( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)

rational number p, p, (q ) q ratio 3.14 = 3 + 1 10 + 4 100

( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)

( a) ( b) a > b > 0 a < nb n A A B B A A, B B A = ( A A )A

( 408 355) V 4: 2 a > 0, b > 0 m, n ma > b, nb > a a, b a : b, m, n

5: 4 a, b, c, d, a : b=c : d m, n, ma > nb = mc > nd, ma = nb = mc = nd, ma < nb = mc < nd, 7: a : b > c : d ma > nb mc nd m, n

a > b > 0 nb > a n b = 1 a > 1 n > a, 1 n < 1 a ϵ = 1 a ϵ N n n > N 1 0 ϵ n lim n 1 n = 0

http://www.math.ubc.ca/ cass/euclid/byrne.html Oliver Byrne s edition of Euclid, http://aleph0.clarku.edu/ djoyce/java/elements/elements.ht David B, Joyce ( ), ( ), ( ), ( ) : ; (2011/5/25) ISBN-10: 4320019652 ISBN-13: 978-4320019652

ブルバキの評価 ニコラ ブルバキ 1886 年生まれ モルダビア出身の架空の人物 数学原論 の著者 1960 年代に構造主義という一大旋風を巻き起こした

(1872 ) A x, B y x < y A B (1) A B (2) A B B A. (3) A B

(1) (2) 2/3 A B (2) (3) A B (2)

S α s S < m R α (1) s α (2) α < α α < s S na α α a < α (2) α a < na, α < (n + 1)a. (1)

a 1 < a 2 < a 3 < < a n < < M a n α α M M

f : X f Y : 1 1 x 1 x 2 f(x 1 ) f(x 2 ) : y y = f(x) x X Y X Y Y X

1, 2, 1 2 3 4 5 1 1 2 3 4 5 1 1 1 1 2 1 2 3 4 5 2 2 2 2 3 1 2 3 4 5 3 3 3 3 4 1 2 3 4 4 4 4 4 4 5 2 1:1 : 1, 2,?

G. (1845 1918) 19

( ) ( )ϕ(x) x, x, x :

( 16-20 ) ( ) 18 ( 38 1905 )

( ) : ( ) ( )

40 44 1 43 : : 53 :

: 2500 1990

S n = 1 + r + r 2 + + r n 1 n 1 + r 2 1 + r + r 2 3 1 r 2 = (1 r)(1 + r), 1 r 3 = (1 r)(1 + r + r 2 ),. 1 r n = (1 r)(1 + r + r 2 + + r n 1 ) 1 1 + r + r 2 + + r n 1 = 1 rn 1 r. a + ar + ar 2 + + ar n 1 = a 1 rn 1 r.

r < 1 r n 0 (n ) S n 1 1 r. n S n S n r = 1 1+( 1)+( 1) 2 +( 1) 3 + = (1 1)+(1 1)+ = 0? ( F = ma ) ζ( 1) = 1 + 2 + 3 + + n + = 1 12

0. 9 = 0.9 (1 + ( 1 10 )1 + ( 1 10 )2 + ( 1 10 )3 + ) = 9 1 10(1 10 1 ) = 1. 0.9, 0.99, 0.999, 0.9999, 1 0. 9 = 1

lim x α f(x) = β ϵ δ x α < δ f(x) β < ϵ

: x x x

. :

: lim f(a + h) = f(a), h 0 f(x) x = a, ϵ > 0 δ( h < δ f(a + h) f(a) < ϵ) 2

{ 0 x f(x) = 1 p x = q p, p, q, p > 0,, x ϵ δ (x δ, x + δ). ϵ ϵ p 0 p 1 < ϵ 0 p 0 < ϵ p 0 q 1 p < x < q p x

: f(x) [a, b], f(a) > 0, f(b) < 0 [a, b] f(c) = 0 f(x) > 0 A A A c f(c) > 0 f(c) < 0 f(x) = 0,,

x 3 5x + 1 = 0 x 1 x c x 2 y(x 1 ) y(x c ) y(x 2 ) 0.000000 0.500000 1.000000 1.000000-1.375000-3.000000 0.000000 0.250000 0.500000 1.000000-0.234375-1.375000 0.000000 0.125000 0.250000 1.000000 0.376953-0.234375 0.125000 0.187500 0.250000 0.376953 0.069092-0.234375 0.187500 0.218750 0.250000 0.069092-0.083282-0.234375 0.187500 0.203125 0.218750 0.069092-0.007244-0.083282 0.187500 0.195312 0.203125 0.069092 0.030888-0.007244 0.195312 0.199219 0.203125 0.030888 0.011813-0.007244 0.199219 0.201172 0.203125 0.011813 0.002282-0.007244 0.201172 0.202148 0.203125 0.002282-0.002482-0.007244 0.201172 0.201660 0.202148 0.002282-0.000100-0.002482 0.201172 0.201416 0.201660 0.002282 0.001091-0.000100 0.201416 0.201538 0.201660 0.001091 0.000496-0.000100 0.201538 0.201599 0.201660 0.000496 0.000198-0.000100 0.201599 0.201630 0.201660 0.000198 0.000049-0.000100 0.201630 0.201645 0.201660 0.000049-0.000026-0.000100 0.201630 0.201637 0.201645 0.000049 0.000012-0.000026 0.201637 0.201641 0.201645 0.000012-0.000007-0.000026 0.201637 0.201639 0.201641 0.000012 0.000002-0.000007

, [a, b] [a, b], ( 1, 1) y = tan( π 2x),?

2 ( ) 2

O. O x x x < r x O r

f : X f Y Y O f 1 (O) y = x x < 0 y = 1, x 0 y = 1

X Q π < p < π A A ( π, π) Q A [ π, π] Q

) ( )

A f B Domain A f B Range Codomain A (arrow)f B x y x (maps to ) y x y

A f B g C A f B A h C g C h = g f g f z = g(f(x)). x A, z C

f 1 f = id A, f f 1 = id B id A A x id x f 1 f : log x e x e log x = x, log e x = x.

( ) fs = id B B? A idb A B B A A B rf = id A A f B ida? f B y = f(x) A B

Fluent:, x Fluxion:, ẋ

ライプニッツの業績: ブルバキ 数学史 より

運動学から始まった数学は 厳密なものではなかったが その応用範 囲は広大無辺のものであった その切れ味は現在でもなまっていない しかし

y x y y dx dy x x x 1 2 x = x 2 x 1, y = y 2 y 1 y (x) = lim x 0 y x = lim x 0 y(x + x) y(x) (x + x) x dy = y dx

(f(x)g(x)) = = lim x 0 = lim x 0 = f g + fg lim x 0 f(x + x)g(x + x) f(x)g(x) (x + x) x (f(x) + f)(g(x) + g) f(x)g(x) f x g(x) + x lim f(x) g x 0 x + lim x 0 ( f)( g) x ( f)( g) 2 lim = 0 x 0 x x, f, g 1 ( f)( g) 2 2 1

y = f(g(x)) dy dx = df dg dg dx d(x 2 + 1) 3 dx = d(x2 + 1) 3 d(x 2 + 1) d(x2 + 1) dx = 3(x 2 + 1) 2 2x f x = f g g x x 0 g = 0 f(z) = f (z) z + ϵ z, z 0 ϵ 0

y = f(x) x x = f 1 (y) x y Inverse y = f(x) f(x) dy dx = 1 dx dy x 1 f (x) x

F (x, y) = 0 y = ϕ(x) 1 y = f(x) (a, f(a)) y f(a) = k(x a). k = lim h 0 f(a + h) f(a) (a + h) a = f (a) dy = f (x)dx 2 z = f(x, y) (a, b, f(a, b)) z f(a, b) = k(x a) + l(y b) k = lim h 0 f(a + h, b) f(a, b) (a + h) a l = lim h 0 f(a, b + h) f(a, b) (b + h) b = f x = f y dz = f f xdx + ydy

f f x x 1 f(x, y) = 0 df = f f xdx + ydy = 0. f y 0 (a, b, f(a, b)) y = ϕ(x) ϕ (x) = dy f dx = x. f y

4 1906,, 1959)

θ = l r ( ) θ r l π = =3.141592653589793238462643383279...

sin θ θ cos θ 1 tan θ < tan θ (cos θ, sin θ) θ lim θ 0 sin θ θ 0 < θ < π 2, = 1 sin θ < θ < tan θ cos θ sin θ < 1 θ < 1 sin θ cos θ < sin θ < 1 θ θ 0 cos θ 1, sin θ θ 1

cos(α β) = cos α cos β + sin α sin β y y A α α β C Β β α β x 1 x 1 D A(cos α, sin α), B(cos β, sin β), C(cos(α β), sin(α β)), D(1, 0). AB = CD (cos α cos β) 2 + (sin α sin β) 2 = (cos(α β) 1) 2 + (sin(α β)) 2 cos 2 α 2 cos α cos β + cos 2 β + sin 2 α 2 sin α sin β + sin 2 β = cos 2 (α β) 2 cos(α β) + 1 + sin 2 (α β) 2 2 cos α cos β 2 sin α sin β = 2 2 cos(α β)

(sin x) sin(x + h) sin x = lim h 0 h 2 cos( x+h+x = lim 2 ) sin(x + h x) h 0 h = lim h 0 cos(x + h 2 )sin h 2 h 2 = cos(x) (cos x) cos(x + h) cos x = lim h 0 h 2 sin( x+h+x = lim 2 ) sin(x + h x) h 0 h = lim h 0 sin(x + h 2 )sin h 2 h 2 = sin(x)

y = a x, x = 0 1 y = e x. x = 0 lim h 0 e 0+h e 0 h e h 1 = lim 1 h 0 h h 0, e h = 1 + h, e = (1 + h) 1 h n e = (1 + 1 n )n (e x ) = lim h 0 e x+h e x h = e x lim h 0 e h 1 h = e x. e = =2.718281828459045...

e x f(x) = e x (1 + x) f (x) = e x 1. x 0 f (x) 0 + f(x) 0 f (x) > 0 x > 0 e x > 1 + x > x ex x > 1. x x 2n e x 2n x 2n e x > 1. e x 2n x > 1 x n > ( 1 2n )2n x n x 2n 2n. (e x )2n > ( 1 2n )2n. (x ).

y = e αx α x = 1/α y = x n lim x e x x n = ( )

y = e x x = log y y = log x x = e y d log x dx = dy dx = 1 dx dy = 1 de y dy = 1 e y = 1 x d log x = dx x log x = x 1 dt t

xy x xy dt log(xy) = 1 t = dt 1 t + dt x t t = xu t u x dt y = t + dt = log x + log y u 1 1 log(x 2 ) = log(xx) = log x + log x = 2 log x, log(x n ) = n log x. log(y) = log(x y x ) = log x + log y x, log( y x ) = log y log x, y = 1 log( x 1 ) = log 1 log x = log x.

d log(x) = dx x

e 10 x = 10 y y = log 10 x. log e x = log e 10 y = y log e 10 = log 10 x log e 10. log e x = 2.3 log 10 x log 10 2 = 0.3, log 10 3 = 0.5, log 10 5 = 0.7 ( ) ( )

y = log x x > 0 x < 0 d log x dx = d log( x) d( x) d( x) dx = 1 x ( 1) = 1 x. x d log f dx = d log f df df dx = 1 f df dx. f (x) = f(x) d log f dx.

y = sin x y x y = sin x x π 2 x π 2 y = arcsin x 1

π 2 x π 2, 0 x π, y = sin x x x = arcsin y y = cos x x x = arccos y π 2 < x < π 2, y = tan x x x = arctan y

f f 1 sin sin 1 sin 2 x = (sin x) 2, sin 2 x = 1 (sin x) 2, sin 1 x 1 sin x, arcsin, arccos, arctan US asin, acos, atan tan tg, arctan arctg, atg

y = arctan x 1.5 y 1 0.5 0-10 -5 0 5 10 x -0.5-1 -1.5

y = arcsin x π 2 y π 2, x = sin y d arcsin x dx = dy dx = 1 dx dy = 1 d sin y dy = 1 cos y π 2 y π 2 cos y 0. d arcsin x dx = 1 1 sin 2 y = 1 1 x 2.

y = arccos x 0 y π, d arccos x = dy dx dx = 1 = dx dy 1 d cos y dy x = cos y = 1 sin y 0 y π sin y 0. d arccos x 1 1 = = dx 1 cos 2 y 1 x 2. y = arctan x π 2 < y < π 2, d arctan x dx = cos 2 y = = dy dx = 1 dx dy = 1 d tan y dy 1 1 + tan 2 y = 1 1 + x 2. x = tan y = 1 1 cos 2 y

e ix = cos x + i sin x sin x = eix e ix, cos x = eix + e ix 2i 2

sinh = ex e x 2 cosh = ex + e x 2 tanh = ex e x e x + e x

t (x, y), t (x, y) $> gnuplot gnuplot> set parametric gnuplot> plot cos(t)**3,sin(t)**3 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8 cos(t)**3, sin(t)**3-1 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 x = f(t), y = g(t) dx = f (t)dt, dy = g (t)dt. t

y (n) (x) = (y (n 1) (x)). d n 1 y d n y dx n = d( dx n 1). dx n [f(x)g(x)] (n) = nc k f (n k) (x)g (k) (x). k=0

n = 1 [f(x)g(x)] = f (x)g(x) + f(x)g (x). n = m [f(x)g(x)] (m) = m mc k f (m k) (x)g (k) (x). k=0 m [f(x)g(x)] (m+1) = mc k [f (m k) (x)g (k) (x)] k=0 m = mc k f (m k+1) (x)g (k) (x) + = k=0 m mc k f (m k+1) (x)g (k) (x) + m mc k f (m k) (x)g (k+1) (x) k=0 m+1 k=0 k=1 mc k + m C k 1 = m+1 C k mc k 1 f (m k+1) (x)g (k) (x). [f(x)g(x)] (m+1) = m+1 k=0 mc k f (m+1 k) (x)g (k) (x). n = m + 1

Rolle : f(x) [a, b], (a, b) f(a) = f(b), a < ξ < b, f (ξ) = 0. : [a, b] f(x) f(x) > 0 (a, b) ξ h > 0 f(ξ+h) f(ξ) h 0, ξ h ξ ξ+ h f(ξ) f(ξ h) h 0 h 0,, 0 f (ξ) = 0.

( ): f(x) [a, b], (a, b), a < ξ < b, f f(b) f(a) (ξ) =. b a f(b) f(a) a ξ b F (x) = f(x) { (f(b) f(a)) (b a) } (x a) + f(a) F (a) = F (b) = 0. F ( ξ) = 0

: f(x), g(x) [a, b], (a, b) g (x) 0 a < ξ < b, f(b) f(a) g(b) g(a) = f (ξ) g (ξ) f(b) f(a) F (x) = f(x) f(a) ( )(g(x) g(a)) g(b) g(a) F (x) F (a) = 0, F (b) = 0 F (ξ) = 0 ξ

f(0) = 0, g(0) = 0 0 < ξ < x, f(x) f(0) g(x) g(0) = f (ξ) g (ξ) lim x 0 f(x) g(x) = lim x 0 f (x) g (x). l Hopital (1661-1704), Cauchy (1789-1857).

f(b) = f(a) + f (a) (b a) + f (2) (a) (b a) 2 + 1! 2! + f (n 1) (a) (n 1)! (b a)n 1 + f (n) ( ξ) (b a) n. n!, f (x) f(x) f (2) (x) f (x) f (n) (x) f (n 1) (x),, n, f (n) ( ξ) (b a) n 0 n! f(x) = f(a) + f (a) 1! (x a) + f (2) (a) (x a) 2 + 2! + f (n) (a) (x a) n +. (n)!

F (x) = f(x) {f(a) + f (a) 1! (x a) + f (2) (a) 2! (x a) 2 + + f (n 1) (a) (x a) (n 1)! n 1 }, G(x) = (x a) n F (x) F (x) F (a) = G(x) G(x) G(a) F (a) = F (a) = = F (n 1) (a) = 0 G(a) = G (a) = = G (n 1) (a) = 0, F (x), G (x),, F (n 1), G (n 1) F (x) G(x) = F (x) F (a) G(x) G(a) = F (ξ 1 ) G (ξ 1 ), ξ 1(x a ) F (ξ 1 ) G(ξ 1 ) = F (ξ 1) F (a) G(ξ 1 ) G(a) = F (ξ 2 ) G (ξ 2 ), ξ 2(x a )...

(1821 )

f(x) = e x x = 0 ( ) R n = eθx n! xn. 0 < θ < 1 x n R n 0. e x x n = n!. n=0 n 1 1 1 x = k=0 x k + xn 1 x x < 1 R n = xn 0 x < 1 1 x

dx 1 x = log(1 x) = x 1 2 x2 1 n xn + log(1) = 0

James Gregory, 1671 arctan x x = 0 (1642 1727) Brook Taylor 1715 (arcsin x) 2 1722 Colin MacLaurin 1742 Lagrange 1772. S.A.J. Lhuilier, 1786 100

1 1 x = 1 + x + x2 + + x n + (1 + x) α α(α 1) = 1 + αx + x 2 + 2! α = 2 (1 + x) 2 = 1 + 2x + x 2. e x = 1 + 1 1! x + 1 2! x2 + + 1 n! xn + log(1 x) = x 1 2 x2 1 n xn + ( ) sin x = x 1 3! x3 + 1 5! x5 1 7! x7 + cos x = 1 1 2! x2 + 1 4! x4 1 6! x6 +

b a f (a) 1! (b a), b a 0 1 f (2) (a) 2! (b a) 2 2 f (3) (a) 3! (b a) 3 3

y(x) = arcsin x y = (1 x 2 )(y ) 2 = 1. 1 1 x 2. 2x(y ) 2 + 2(1 x 2 )y y = 0, (1 x 2 )y xy = 0. n (1 x 2 )y (n+2) (2n + 1)xy (n+1) n 2 y (n) = 0 x = 0 y(0) = 0, y (0) = 1. y (2n) (0) = 0, y (2n+1) (0) = 1 2 3 2 (2n 1) 2. y = x + 1 6 x3 + 3 40 x5 + 5 112 x7 + 35 1152 x9 +.

x tan(x) sin(x) x 4 f(x) = tan x = sin x f(0) = 0. cos x f (cos x)(cos x) (sin x)( cos x) 1 (x) = (cos x) 2 = (cos x) 2, f (0) = 1 f d(cos x) 2 d(cos x) 2 d(cos x) (x) = = dx d(cos x) dx = 2(cos x) 3 ( sin x) = 2(sin x)(cos x) 3, f (0) = 0. f (x) = 2(cos x)(cos x) 3 + 2(sin x)( 3 cos x) 4 ( sin x) = 2(cos x) 2 + 6(sin x) 2 (cos x) 4, f (0) = 2. f (4) (x) = 4(cos x) 3 (sin x)+12(sin x)(cos x) 4 +6(sin x) 2 ( 4) f (4) = 0.

tan x = x + 2 3! x3 + O(x 5 ). O(x 5 ) x 5 sin x = x 1 3! x3 + O(x 5 ). tan x sin x = 1 2 x3 + O(x 5 ) lim x 0 tan x sin x x 3 = 1 2.

y = 1 y = x y = x 2 y = x 3 x = 0 1 : 2 : 3 :

f (x) > 0 f(x) : x < x, f(x ) f(x) = (x x)f (ξ), x < ξ < x. f(x ) > f(x).

(x0,y0) (x2,y2) (x1,y1) (x0,y0) (x1,y1) (x2,y2) : y = f(x) x 0 < x 1 < x 2 y 1 < (x 2 x 1 ) (x 2 x 0 ) y 0 + (x 1 x 0 ) (x 2 x 0 ) y 2 y 1 y 0 x 1 x 0 < y 2 y 1 x 2 x 1

f (x) > 0 f(x) : x 0 < x 1 < x 2 f (x) > 0 y 1 y 0 x 1 x 0 < y 2 y 1 x 2 x 1 x 0 x 1, x 1 x 2

I 2 I

A A

f (x 0 ) x = x 0 dx, dy dy = f (x 0 ) dx (y y 0 ) = f (x 0 ) (x x 0 ) (dx, dy) III df dx = df(y(x)) dy dy dx

( ) ( 4 ) ( 287 212 )

( ) : III 4