f(x) = e x2 25 d f(x) 0 x d2 dx f(x) 0 x dx2 f(x) (1 + ax 2 ) 2 lim x 0 x 4 a 3 2 a g(x) = 1 + ax 2 f(x) g(x) 1/2 f(x)dx n n A f(x) = Ax (x R
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1 29 ( )
2 f(x) = e x2 25 d f(x) 0 x d2 dx f(x) 0 x dx2 f(x) (1 + ax 2 ) 2 lim x 0 x 4 a 3 2 a g(x) = 1 + ax 2 f(x) g(x) 1/2 f(x)dx n n A f(x) = Ax (x R n ) 25 A 2 = A A 2 A 2 = A t A = A A t A A 3 n = 2 A 2 = O A f Ker(f) m(f) 1
3 n (n > 2) x 1, x 2,, x n G A A k (i, j) (A k ) ij 3 G G 50 G A x l x l 3 A = G A 3 (A) ij (A 2 ) ij 0 i, j G 2
4 4 C C i, N i, P i, S i C = 10 i N i FUN 1 FUN 2 FUN 3 FUN 4 P i S i C ( ) C (j = 1, 2,, C) j (V 4 j ) 4 4 (V i j ) j = 1, 2,, 10 i = 1,, j Vj 1 0 Vj 2 0 Vj 3 0 Vj 4 0 3
5 2 3 4 i = 1 N 1 j(= 1, 2,,C) (V 1 j ) L j ( ) V 1 j 0 L j i = 2 N 1 N 2 j(= 1, 2,, C) (V 2 j ) L j 2 i = 3 N 1 N 2 N 3 j(= 1, 2,, C) (V 3 j ) L j 2 i = 4 N 1 N 2 N 3 N 4 j(= 1, 2,, C) (V 4 j ) L j 2 C 2 Vj i = Vj i 1 j j S i 0 P i + V i j S i j Vj i V j i P i + Vj S i i L j L j Si N i V k 0 = 0 (k = 1,, 4) 4
6 T1 10 T2 T1 T
7 29 ( ) CT
8 f(x) = e x2 25 d f(x) 0 x d2 dx f(x) 0 x dx2 f(x) (1 + ax 2 ) 2 lim x 0 x 4 a 3 2 a g(x) = 1 + ax 2 f(x) g(x) 1/2 f(x)dx n n A f(x) = Ax (x R n ) 25 A 2 = A A 2 A 2 = A t A = A A t A A 3 n = 2 A 2 = O A f Ker(f) m(f) 1
9 n (n > 2) x 1, x 2,, x n G A A k (i, j) (A k ) ij 3 G G 50 G A x l x l 3 A = G A 3 (A) ij (A 2 ) ij 0 i, j G 2
10 4 C C i, N i, P i, S i C = 10 i N i FUN 1 FUN 2 FUN 3 FUN 4 P i S i C ( ) C (j = 1, 2,, C) j (V 4 j ) 4 4 (V i j ) j = 1, 2,, 10 i = 1,, j Vj 1 0 Vj 2 0 Vj 3 0 Vj 4 0 3
11 2 3 4 i = 1 N 1 j(= 1, 2,,C) (V 1 j ) L j ( ) V 1 j 0 L j i = 2 N 1 N 2 j(= 1, 2,, C) (V 2 j ) L j 2 i = 3 N 1 N 2 N 3 j(= 1, 2,, C) (V 3 j ) L j 2 i = 4 N 1 N 2 N 3 N 4 j(= 1, 2,, C) (V 4 j ) L j 2 C 2 Vj i = Vj i 1 j j S i 0 P i + V i j S i j Vj i V j i P i + Vj S i i L j L j Si N i V k 0 = 0 (k = 1,, 4) 4
12 T1 10 T2 T1 T
13 29 ( )
14
15 50 Atkinson Shiffrin (1968) Craik Lockhart (1972) 250 Atkinson, R. C., & Shiffrin, R. M. (1968). Human memory: A proposed system and its control processes. n K. W. Spence & J. T. Spence (Eds.), The psychology of learning and motivation, Vol. 2. New York: Academic Press, pp Craik, F.. M., & Lockhart, R. S. (1972). Levels of processing: A framework for memory research. Journal of Verbal Learning and Verbal Behavior, 11, pp
16 pp
17 4 C C i, N i, P i, S i C = 10 i N i FUN 1 FUN 2 FUN 3 FUN 4 P i S i C ( ) C (j = 1, 2,, C) j (V 4 j ) 4 4 (V i j ) j = 1, 2,, 10 i = 1,, j Vj 1 0 Vj 2 0 Vj 3 0 Vj 4 0 4
18 2 3 4 i = 1 N 1 j(= 1, 2,,C) (V 1 j ) L j ( ) V 1 j 0 L j i = 2 N 1 N 2 j(= 1, 2,, C) (V 2 j ) L j 2 i = 3 N 1 N 2 N 3 j(= 1, 2,, C) (V 3 j ) L j 2 i = 4 N 1 N 2 N 3 N 4 j(= 1, 2,, C) (V 4 j ) L j 2 C 2 Vj i = Vj i 1 j j S i 0 P i + V i j S i j Vj i V j i P i + Vj S i i L j L j Si N i V k 0 = 0 (k = 1,, 4) 5
19 29 ( )
20 f(x) = e x2 25 d f(x) 0 x d2 dx f(x) 0 x dx2 f(x) (1 + ax 2 ) 2 lim x 0 x 4 a 3 2 a g(x) = 1 + ax 2 f(x) g(x) 1/2 f(x)dx n n A f(x) = Ax (x R n ) 25 A 2 = A A 2 A 2 = A t A = A A t A A 3 n = 2 A 2 = O A f Ker(f) m(f) 1
21 n (n > 2) x 1, x 2,, x n G A A k (i, j) (A k ) ij 3 G G 50 G A x l x l 3 A = G A 3 (A) ij (A 2 ) ij 0 i, j G 2
22 D z = x + iy w(z) = u(x, y) + iv(x, y) w u, v Cauchy-Riemann u x = v y u y = v x u, v, x, y R, i = 1 50 u, v 2 u x u y 2 = 0 2 v x v y 2 = 0 2 x = r cos θ, y = r sin θ Cauchy-Riemann u r = 1 r v θ v r = 1 r u θ 3 w u u = x 3 + axy 2 a w a v(1, 1) = 0 w v 3
23 4 C C i, N i, P i, S i C = 10 i N i FUN 1 FUN 2 FUN 3 FUN 4 P i S i C ( ) C (j = 1, 2,, C) j (V 4 j ) 4 4 (V i j ) j = 1, 2,, 10 i = 1,, j Vj 1 0 Vj 2 0 Vj 3 0 Vj 4 0 4
24 2 3 4 i = 1 N 1 j(= 1, 2,,C) (V 1 j ) L j ( ) V 1 j 0 L j i = 2 N 1 N 2 j(= 1, 2,, C) (V 2 j ) L j 2 i = 3 N 1 N 2 N 3 j(= 1, 2,, C) (V 3 j ) L j 2 i = 4 N 1 N 2 N 3 N 4 j(= 1, 2,, C) (V 4 j ) L j 2 C 2 Vj i = Vj i 1 j j S i 0 P i + V i j S i j Vj i V j i P i + Vj S i i L j L j Si N i V k 0 = 0 (k = 1,, 4) 5
25 29 ( )
26 f(x) = e x2 25 d f(x) 0 x d2 dx f(x) 0 x dx2 f(x) (1 + ax 2 ) 2 lim x 0 x 4 a 3 2 a g(x) = 1 + ax 2 f(x) g(x) 1/2 f(x)dx n n A f(x) = Ax (x R n ) 25 A 2 = A A 2 A 2 = A t A = A A t A A 3 n = 2 A 2 = O A f Ker(f) m(f) 1
27 n (n > 2) x 1, x 2,, x n G A A k (i, j) (A k ) ij 3 G G 50 G A x l x l 3 A = G A 3 (A) ij (A 2 ) ij 0 i, j G 2
28
29 4 C C i, N i, P i, S i C = 10 i N i FUN 1 FUN 2 FUN 3 FUN 4 P i S i C ( ) C (j = 1, 2,, C) j (V 4 j ) 4 4 (V i j ) j = 1, 2,, 10 i = 1,, j Vj 1 0 Vj 2 0 Vj 3 0 Vj 4 0 4
30 2 3 4 i = 1 N 1 j(= 1, 2,,C) (V 1 j ) L j ( ) V 1 j 0 L j i = 2 N 1 N 2 j(= 1, 2,, C) (V 2 j ) L j 2 i = 3 N 1 N 2 N 3 j(= 1, 2,, C) (V 3 j ) L j 2 i = 4 N 1 N 2 N 3 N 4 j(= 1, 2,, C) (V 4 j ) L j 2 C 2 Vj i = Vj i 1 j j S i 0 P i + V i j S i j Vj i V j i P i + Vj S i i L j L j Si N i V k 0 = 0 (k = 1,, 4) 5
36 3 D f(z) D z f(z) z Taylor z D C f(z) z C C f (z) C f(z) f (z) f(z) D C D D z C C 3.: f(z) 3. f (z) f 2 (z) D D D D D f (z) f 2 (z) D D f (z) f 2 (
3 3. D f(z) D D D D D D D D f(z) D f (z) f (z) f(z) D (i) (ii) (iii) f(z) = ( ) n z n = z + z 2 z 3 + n= z < z < z > f (z) = e t(+z) dt Re z> Re z> [ ] f (z) = e t(+z) = (Rez> ) +z +z t= z < f(z) Taylor
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0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
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5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0
I y = f(x) a I a x I x = a + x 1 f(x) f(a) x a = f(a + x) f(a) x (11.1) x a x 0 f(x) f(a) f(a + x) f(a) lim = lim x a x a x 0 x (11.2) f(x) x
11 11.1 I y = a I a x I x = a + 1 f(a) x a = f(a +) f(a) (11.1) x a 0 f(a) f(a +) f(a) = x a x a 0 (11.) x = a a f (a) d df f(a) (a) I dx dx I I I f (x) d df dx dx (x) [a, b] x a ( 0) x a (a, b) () [a,
4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X
4 4. 4.. 5 5 0 A P P P X X X X +45 45 0 45 60 70 X 60 X 0 P P 4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P 0 0 + 60 = 90, 0 + 60 = 750 0 + 60 ( ) = 0 90 750 0 90 0
( ) ( ) ( ) i (i = 1, 2,, n) x( ) log(a i x + 1) a i > 0 t i (> 0) T i x i z n z = log(a i x i + 1) i=1 i t i ( ) x i t i (i = 1, 2, n) T n x i T i=1 z = n log(a i x i + 1) i=1 x i t i (i = 1, 2,, n) n
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38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: 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
![(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](/thumbs/67/56612994.jpg "(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 ]
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y
017 8 10 f : R R f(x) = x n + x n 1 + 1, f(x) = sin 1, log x x n m :f : R n R m z = f(x, y) R R R R, R R R n R m R n R m R n R m f : R R f (x) = lim h 0 f(x + h) f(x) h f : R n R m m n M Jacobi( ) m n
(, ) (, ) S = 2 = [, ] ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) ( ) (
![(, ) (, ) S = 2 = [, ] ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) ( ) (](/thumbs/92/107866524.jpg "(, ) (, ) S = 2 = [, ] ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) ( ) (")
B 4 4 4 52 4/ 9/ 3/3 6 9.. y = x 2 x x = (, ) (, ) S = 2 = 2 4 4 [, ] 4 4 4 ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, 4 4 4 4 4 k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) 2 2 + ( ) 3 2 + ( 4 4 4 4 4 4 4 4 4 ( (
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (
![B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (](/thumbs/97/133036028.jpg "B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (")
. 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1
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1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
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