(1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9"

Transcription

1 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n m n m n 0 2 = π =

2 (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9) 0 = 0, (10) ( ) ( b) = b b + x = b x b b x = b x P (x) x α α P (x) = n ( + b) n = n + n n 1 b + 2 n(n 1) n 2 b 2 n! k!(n k)! k b n k + + b n 1-9. A A x x A minimum A 2

3 A x x A mximum A c A upper bound A x x c A A A A A sup A 2-1. f y = f(x) x y y = f(x) c, A x c f(x) A f(x) A x c lim f(x) = A x c A x c f(x) limit x c+ x c c x c x c c lim x c+ f(x), lim x c f(x) x = c f(x) x = c lim f(x) = f(c) x c f(x) x = c lim f(x) = f(c) x c+ f(x) x = c lim x c f(x) = f(c) f(x) [, b] < c < b c x = x = b 3

4 [, b] f(x) f() f(b) M f(c) = M c [, b] 2-5. x x x x y = x+y ( x ) y = xy 2-6. x x > 1 0 < < > 1 0 < < 1 x = y (, ) (0, ) x y = x y 1 y log x x log x + log y = log xy, log x y = y log x 2-8. y = log x (0, ) (, ) log x > 1 0 < < (1 + 1 n )n n e = lim n (1 + 1 n )n e x 1 lim = 1 x 0 x 4

5 x e xi x = i e 0i = 1 x > i x e xi x < i x e xi x e xi e xi e yi = e (x+y)i, e xi 1 lim x 0 x = i sin x cos x e xi e xi = cos x + i sin x 3-1. y = f(x) f (x) y dy dx (f(x)) 3-2. y = f(x) f (x) x = f () f () = lim x f(x) f() x f () = lim h 0 f( + h) f() h f(x) f () f(x) x = f () f(x) x = f(x) x = x = f(x) 5

6 f (x) f (x) f(x) 3-3. (x ) = x 1 2 (cf(x) + dg(x)) = cf (x) + dg (x), (f(x)g(x)) = f (x)g(x) + f(x)g (x), ( g(x) f(x) ) = f(x)g (x) f (x)g(x) f(x) 2 (f(x) 0 x 3-4. y = f(t) t = g(x) y = f(g(x)) y = f(t) t = g(x) (f(g(x)) = f (g(x))g (x), dy dx = dy dt dt dx 3-5. (e x ) = e x, (log x) = 1 x, (sin x) = cos x, (cos x) = sin x e x 1 lim = 1 x 0 x e ix 1 lim = i x 0 x (e xi ) = ie xi 3-6. f(x) x = f () x = x = +h f( + h) f() x f(x) ( + h) 6

7 h 0 f(x) x = f () > 0 x = f () < 0 x = 3-7. y x x = f(y) x x = f(y) y 1 x = f(y) 1 y y = g(x) x y y = g(x) x = f(y) x = g(y) g (y) > y = f(x) y = f(x) y = 1 g (f(x)), dy dx = 1 dx dy 3-8. x = sin y [ π 2, π ] [ 1, 1] 2 dx dy = cos y > 0 ( π 2 < y < π 2 ) y = sin 1 x x = cos y [0, π] [ 1, 1] dx dy = sin y < 0 (0 < y < π) y = cos 1 x x = tn y = sin x cos x ( π 2, π 2 ) (, ) dx dy = 1 cos 2 y > 0 y = tn 1 x (sin 1 x) = 1, 1 x 2 (cos 1 x) = 1, 1 x 2 (tn 1 x) = x 2 7

8 3-9. y = f(x) f (x), (f(x)), y dy, dx f (x), (f(x)), y d 2 y, dx 2 f () > 0 y = f(x) x = f () < 0 y = f(x) x = n y = f(x) n f (n) (x), y (n) d n y, dx n 4-1. [, b] f(x) (, b) lim f(x) = x + f() x = lim f(x) = f(b) x b f(b) f() x = b = f (ξ) b b ξ x = f(b) f() x = b f (ξ) x = ξ b ξ b 4-2. (, b) f(x) f (x) > 0 ( < x < b) f (x) < 0 ( < x < b) f(x) f (x) C 1 C 1 f(x), x f(x) = f() + f (ξ)(x ) ξ x f(x) x = C 1 f(x) = f() + f ()(x ) + o( x ) o( x ) R(x) lim x x = 0 1 f(x) f() + f ()(x ) y = f() + f ()(x ) y = f(x) 8

9 (, f()) 4-3. f(x) f (x) C 2 C 2 f(x), x f(x) = f() + f ()(x ) f (ξ)(x ) 2 ξ x f(x) x = C 2 f(x) = f() + f ()(x ) + f () (x ) 2 + o((x ) 2 ) 2 o( x 2 ) f(x) x = f(x) f() + f ()(x ) + f () (x ) f(x) n f (n) (x) C n C n f(x), x f(x) = f() + 1 1! f ()(x ) + 1 2! f ()(x ) ! f (3) ()(x ) (n 1)! f (n 1) ()(x ) n n! f (n) (ξ)(x ) n ξ x 4-5. f(x) g(x) f() = g() = 0 x = f(x), g(x) f (x) lim x g (x) = A lim f(x) x g(x) = A lim x f(x) =, lim g(x) = x f (x) f(x), g(x) lim x g (x) = A f(x) lim x g(x) = A 9

10 5-1. F (x) = f(x) F (x) f(x) f(x) 1 F (x) f(x) { F (x)+c c } f(x) f(x)dx f(x) 5-2. f(x) f (x) f (x)dx = f(x) + c x dx = x+1 ( 1 ) 1 dx = log x, e x dx = e x x cos xdx = sin x, sin xdx = cos x, (cf(x) + dg(x))dx = c f(x)dx + d g(x)dx, f (x)g(x)dx = f(x)g(x) f(x)g (x)dx, f(x)dx = f(g(t))g (t)dt x = g(t) 1 dx = tn x, cos 2 x y x 5-6. y = f( y x ) u = y x 10

11 5-7. y + P (x)y = Q(x) 1 1 y = e P (x)dx ( e P (x)dx Q(x)dx + c) 6-1. b f(x)dx f(x) x = x = b F (x) f(x) b b f(x)dx, f(x)dx = F (b) F () [, b] C 1 f(x), g(x) b f (x)g(x)dx = [ f(x)g(x) ] b b f(x)g (x)dx C 1 x = g(t) g(α) =, g(β) = b b f(x)dx = β α f(g(t))g (t)dt 6-2. [, b] f(x) [, b] n : = x 0 < x 1 < x 2 < < x n 1 < x n = b i [x i 1, x i ] f(x) m i M i x i x i 1 m i n i s( ) x i x i 1 M i n i S( ) s( ) s S( ) S s S s = S f(x) [, b] [, b] f(x) f(x) [, b] b f(x)dx [, b] 6-3. [, b] f(x) G(t) = 11 t f(x)dx G (t) =

12 f(t) G(x) f(x) b f(x)dx = G(b) f(x) 6-4. [α, β] x = f(t), y = g(t) (x, y) (f(t), g(t)) t β f (t) 2 + g (t) 2 dt α 7-1. (x, y) z = f(x, y) 2 z = f(x, y) y x 1 z z x x x y 1 z y z y z x, z y f x(x, y), f y (x, y) z = f(x, y) f x (x, y) f y (x, y) (x, y) = (.b) f( + h, b) f(, b) f(, b + k) f(, b) f x (, b) = lim, f y (, b) = lim h 0 h k 0 k f x (, b) f y (, b) z = f(x, y) (x, y) = (, b) f x (, b) f y (, b) (x, y) = (, b) x y f x (, b) (, b) x f y (, b) (, b) y 7-2. z = f(x, y) x = h(t), y = k(t) z = f(h(t), k(t)) z = f(x, y) x = h(t) y = k(t) z = f(h(t), k(t)) 12

13 ϵ(x, y) ϵ(x, y) (x ) 2 + (y b) 2 0 (x )2 + (y b) 2 0 (x ) 2 + (y b) 2 ϵ(x, y) = o( (x ) 2 + (y b) 2 ) 2 f(x, y) f(x, y) = f(, b) + A(x ) + B(y b) + o( (x ) 2 + (y b) 2 ) A B (, b) f(x, y) (, b) (, b) f(x, y) = f(, b) + f x (, b)(x ) + f y (, b)(y b) + o( (x ) 2 + (y b) 2 ) z = f(x, y) z x, z y C 1 f(x, y) (, b) C 1 (, b) z = f(x, y) C 1 x = g(t), y = h(t) z = f(g(t), h(t)) dz dt = z dx x dt + z dy dy dt z = f(x, y) C 1 x = g(u, v), y = h(u, v) z = f(g(u, v), h(u, v)) z u = z x x u + z y y u, z v = z x x v + z y y v 7-3. z = f(x, y) x z x f x(x, y) x 2 z x 2 f xx(x, y) y 2 z y x f xy(x, y) z = f(x, y) y z y f y(x, y) x 2 z x y f yx (x, y) y 2 z y f yy(x, y) 2 13

14 2 z x, 2 z 2 y x, 2 z x y, 2 z y 2 f xx(x, y), f xy (x, y), f yx (x, y), f yy (x, y) z = f(x, y) z = f(x, y) C 2 z = f(x, y) (, b) C 2 f yx (, b) = f xy (, b) 7-4. (, b) C 2 f(x, y) f(x, y) f(, b) + f x (, b)(x ) + f y (, b)(y b) f xx(, b)(x ) 2 + f xy (, b)(x )(y b) f yy(, b)(y b) C 2 f(x, y) (x, y) = (, b) f x (, b) = f y (, b) = 0 f xx (, b)f yy (, b) f xy (, b) 2 > 0 f xx (, b) > 0 f(x, y) (, b) f xx (, b)f yy (, b) f xy (, b) 2 > 0 f xx (, b) < 0 f(x, y) (, b) 7-6. (, b) C 1 F (x, y) F (, b) = 0 F y (, b) 0 x = y = f(x) f() = b F (x, f(x)) = 0 C 1 F (x, y), G(x, y) F (x, y) Λ = { (x, y) G(x, y) = 0 } (, b) G x (, b) 2 + G y (, b) 2 0 H(x, y, λ) = F (x, y) λg(x, y) H x (, b, λ) = H y (, b, λ) = 0 λ G(x, y) = 0 F (x, y) H(x, y, λ) 8-1. [, b] g(x), h(x) h(x) g(x) ( x b) D = { (x, y) h(x) y g(x), x b } f(x, y) f(x, y) dxdy f(x, y) D D 14

15 2 D f(x, y) dxdy = b { g(x) h(x) f(x, y) dy}dx f(x, y) 0 z = f(x, y) (x, y) D [, b] [c, d] = { (x, y) x b, c y d } 2 f(x, y) [, b] = x 0 < x 1 < x 2 < < x n = b [c, d] c = y 0 < y 1 < y 2 < < y m = d [, b] [c, d] : = x 0 < x 1 < x 2 < < x n = b, c = y 0 < y 1 < y 2 < < y m = d (i, j) [x i 1, x i ] [y j 1, y j ] = { (x, y) x i 1 x x i, y j 1 y y j } f(x, y) M ij m i,j (x i x i 1 )(y j y j 1 ) M ij S( ) m ij s( ) S( ) S s( ) s s S s = S f(x.y) [, b] [c, d] f(x.y) [, b] [c, d] f(x, y) dxdy [,b] [c,d] D D 1 D 0 1 D (x, y) D [, b] g(x), h(x) D f(x, y) f(x, y)1 D (x, y) f(x, y)1 D (x, y)dxdy f(x, y) dxdy [,b] [c,d] D C 1 u = g(x, y), v = h(x, y) (x, y) D (u, v) E 1 1 E f(u, v) f(g(x, y), h(x, y)) E f(u, v) dudv = D (u, v) f(g(x, y), h(x, y)) (x, y) dxdy 2 15

16 (u, v) (x, y) = (x, y) u v (x, y) x u v (x, y) x (x, y) x (x, y) y = u v x y u v y x u = g(x, y), v = h(x, y) 8-4. (u, v) D = { (u, v) u b, c y d } x = f(u, v), y = g(u, v), z = h(u, v) C 1 S : x = f(u, v), y = g(u, v), z = h(u, v) ((u, v) D) (y, z) (z, x) (x, y) ( (u, v) )2 + ( (u, v) )2 + ( (u, v) )2 dudv D 8-5. n f(x 1, x 2,, x n ) n D f(x 1, x 2,, x n )dx 1 dx 2 dx n n n n n 9-2. lim x f(x) = A ϵ-δ ϵ > 0 0 < x < δ x f(x) A < ϵ δ > lim f(x) = A lim g(x) = B (1) lim(f(x)+g(x)) = x x x A + B (2) lim cf(x) = ca (c ) (3) lim f(x)g(x) = AB (4) A 0 x x g(x) lim x f(x) = B A 9-5. lim n n = ϵ-n ϵ > 0 n N n < ϵ N 16

17 9-6. lim n n = lim n b n = b (1) lim n ( n + b n ) = + b (2) lim c n n = c (3) lim n b n n = b (4) 0 lim b n n n = b lim n n = lim = n n n ϵ N ϵ-δ 9-7. lim f(x) = A lim n =, n (n = x n 1, 2, 3, ) n lim f( n ) = A n (x, y) (, b) (x ) 2 + (y b) 2 0 ( n, b n ), n = 1, 2, 3, ( 0, b 0 ) lim (n 0 ) 2 + (b n b 0 ) 2 = 0 n lim f(x, y) = A ϵ δ ϵ > 0 0 < (x,y) (,b) (x )2 + (y b) 2 < δ (x, y) f(x, y) A < ϵ δ , 2,, n, ϵ n, m N n m < ϵ N 17

18 10-4. [, b] x x δ(x) U δ(x) (x) = (x δ(x), x+δ(x)) [, b] U δ(x1 )(x 1 ), U δ(x2 )(x 2 ),, U δ(xk )(x k ) (1),(2),(3),(4) R R + (1) R R (2) R + R + (3) R R + (R R + = ) (4) R R + (R R + = R) R, R + R R [, b] f(x) (1) (2)[, b] (3)f() < M < f(b) f() > M > f(b) f(c) = M c ( c b) (4) ϵ x x < δ, x, x b f(x) f(x ) < ϵ ϵ δ ( D D D D (x, y) (x, y) δ(x, y) U δ(x,y) (x, y) U δ(x1,y 1 )(x 1, y 1 ), U δ(x2,y 2 )(x 2, y 2 ),, U δ(xk,y k )(x k, y k ) D D f(x, y) (1) (2) D (3) ϵ (x x ) 2 + (y y ) 2 < δ, (x, y) D, (x, y ) D f(x, y) f(x, y ) < ϵ ϵ δ 18

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 1 1 1.1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k=1 1.1.7 ɛ-n 1.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ

More information

( ) 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 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 =

More information

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................ 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)

More information

f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x >

f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x > 5.1 1. x = a f (x) a x h f (a + h) f (a) h (5.1) h 0 f (x) x = a f +(a) f (a + h) f (a) = lim h +0 h (5.2) x h h 0 f (a) f (a + h) f (a) f (a h) f (a) = lim = lim h 0 h h 0 h (5.3) f (x) x = a f (a) =

More information

http://know-star.com/ 3 1 7 1.1................................. 7 1.2................................ 8 1.3 x n.................................. 8 1.4 e x.................................. 10 1.5 sin

More information

body.dvi

body.dvi ..1 f(x) n = 1 b n = 1 f f(x) cos nx dx, n =, 1,,... f(x) sin nx dx, n =1,, 3,... f(x) = + ( n cos nx + b n sin nx) n=1 1 1 5 1.1........................... 5 1.......................... 14 1.3...........................

More information

1 1 1 1 1 1 2 f z 2 C 1, C 2 f 2 C 1, C 2 f(c 2 ) C 2 f(c 1 ) z C 1 f f(z) xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q) 1 1 (b, d) (a, c) 2 3 2 3 a = d, c = b

More information

2 p T, Q

2 p T, Q 270 C, 6000 C, 2 p T, Q p: : p = N/ m 2 N/ m 2 Pa : pdv p S F Q 1 g 1 1 g 1 14.5 C 15.5 1 1 cal = 4.1855 J du = Q pdv U ( ) Q pdv 2 : z = f(x, y). z = f(x, y) (x 0, y 0 ) y y = y 0 z = f(x, y 0 ) x x =

More information

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

More information

Microsoft Word - 触ってみよう、Maximaに2.doc

Microsoft Word - 触ってみよう、Maximaに2.doc i i e! ( x +1) 2 3 ( 2x + 3)! ( x + 1) 3 ( a + b) 5 2 2 2 2! 3! 5! 7 2 x! 3x! 1 = 0 ",! " >!!! # 2x + 4y = 30 "! x + y = 12 sin x lim x!0 x x n! # $ & 1 lim 1 + ('% " n 1 1 lim lim x!+0 x x"!0 x log x

More information

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt

More information

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1 II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2

More information

α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn A, B A B A B A B A B B A A B N 2

α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn A, B A B A B A B A B B A A B N 2 1. 2. 3. 4. 5. 6. 7. 8. N Z 9. Z Q 10. Q R 2 1. 2. 3. 4. Zorn 5. 6. 7. 8. 9. x x x y x, y α = 2 2 α x = y = 2 1 α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn

More information

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory 10.3 Fubini 1 Introduction [1],, [2],, [3],, [4],, [5],, [6],, [7],, [8],, [1, 2, 3] 1980

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory 10.3 Fubini 1 Introduction [1],, [2],, [3],, [4],, [5],, [6],, [7],, [8],, [1, 2, 3] 1980 % 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2006.11.20 1 8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory

More information

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1

More information

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6 26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7

More information

3 0 4 3 5 6 6 7 7 8 4 9 6 0 30 33 34 3 36 4 4 5 44 6 47 7 54 8 56 9 60 0 6 64 67 3 70 4 7 5 75 6 80

3 0 4 3 5 6 6 7 7 8 4 9 6 0 30 33 34 3 36 4 4 5 44 6 47 7 54 8 56 9 60 0 6 64 67 3 70 4 7 5 75 6 80 3 0 4 3 5 6 6 7 7 8 4 9 6 0 30 33 34 3 36 4 4 5 44 6 47 7 54 8 56 9 60 0 6 64 67 3 70 4 7 5 75 6 80 7 8 3 elemet, set A, A A, A A, A A, b, c, {, b, c, }, x P x, P x x {x P x}, A x, P x {x A P x} 3 { {,,

More information

untitled

untitled Y = Y () x i c C = i + c = ( x ) x π (x) π ( x ) = Y ( ){1 + ( x )}( 1 x ) Y ( )(1 + C ) ( 1 x) x π ( x) = 0 = ( x ) R R R R Y = (Y ) CS () CS ( ) = Y ( ) 0 ( Y ) dy Y ( ) A() * S( π ), S( CS) S( π ) =

More information

example2_time.eps

example2_time.eps Google (20/08/2 ) ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67 Introduction ( ) Random Walk & Google Page Rank Agora on Aug. 20 2 / 67 Introduction Google ( ) Random Walk & Google Page Rank

More information

untitled

untitled 1 1 1. 2. 3. 2 2 1 (5/6) 4 =0.517... 5/6 (5/6) 4 1 (5/6) 4 1 (35/36) 24 =0.491... 0.5 2.7 3 1 n =rand() 0 1 = rand() () rand 6 0,1,2,3,4,5 1 1 6 6 *6 int() integer 1 6 = int(rand()*6)+1 1 4 3 500 260 52%

More information

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1) D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y

More information

5 36 5................................................... 36 5................................................... 36 5.3..............................

5 36 5................................................... 36 5................................................... 36 5.3.............................. 9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................

More information

2 1 17 1.1 1.1.1 1650

2 1 17 1.1 1.1.1 1650 1 3 5 1 1 2 0 0 1 2 I II III J. 2 1 17 1.1 1.1.1 1650 1.1 3 3 6 10 3 5 1 3/5 1 2 + 1 10 ( = 6 ) 10 1/10 2000 19 17 60 2 1 1 3 10 25 33221 73 13111 0. 31 11 11 60 11/60 2 111111 3 60 + 3 332221 27 x y xy

More information

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

More information

lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d

lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d lim 5. 0 A B 5-5- A B lim 0 A B A 5. 5- 0 5-5- 0 0 lim lim 0 0 0 lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d 0 0 5- 5-3 0 5-3 5-3b 5-3c lim lim d 0 0 5-3b 5-3c lim lim lim d 0 0 0 3 3 3 3 3 3

More information

Z: Q: R: C:

Z: Q: R: C: 0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x

More information

1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1

1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1 I, A 25 8 24 1 1.1 ( 3 ) 3 9 10 3 9 : (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4), (3,3,3) 10 : (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4) 6 3 9 10 3 9 : 6 3 + 3 2 + 1 = 25 25 10 : 6 3 + 3 3

More information

IV.dvi

IV.dvi IV 1 IV ] shib@mth.hiroshim-u.c.jp [] 1. z 0 ε δ := ε z 0 z

More information

z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z

z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z Tips KENZOU 28 6 29 sin 2 x + cos 2 x = cos 2 z + sin 2 z = OK... z < z z < R w = f(z) z z w w f(z) w lim z z f(z) = w x x 2 2 f(x) x = a lim f(x) = lim f(x) x a+ x a z z x = y = /x lim y = + x + lim y

More information

ィェィ ィョ02ィヲィー ィェ ィャ0200ィ ィェ 08ィ ィィ ィョ07ィー D ィョ0007 T, ィヲィ 06ィョ0002: D 6メ6 (x; y) 6モ1 f (x; y

ィェィ ィョ02ィヲィー ィェ ィャ0200ィ ィェ 08ィ ィィ ィョ07ィー D ィョ0007 T, ィヲィ 06ィョ0002: D 6メ6 (x; y) 6モ1 f (x; y 130005ィィ04ィャィ 14 0709010905080507030707 040309090201 00030809000905080201 14.1 03ィヲィョィ 00ィエ00ィヲィコ06ィー 06ィェィェ07ィヲ02ィー 070007 ィャ05ィィ04ィャィ ィ 0100ィケ ィィィ 0008ィェ02ィヲ ィャィヲィ 0002ィェ08ィコ0201ィョ04 0004ィー 070104 00ィェィエィョ0007ィー

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,,

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,, 2012 10 13 1,,,.,,.,.,,. 2?.,,. 1,, 1. (θ, φ), θ, φ (0, π),, (0, 2π). 1 0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ).

More information

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46..

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46.. Cotets 6 6 : 6 6 6 6 6 6 7 7 7 Part. 8. 8.. 8.. 9..... 3. 3 3.. 3 3.. 7 3.3. 8 Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3.

More information

Z: Q: R: C: 3. Green Cauchy

Z: Q: R: C: 3. Green Cauchy 7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................

More information

1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3.....................................

1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3..................................... 1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3........................................... 1 17.1................................................

More information

「数列の和としての積分 入門」

「数列の和としての積分 入門」 7 I = 5. introduction.......................................... 5........................................... 7............................................. 9................................................................................................

More information

統計学のポイント整理

統計学のポイント整理 .. September 17, 2012 1 / 55 n! = n (n 1) (n 2) 1 0! = 1 10! = 10 9 8 1 = 3628800 n k np k np k = n! (n k)! (1) 5 3 5 P 3 = 5! = 5 4 3 = 60 (5 3)! n k n C k nc k = npk k! = n! k!(n k)! (2) 5 3 5C 3 = 5!

More information

34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 1.2 1 関 数 値 0.8 0.6 0.4 0.2 0 15 10 5 0 5 10

34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 1.2 1 関 数 値 0.8 0.6 0.4 0.2 0 15 10 5 0 5 10 33 2 2.1 2.1.1 x 1 T x T 0 F = ma T ψ) 1 x ψ(x) 2.1.2 1 1 h2 d 2 ψ(x) + V (x)ψ(x) = Eψ(x) (2.1) 2m dx 2 1 34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2

More information

1. 2. ( ) 3. ( ) 2

1. 2. ( ) 3. ( ) 2 IV 3 : 1 2013 10 14 1 1. 2. ( ) 3. ( ) 2 1. 3 (procurement auctions) etc. ( ) : : 4 5 : (sealed-bid auctions) : (1st-price auctions) (2nd-price auctions) 6 : ( ) (open auctions) (English auctions) (Dutch

More information

i ( ) PDF http://moodle.sci.u-toyama.ac.jp/kyozai/ I +α II II III A: IV B: V C: III V I, II III IV V III IV 8 5 6 krmt@sci.u-toyama.ac.jp

i ( ) PDF http://moodle.sci.u-toyama.ac.jp/kyozai/ I +α II II III A: IV B: V C: III V I, II III IV V III IV 8 5 6 krmt@sci.u-toyama.ac.jp 8 5 6 i ( ) PDF http://moodle.sci.u-toyama.ac.jp/kyozai/ I +α II II III A: IV B: V C: III V I, II III IV V III IV 8 5 6 krmt@sci.u-toyama.ac.jp ii I +α 3.....................................................

More information

12 2 E ds = 1 ρdv ε 1 µ D D S S D B d S = 36 E d B l = S d S B d l = S ε E + J d S 4 4 div E = 1 ε ρ div B = rot E = B 1 rot µ E B = ε + J 37 3.2 3.2.

12 2 E ds = 1 ρdv ε 1 µ D D S S D B d S = 36 E d B l = S d S B d l = S ε E + J d S 4 4 div E = 1 ε ρ div B = rot E = B 1 rot µ E B = ε + J 37 3.2 3.2. 213 12 1 21 5 524 3-5465-74 nkiyono@mail.ecc.u-tokyo.ac.jp http://lecture.ecc.u-tokyo.ac.jp/~nkiyono/index.html 3 2 1 3.1 ρp, t EP, t BP, t JP, t 35 P t xyz xyz t 4 ε µ D D S S 35 D H D = ε E B = µ H E

More information

17 3 31 1 1 3 2 5 3 9 4 10 5 15 6 21 7 29 8 31 9 35 10 38 11 41 12 43 13 46 14 48 2 15 Radon CT 49 16 50 17 53 A 55 1 (oscillation phenomena) e iθ = cos θ + i sin θ, cos θ = eiθ + e iθ 2, sin θ = eiθ e

More information

., a = < < < n < n = b, j = f j j =,,, n, C P,, P,,, P n n, n., P P P n = = n j= n j= j j + j j + { j j / j j } j j, j j / j j f j 3., n., Oa, b r > P

., a = < < < n < n = b, j = f j j =,,, n, C P,, P,,, P n n, n., P P P n = = n j= n j= j j + j j + { j j / j j } j j, j j / j j f j 3., n., Oa, b r > P . ϵριµϵτρoζ perimetros 76 Jones, Euler. =.,.,,,, C, C n+ P, P,, P n P, P n P n, P P P P n P n n P n,, C P, P j P j j =,,, n P n P., C.,, C. f [a, b], f. C = f a b, C l l = b a + f d P j P j a b j j j j

More information

6 1

6 1 (c) Masaya Kasuga Shaltics 2001 6 1 1 1 2 1 1.1 USO 2 3 4 EPR 5 6 27 2 (Unfinded Superconducting Object) 3 - - - 342K (Physica C 351 (2001) pp.78-81) 4 40K 5 Einstein-Podolsky-Rosen s paradox 1/2 ( 0)

More information

. sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s

. sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s . 00 3 9 [] sinh x = ex e x, cosh x = ex + e x ) sinh cosh 4 hyperbolic) hyperbola) = 3 cosh x cosh x) = e x + e x = cosh x ) . sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y =

More information

untitled

untitled 3,,, 2 3.1 3.1.1,, A4 1mm 10 1, 21.06cm, 21.06cm?, 10 1,,,, i),, ),, ),, x best ± δx 1) ii), x best ), δx, e,, e =1.602176462 ± 0.000000063) 10 19 [C] 2) i) ii), 1) x best δx

More information

A B 5 C 9 3.4 7 mm, 89 mm 7/89 = 3.4. π 3 6 π 6 6 = 6 π > 6, π > 3 : π > 3

A B 5 C 9 3.4 7 mm, 89 mm 7/89 = 3.4. π 3 6 π 6 6 = 6 π > 6, π > 3 : π > 3 π 9 3 7 4. π 3................................................. 3.3........................ 3.4 π.................... 4.5..................... 4 7...................... 7..................... 9 3 3. p

More information

l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r

l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r 2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2

More information

/ 2 ( ) ( ) ( ) = R ( ) ( ) 1 1 1/ 3 = 3 2 2/ R :. (topology)

/ 2 ( ) ( ) ( ) = R ( ) ( ) 1 1 1/ 3 = 3 2 2/ R :. (topology) 3 1 3.1. (set) x X x X x X 2. (space) Hilbert Teichmüller 2 R 2 1 2 1 / 2 ( ) ( ) ( ) 1 0 1 + = R 2 0 1 1 ( ) ( ) 1 1 1/ 3 = 3 2 2/ R 2 3 3.1:. (topology) 3.2 30 3 3 2 / 3 3.2.1 S O S (O1)-(O3) (O1) S

More information

phs.dvi

phs.dvi 483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....

More information

取扱説明書 [F-12C]

取扱説明書 [F-12C] F-12C 11.7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 a bc b c d d a 15 a b cd e a b c d e 16 17 18 de a b 19 c d 20 a b a b c a d e k l m e b c d f g h i j p q r c d e f g h i j n o s 21 k l m n o p q r s a X

More information

d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x =

d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x = 3 MATLAB Runge-Kutta Butcher 3. Taylor Taylor y(x 0 + h) = y(x 0 ) + h y (x 0 ) + h! y (x 0 ) + Taylor 3. Euler, Runge-Kutta Adams Implicit Euler, Implicit Runge-Kutta Gear y n+ y n (n+ ) y n+ y n+ y n+

More information

2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a

More information

untitled

untitled .. 3. 3 3. 3 4 3. 5 6 3 7 3.3 9 4. 9 0 6 3 7 0705 φ c d φ d., φ cd, φd. ) O x s + b l cos s s c l / q taφ / q taφ / c l / X + X E + C l w q B s E q q ul q q ul w w q q E E + E E + ul X X + (a) (b) (c)

More information

Fourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a 0 2 + (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin

Fourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a 0 2 + (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin ( ) 205 6 Fourier f : R C () (2) f(x) = a 0 2 + (a n cos nx + b n sin nx), n= a n = f(x) cos nx dx, b n = π π f(x) sin nx dx a n, b n f Fourier, (3) f Fourier or No. ) 5, Fourier (3) (4) f(x) = c n = n=

More information

a (a + ), a + a > (a + ), a + 4 a < a 4 a,,, y y = + a y = + a, y = a y = ( + a) ( x) + ( a) x, x y,y a y y y ( + a : a ) ( a : a > ) y = (a + ) y = a

a (a + ), a + a > (a + ), a + 4 a < a 4 a,,, y y = + a y = + a, y = a y = ( + a) ( x) + ( a) x, x y,y a y y y ( + a : a ) ( a : a > ) y = (a + ) y = a [] a x f(x) = ( + a)( x) + ( a)x f(x) = ( a + ) x + a + () x f(x) a a + a > a + () x f(x) a (a + ) a x 4 f (x) = ( + a) ( x) + ( a) x = ( a + a) x + a + = ( a + ) x + a +, () a + a f(x) f(x) = f() = a

More information

semi4.dvi

semi4.dvi 1 2 1.1................................................. 2 1.2................................................ 3 1.3...................................................... 3 1.3.1.............................................

More information

120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2

120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2 9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0

More information

USB 0.6 https://duet.doshisha.ac.jp/info/index.jsp 2 ID TA DUET 24:00 DUET XXX -YY.c ( ) XXX -YY.txt() XXX ID 3 YY ID 5 () #define StudentID 231

USB 0.6 https://duet.doshisha.ac.jp/info/index.jsp 2 ID TA DUET 24:00 DUET XXX -YY.c ( ) XXX -YY.txt() XXX ID 3 YY ID 5 () #define StudentID 231 0 0.1 ANSI-C 0.2 web http://www1.doshisha.ac.jp/ kibuki/programming/resume p.html 0.3 2012 1 9/28 0 [ 01] 2 10/5 1 C 2 3 10/12 10 1 2 [ 02] 4 10/19 3 5 10/26 3 [ 03] 6 11/2 3 [ 04] 7 11/9 8 11/16 4 9 11/30

More information

C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 新装版 1 刷発行時のものです.

C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.  このサンプルページの内容は, 新装版 1 刷発行時のものです. C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009383 このサンプルページの内容は, 新装版 1 刷発行時のものです. i 2 22 2 13 ( ) 2 (1) ANSI (2) 2 (3) Web http://www.morikita.co.jp/books/mid/009383

More information

3 - { } / f ( ) e nπ + f( ) = Cne n= nπ / Eucld r e (= N) j = j e e = δj, δj = 0 j r e ( =, < N) r r r { } ε ε = r r r = Ce = r r r e ε = = C = r C r e + CC e j e j e = = ε = r ( r e ) + r e C C 0 r e =

More information

05-5.dvi

05-5.dvi 131 71 71 71 71 71 71 71 7 1 71 71 71 71 71 7 1 71 71 71 71 71 71 7 1 71 71 71 71 71 71 71 71 7 1 1 71 71 71 71 71 71 71 71 71 7 1 75(468) 1 71 71 7 517.95 1 7.1 7. 1 71 71 71 71 71 71 71 7, 1 7.1 7. 1

More information

i 0 1 0.1 I................................................ 1 0.2.................................................. 2 0.2.1...........................

i 0 1 0.1 I................................................ 1 0.2.................................................. 2 0.2.1........................... 2008 II 21 1 31 i 0 1 0.1 I................................................ 1 0.2.................................................. 2 0.2.1............................................. 2 0.2.2.............................................

More information

I.2 z x, y i z = x + iy. x, y z (real part), (imaginary part), x = Re(z), y = Im(z). () i. (2) 2 z = x + iy, z 2 = x 2 + iy 2,, z ± z 2 = (x ± x 2 ) +

I.2 z x, y i z = x + iy. x, y z (real part), (imaginary part), x = Re(z), y = Im(z). () i. (2) 2 z = x + iy, z 2 = x 2 + iy 2,, z ± z 2 = (x ± x 2 ) + I..... z 2 x, y z = x + iy (i ). 2 (x, y). 2.,,.,,. (), ( 2 ),,. II ( ).. z, w = f(z). z f(z), w. z = x + iy, f(z) 2 x, y. f(z) u(x, y), v(x, y), w = f(x + iy) = u(x, y) + iv(x, y).,. 2. z z, w w. D, D.

More information

0 (1 ) 0 (1 ) 01 Excel Excel ( ) = Excel Excel =5+ 5 + 7 =5-5 3 =5* 5 10 =5/ 5 5 =5^ 5 5 ( ), 0, Excel, Excel 13E+05 13 10 5 13000 13E-05 13 10 5 0000

0 (1 ) 0 (1 ) 01 Excel Excel ( ) = Excel Excel =5+ 5 + 7 =5-5 3 =5* 5 10 =5/ 5 5 =5^ 5 5 ( ), 0, Excel, Excel 13E+05 13 10 5 13000 13E-05 13 10 5 0000 1 ( S/E) 006 7 30 0 (1 ) 01 Excel 0 7 3 1 (-4 ) 5 11 5 1 6 13 7 (5-7 ) 9 1 1 9 11 3 Simplex 1 4 (shadow price) 14 5 (reduced cost) 14 3 (8-10 ) 17 31 17 3 18 33 19 34 35 36 Excel 3 4 (11-13 ) 5 41 5 4

More information

6. [1] (cal) (J) (kwh) ( 1 1 100 1 ( 3 t N(t) dt dn ( ) dn N dt N 0 = λ dt (3.1) N(t) = N 0 e λt (3.2) λ (decay constant), λ [λ] = 1/s 1947 2

6. [1] (cal) (J) (kwh) ( 1 1 100 1 ( 3 t N(t) dt dn ( ) dn N dt N 0 = λ dt (3.1) N(t) = N 0 e λt (3.2) λ (decay constant), λ [λ] = 1/s 1947 2 filename=decay-text141118.tex made by R.Okamoto, Emeritus Prof., Kyushu Inst.Tech. * 1, 320 265 radioactive ray ( parent nucleus) ( daughter nucleus) disintegration, decay 2 1. 2. 4 ( 4 He) 3. 4. X 5.,

More information

140 120 100 80 60 40 20 0 115 107 102 99 95 97 95 97 98 100 64 72 37 60 50 53 50 36 32 18 H18 H19 H20 H21 H22 H23 H24 H25 H26 H27 1 100 () 80 60 40 20 0 1 19 16 10 11 6 8 9 5 10 35 76 83 73 68 46 44 H11

More information

学習内容と日常生活との関連性の研究-第2部-第6章

学習内容と日常生活との関連性の研究-第2部-第6章 378 379 10% 10%10% 10% 100% 380 381 2000 BSE CJD 5700 18 1996 2001 100 CJD 1 310-7 10-12 10-6 CJD 100 1 10 100 100 1 1 100 1 10-6 1 1 10-6 382 2002 14 5 1014 10 10.4 1014 100 110-6 1 383 384 385 2002 4

More information

診療ガイドライン外来編2014(A4)/FUJGG2014‐01(大扉)

診療ガイドライン外来編2014(A4)/FUJGG2014‐01(大扉) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

More information

04年度LS民法Ⅰ教材改訂版.PDF

04年度LS民法Ⅰ教材改訂版.PDF ?? A AB A B C AB A B A B A B A A B A 98 A B A B A B A B B A A B AB AB A B A BB A B A B A B A B A B A AB A B B A B AB A A C AB A C A A B A B B A B A B B A B A B B A B A B A B A B A B A B A B

More information

…K…E…X„^…x…C…W…A…fi…l…b…g…‘†[…N‡Ì“‚¢−w‘K‡Ì‹ê™v’«‡É‡Â‡¢‡Ä

…K…E…X„^…x…C…W…A…fi…l…b…g…‘†[…N‡Ì“‚¢−w‘K‡Ì‹ê™v’«‡É‡Â‡¢‡Ä 2009 8 26 1 2 3 ARMA 4 BN 5 BN 6 (Ω, F, µ) Ω: F Ω σ 1 Ω, ϕ F 2 A, B F = A B, A B, A\B F F µ F 1 µ(ϕ) = 0 2 A F = µ(a) 0 3 A, B F, A B = ϕ = µ(a B) = µ(a) + µ(b) µ(ω) = 1 X : µ X : X x 1,, x n X (Ω) x 1,,

More information

1 Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier analog digital Fourier Fourier Fourier Fourier Fourier Fourier Green Fourier

1 Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier analog digital Fourier Fourier Fourier Fourier Fourier Fourier Green Fourier Fourier Fourier Fourier etc * 1 Fourier Fourier Fourier (DFT Fourier (FFT Heat Equation, Fourier Series, Fourier Transform, Discrete Fourier Transform, etc Yoshifumi TAKEDA 1 Abstract Suppose that u is

More information

Note5.dvi

Note5.dvi 12 2011 7 4 2.2.2 Feynman ( ) S M N S M + N S Ai Ao t ij (i Ai, j Ao) N M G = 2e2 t ij 2 (8.28) h i μ 1 μ 2 J 12 J 12 / μ 2 μ 1 (8.28) S S (8.28) (8.28) 2 ( ) (collapse) j 12-1 2.3 2.3.1 Onsager S B S(B)

More information

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si I 8 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No. : I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin.

More information

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9

More information

a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p

a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p a a a a y y ax q y ax q q y ax y ax a a a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p y a xp q y a x p q p p x p p q p q y a x xy xy a a a y a x

More information

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N

More information

96 7 1m =2 10 7 N 1A 7.1 7.2 a C (1) I (2) A C I A A a A a A A a C C C 7.2: C A C A = = µ 0 2π (1) A C 7.2 AC C A 3 3 µ0 I 2 = 2πa. (2) A C C 7.2 A A

96 7 1m =2 10 7 N 1A 7.1 7.2 a C (1) I (2) A C I A A a A a A A a C C C 7.2: C A C A = = µ 0 2π (1) A C 7.2 AC C A 3 3 µ0 I 2 = 2πa. (2) A C C 7.2 A A 7 Lorentz 7.1 Ampère I 1 I 2 I 2 I 1 L I 1 I 2 21 12 L r 21 = 12 = µ 0 2π I 1 I 2 r L. (7.1) 7.1 µ 0 =4π 10 7 N A 2 (7.2) magnetic permiability I 1 I 2 I 1 I 2 12 21 12 21 7.1: 1m 95 96 7 1m =2 10 7 N

More information

(w) F (3) (4) (5)??? p8 p1w Aさんの 背 中 が 壁 を 押 す 力 垂 直 抗 力 重 力 静 止 摩 擦 力 p8 p

(w) F (3) (4) (5)??? p8 p1w Aさんの 背 中 が 壁 を 押 す 力 垂 直 抗 力 重 力 静 止 摩 擦 力 p8 p F 1-1................................... p38 p1w A A A 1-................................... p38 p1w 1-3................................... p38 p1w () (1) ()?? (w) F (3) (4) (5)??? -1...................................

More information

* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................

More information

dynamics-solution2.dvi

dynamics-solution2.dvi 1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj

More information

‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í

‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í Markov 2009 10 2 Markov 2009 10 2 1 / 25 1 (GA) 2 GA 3 4 Markov 2009 10 2 2 / 25 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov 2009 10 2 3 / 25 (GA) ρ(i, j), i, j I

More information

2010 4 3 0 5 0.1......................................... 5 0.2...................................... 6 1 9 2 15 3 23 4 29 4.1............................................. 29 4.2..............................

More information

7 9 7..................................... 9 7................................ 3 7.3...................................... 3 A A. ω ν = ω/π E = hω. E

7 9 7..................................... 9 7................................ 3 7.3...................................... 3 A A. ω ν = ω/π E = hω. E B 8.9.4, : : MIT I,II A.P. E.F.,, 993 I,,, 999, 7 I,II, 95 A A........................... A........................... 3.3 A.............................. 4.4....................................... 5 6..............................

More information

繖 7 縺6ァ80キ3 ッ0キ3 ェ ュ ョ07 縺00 06 ュ0503 ュ ッ 70キ ァ805 ョ0705 ョ ッ0キ3 x 罍陦ァ ァ 0 04 縺 ァ タ0903 タ05 ァ. 7

繖 7 縺6ァ80キ3 ッ0キ3 ェ ュ ョ07 縺00 06 ュ0503 ュ ッ 70キ ァ805 ョ0705 ョ ッ0キ3 x 罍陦ァ ァ 0 04 縺 ァ タ0903 タ05 ァ. 7 30キ36ヲ0 7 7 ュ6 70キ3 ョ6ァ8056 50キ300 縺6 5 ッ05 7 07 ッ 7 ュ ッ04 ュ03 ー 0キ36ヲ06 7 繖 70キ306 6 5 0 タ0503070060 08 ョ0303 縺0 ァ090609 0403 閨0303 003 ァ 0060503 陦ァ 06 タ09 ァ タ04 縺06 閨06-0006003 ァ ァ 04 罍ァ006 縺03 0403

More information