo 1170?1250? Descartes René 15961650: founder of analytic geometry TartagliaCardano: solution of third degree equation or cubic equation Ferrari L 15221565: solution of biquadratic equation Barrow Isaac 16301677: reverse tangent (逆接線) → find out the original curve by detecting the characteristics of tangent = the beginning of infinitesimal calculus Newton: differentiation (called fluxion at that time) and integrationLeibniz, Gottfried Wilhelm 16461716, Germany: competition to the priority of differential and integral calculus Alembert (d'), Jean Le Rond 17171783, France, mathematics, physics and enlightenment:edited Encyclopedie universelle des arts et sciences (百科全書) with Diderot D (17131784) Laplace, Pierre Simon, Marquis de 17491827, France, mathematics, astronomy and physics (probability, equation and geodesy)left his name Laplace transformation or transform Gauss, Karl Friedrich 17771855, Germany: algebra, geometry, analysis
left his name gauss (G) on the electromagnetic unit (EMU), gauss proposed the analytical theory of heat or thermodynamics (contributed the development of mathematical physics) 
[ calculus ] Probability theory and static electromagnetics Poncelet, Jean Victor 17881867, France
Study on geometry when he was in jail because he became captive during joining an army for the conquest to Russia by Napoleon I
Lobachevskian geometry → hyperbolic geometry Researches on the bacground of theory proposed by Abel → establishing group theory (群論) Riemann, Georg Friedrich Bolyai 18261866, GermanyNonEuclidean axiom: no pararell lines exist Poincaré, Jules Henri 18541912, France: mathematics, physics and philosophy
Cousin of The president Poincaré R (18601934)
Prime number theorem (PNT) 
Def. Algebraic function (代数関数): a function defined as the root of a polynomial equation
= 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)function that is not algebraicEx. trigonometric function (f), exponent f, logarithmic f Elementary f = algebraic f + elementary transcendental fExponent function (指数関数)Law. exponential law (指数法則): b > 0, a^{x} > 0 ⇒
a^{x}a^{y} = a^{x+y} → a^{x}/a^{y} = a^{xy} Logarithmic function (対数関数)Def. Power function (冪関数)Ex. lim_{x→∞}(1 + a/x)^{x} = e^{a}Pr. a/x = 1/y → (1 + a/x)^{x} = (1+ 1/y)^{ay} = ((1 + 1/y)^{y})^{a} continuous
x → +∞, a > 0 → y → +∞, a < 0 → y → –∞ Def. Hyperbolic function (双曲線関数)x ∈ ℜ (∞ < x < ∞) ⇒coshx = (e^{x} + e^{x})/2 (hyperbolic cosine 双曲線余弦) sinhx = (e^{x}  e^{x})/2 (hyperbolic sine 双曲線正弦) ⇒ graph: catenary (懸垂線) tanhx = sinhx/coshxcothx = coshx/sinhx sechx = 1/coshx coshx (or cosechx) = 1/sinhx ⇒ inverse hyperbolic function (逆双曲線関数) Prop. (1) cosh^{2}x  sinh^{2}x = 1
(2) tanhx = coshx/sinhx
sinh(x ± y) = sinhxcoshy ± coshxsinhy (4) lim_{x→∞}tanhx = 1, and lim_{x→∞}tanhx = 1 Pr._(1) cosh^{2}x  sinh^{2}x = (coshx + sinhx)(coshx  sinhx) = e^{x}·e^{x} = 1 //
(2) trivial //
= (e^{x} + e^{x})/2·(e^{y} + e^{y})/2 + (e^{x}  e^{x})/2·(e^{y}  e^{y})/2
(4) lim_{x→∞}tanhx = lim_{x→∞}((1  e^{2x})/(1 + e^{2x})) = 1 Def. Trigonometric function (三角関数)sine (正弦): sinθ = y/rcosine (余弦): cosθ = x/r tangent (正接): tanθ = x/y = sinθ/cosθ cotangent (余接): cotθ = x/y, = 1/tanθ secant (正割): secθ = r/x, = 1/cosθ cosecant (余割): cosecθ = y/x = 1/sinθ tanθ = sinθ/cosθ, cotθ = cosθ/sinθ Cycle (周期): sinθ, cosθ, secθ, cosecθ → 2π, tanθ, cotθ → π → sin(π/2 – θ) = cosθ, cos(π/2 – θ) = sinθ, sinnπ = 0 lim_{θ→0}(sinθ/θ) = 1, θ ∠ 1 → sinθ ≈ θ sin(–θ) = –sinθ, cos(–θ) = cosθ, tan(–θ) = –tanθsin^{2}θ + cos^{2}θ = 1, 1 + tan^{2}θ = sec^{2}θ, 1 + cot^{2}θ = cosec^{2}θ Th. lim(sinx/x) = 1 Ex. lim(tanx/x) = lim(sinx/x)·(1/cosx) = 1 Ex. limsin(sinx)/x = lim(sin(sinx)/sinx)·(sinx/x) = 1·1 = 1 (∵ x → 0 ⇒ sinx → 0) 
Equations of trigonometric function
ΔABC, AB = a, BC = b, CA = c, ∠A, ∠B, ∠C ⇒ Th. sine theorem (正弦定理): a/sinA = b/sinB = c/sinC= 2R (R: radius of circumcircle) Th. cosine theorem (余弦定理)
a^{2} = b^{2} + c^{2} + bccosA
sin(x ± y) = sinxcosy ± cosxsiny
sin2x = 2sinxcosx
sin3x =3sinx – 4sin3x
sin^{2}x/2 = (1 – cosx/2)
sinαcosβ = 1/2{sin(α + β) + sin(α – β)}
sinα + sinβ = 2sin((α + β)/2)cos((α – β)/2) Pr. ^{∀}ε > 0, ^{∃}δ > 0, x – a < δ → sinx – sina < ε
sinx – sina = 2sin((x – a)/2)·cos((x + a)/2) < x – a < δ (= ε) asinθ + bcosθ = √(a^{2} + b^{2})·sin(θ + α) Here, α is obtained by: cosα = a/√(a^{2} + b^{2}), sinα = b/√(a^{2} + b^{2}) Pr. asinθ + bcosθ = √(a^{2} + b^{2})·((a/√(a^{2} + b^{2}))·sinθ + (b/√(a^{2} + b^{2}))·cosθ)= √(a^{2} + b^{2})·(cosαsinθ + sinαcosθ) = √(a^{2} + b^{2})·sin(θ + α) // _____________________from addition theorem Ax. t := tan(x/2) ⇒cosx = (1  t^{2})/(1 + t^{2}), sinx = 2t/(1 + t^{2}), dx = 2/(1 + t^{2})·dt Pr. cos^{2}(x/2) = 1/(1 + tan^{2}(x/2)) = 1/(1 + t^{2})
From doubleangle formula, = 2·1/(1 + t^{2})·t = 2t/(1 + t^{2})
t = tan(x/2), differentiate both sides Def. Inverse trigonometric function (逆三角関数)= cyclometric functionsin^{1}x = arcsinx: x = siny (1 ≤ x ≤ +1) cos^{1}x = arccosx: x = cosy (1 ≤ x ≤ +1) tan^{1}x = arctanx: x = tany (^{∀}ℜ) cot^{1}x[∞, +∞] = arccotx: x = coty (^{∀}ℜ) sec^{1}x = arcsecx: x = secy (x ≤ 1 or 1 ≤ x) csc^{1}x = arccscx: x = cscy (x ≤ 1 or 1 ≤ x) Q. Solve (1) sin^{1}(1/2), (2) tan^{1}(√3) and (3) sin^{1}(sin(3/5·π)) (π/2 ≤ θ ≤ π/2) A._(1) θ := sin^{1}(1/2), sinθ = 1/2 ∴ θ = π/6 ∴ sin^{1}(1/2) = π/6__ (2) θ := tan^{1}(√3), tanθ = √3 ∴ θ = π/3 ∴ tan^{1}(√3) = π/3 __ (3) θ := sin^{1}(sin(3/5·π)), sinθ = sin(3/5·π) ∴ θ = 2/5·π ∴ sin^{1}(sin(3/5·π)) = θ = 2/5·π 
1874 Cantor Georg (18451918): established set theory 
Law. Basic formulae (基本公式)Idempotent law (冪等法則)A∪A = A, A∩A = A Commutative law 交換法則A∪B = B∪A, A∩B = B∩A Distributive law (分配法則)
A∪(B∪C) = (A∪B)∪C
A∪(A∩B) = A (A^{C})^{C} = A De Morgan's law (ド・モルガンの法則)
(A∪B)^{C} = A^{C}∩B^{C} 
Def. Gauss notation (ガウス記号), [•]: x ⊂ ℜ, a ⊂ Z, a < x
⇒ max(a) ≡ [x] Ex. [2] = 2, [π] = 3, [1  √5] = 2Ax. ^{∀}x ⇒ x  1 < [x] or [x] ≤ x < [x] + 1 Ex. [2 + √5] + [2  √5] = 4 + (1) 
Fermat's Last Theorem (フェルマーの最終定理) 1670= Fermat's conjectureTh. No three positive integers a, b and c satisfy the equation a^{n} + b^{n} = c^{n} for any integer value of n greater than 2Prior to prove the theory: n = 1: a^{1} + b^{1} = c^{1} ⇒ a + b = c ⇒ infinite solution n = 2: a^{2} + b^{2} = c^{2} (≡ the Pythagorean theorem) ⇒ infinite solution Pr. Case. n = 3 (proven by Euler 1753) Case. n = 4 (proven by Fermat ≈1630) Case. n = 5 (proven by Dirichlet & Legendre 1825) Case. n = 7 (proven by Lamé 1839)

Concerned with shape, size, relative position of figures, and the properties of space Euclidean geometry (ユークリッド幾何学): the study of points, lines, shapes, figures and spaces based on the Euclidean five axioms
Plane geometry (平面幾何学) = twodimensional Euclidean geometry 
Differential geometry (微分幾何学): using calculus and linear algebra to study problems in geometry Topology (位相幾何学, トポロジー): dealing with the properties of geometry that are unchanged by continuous function Algebraic geometry (代数幾何学): studies geometry through the use of multivariate polynomials and other algebraic techniques 
Euclid's five axioms (5公理)
Euclid's five postulates (5公準)
Trochoid (トロコイド)a roulette formed by a circle rolling along a line (s.l.), including curve and circle
x = aθ  bsinθ
L: a line (the circle center C moves parallel to L) 
Eepitrochoid (外トロコイド)/Hypotrochoid (内トロコイド)a roulette traced by a point attached to a circle of radius r rolling around the outside/inside of a fixed circle of radius R
Cycloid (サイクロイド)= common trochoid
If the cycloid has a cusp (尖点) at the origin and its humps are oriented upward, its parametric equation is: Generation of the involute of the cycloid unwrapping a tense wire placed on half cycloid arc shown in red Astroid (アステロイド, 星芒形) = tetracuspid (四尖点形), cubocycloid, paracycle: a hypocycloid with four cusps If the radius of the fixed circle is a then the equation is given by:
x^{2/3} + y^{2/3} = a^{2/3} ⊂ superellipse
polar coordinates: r = a(1 + cosθ) S = 3/2·πa^{2}, L = 8a NonEuclidean geometry (非ユークリッド幾何学)Hyperbolic_______Euclidean_______Elliptic 
[ combination and permutation ]
a scalar that is computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix (linear albegra)
Expression: det(A), det A, or A 
Two dimensions (二次元)1) Orthogonal coordinates (直交座標)2) Oblique coordinate (斜交座標) 3) Polar coordinate (極座標) Terms: coordinates 座標 axis (pl. axes) 軸 horizontal plane 水平面 horizontal axis 水平軸 hypotenuse 斜辺 normal line 法線 (adj. normal 法線、法線の) orthogonal 直角な parabola 放物線(放物線の) 