(5) 75 (a) (b) ( 1 ) v ( 1 ) E E 1 v (a) ( 1 ) x E E (b) (a) (b)
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1 (5) 74 Re, bondar laer (Prandtl) Re z ω z = x
2 (5) 75 (a) (b) ( 1 ) v ( 1 ) E E 1 v (a) ( 1 ) x E E (b) (a) (b)
3 (5) 76 l V x ) 1/ 1 ( δ δ = x Re x p V x t V l l (1-1) 1/ 1 δ δ δ δ = x Re p V x t V l l (1- ) 1 1 = x V l (1-3) t V t = l v V Re l = (1-4) V 1 / = V x δ δ l / δ = δ l
4 (5) 77 δ 1 x1 1 v = x v δ 1 1 δ 1 x 1 x v 1 v δ v δ x v δ x v 1 δ v δ v 1 δ
5 (5) 78 Re 1 Re δ v l l (1-5) δ p δ 1 p = v x ρ x (1-6) p = (1-7) = x (1-8) 1 p = (1-9) ρ x x =, = ( = ) ( x) ( ) (1-1) p = λρ (1-11) x λ λ 1/ l
6 (5) 79 v x
7 (5) 8 x η ψ, ψ = f n (η) (1-1) 1/ η = = R x (1-13) vx / x x R x = v ψ ( x, ) = v x f ( η) (1-14) ψ n ψ ψ η = = = f ( η) η ψ 1 v = = { ηf ( η) f ( η) } x x η η = = f ( η) x η x x η = = f ( η) η vx η = ( η) = f η vx vx (1-15) f ( η)
8 (5) 81 / 1 x /
9 (5) 8 1 p = v x ρ x = v (1-16) x f ff = (1-17) η f f (1-18) η f v =.865 (1-19) x
10 (5) 83 l 1-5 f, f, f 7 = Rx 1/ x = Rx 1/ x (a).4.8 (b) R x 1/
11 (5) 84 δ(x) = f ( η) =. 99 (11-1) η η δ = 5. vx = 5.xR 1/ x (11- ) = [ ) ] q ( d (11-3) * δ * δ = * δ = q (11-4) 1 d * δ (displacement thickness) vx x * δ = 1.73 = / R x (11-5)
12 (5) () d
13 [ ] (5) 86 ρ d ρ θ (momentm thickness) θ = 1 d (11-6) vx x θ =.664 =.664 1/ R x (11-7) l D l D = µ dx =.664 µρl = D( l) C f =, ( A = bl) (1/ ) ρa (11-8) C f = (11-9) R l R l = l / v Rl Rl
14 (5) 87 l (Dail-Harleman, Addison-Wesle )
15 (5) 88 d p / d x (adverse pressre gradient) (separation) p = const. ρ PA A = ρ PC PB 1 = ρ ρ PB = ρ ( ) B C B = P C ρ C (11-1) (11-11) E = p ρ A A = p ρ B B E A B = p C ρ C E A C (11-1)
16 (5) 89 (a) (b) (c) A A B B C C D D
17 (5) 9 C p pb pc B = EB C (11-13) p = ρ ρ 1 C B ( B C ) B C ρ E (11-14) d p / d x =, ( = ) (bondar laer control)
18 (5) 91 = = h x h h h h h 1 p d d v d d t x = ν ρ x (1-1) h h d = d x h d v = x (1- ) h 1 p = d d d ρ x t x (1-3) τ µ / ( / ) = h = ( ) = µ τ ν = = (1-4) ρ ρ = h h h v d = d d x (1-5) h h = d d x x h h h = h 1 p τ d d d d (1-6) t x x ρ x ρ δx h h dδx δx d vδx t x =
19 (5) 9
20 (5) 93 x x δx h ( ) d [ ] d τ ( ) ( ) d = (1-7) t x x ρ h x d a da db a f ( x, α) dx = f ( a, α) f ( b, α) b fdx dα dα dα b α (1-8) ( δ * ) * τ ( θ ) δ = t x x ρ (1-9) = : = = : =, =
21 (5) 94 B v d x = D δx x A C x
22 (5) 95 d/dx t dθ ( x) τ ( x) = d x ρ (1-1) θ (x) τ ( x) = f (η ) (1-11) η = / δ ( x) 1 θ = ( )d = δ ( x) f (1 f )dη α1δ ( x) = (1-1) α = 1 f ( 1 f ) dη (1-13) τ ν ( / ) ν = ν = = β1 ρ δ η δ = η = (1-14) β1 = f () (1-15) * δ f * δ = α δ ( x) (1-16) α ( 1 f ) η (1-17) = 1 d dδ β1 ν δ = d x α1 (1-18) x =, δ = δ ( x) = β1 νx α1 (1-19) * δ * β1 νx δ = α α1 (1- )
23 (5) 96 θ = α1 β1 νx α1 (1-1) D(l) C f ( = D( l) /{ bl( ρ / )}) α1β1 x τ ( x) = µ x ν (1- ) 3 D ( l) = b α 1β1 µρl (1-3) C f α 1β1 = l / ν (1-4)
24 (5) 97 f ( η) = η ( < η < 1) (1-5) f ( ) = f ( 1) = 1 f (1) = = 1 ( < η < 1) f ( η) = ( η = 1) (1-6) 1 1 α 1 = η(1 η)dη = α = (1 η)dη = (1-7) β 1 = f () = 1 * νx δ = (1-8) C f ν = (1-9) l
25 (5) 98 3
26 (5) f ( η) = A A1η Aη A3η A4η (1-3) = : = ( f () = ) 1 d p d = : v = = ( f () = ) ρ d x d x = δ : = ( f (1) = 1) = δ : = ( f (1) = ) = δ : = ( f (1) = ) (1-31) A =, A1 =, A =, A3 =, A4 = 1 (1-3) 3 4 f ( η ) = η η η (1-33) α f (1 f )dη = d d = 1 f η f η β 1 = f () = (1-34) α = (1 f )dη = 1 d = f η 1 = x = * δ (1-35) C f = 4 37 ν 315 l (1-36)
27 (5) 1 1-1
28 (5) 11 Re = D / ν = D ν Re C < Re < 1 C
29 (5) 1
30 (5) 13 = v = v v (13-1) w = w w, v, w, v, w = v = v (13- ) w = ρ v d A ρ v d A τ = ρv (13-3) σxx τx τxz ρ ρv ρw τx σ τz = ρv ρvv ρvw τzx τz σ zz ρw ρwv ρww (13-4) p = p p ρ ρ µ ρ v w p v w = Fx t x z x x z ρ v ρ µ ρ v v v w v p v v vw = F v t x z x z (13-5) ρ w ρ µ ρ w v w w w p w vw w = Fz w t x z z x z
31 (5) 14
32 (5) 15 ρνd /d ρ v = ρν T d /d (13-6) ν ν T T ν d v l (13-7) d d d ρ v = ρl d d (13-8) d ν T = l d l l = k (13-9) ρv = ρk d d d d (13-1)
33 (5) 16 τ ρ τ / ρ 14-1 * = * η η * = 14- ν ( ) * = fn 14-3 * ν max ( ) = fn 14-4 * a
34 (5) 17 l E (a) l E (b) (Dail-Harleman, Addison-Wesle)
35 (5) 18 d τ τ = µ 14-5 d ( ) * = 14-6 * ν τ l = k τ τ τ = τ = const. l = k τ d = ( k) d ρ * = τ / ρ d * = d k 14-7 ( ) 1 = ln C1 * k 14-8 ( ) 1 * = ln AS * k ν 14-9 AS k =.4 AS ( ) * = 5.75log1 * ν 14-1
36 (5) 19 log S ( ) (a)
37 (5) 11 * 4 < < ν 4 < * < ν * < ν = a max 1 * a = ln AS * k ν 14-1 max ( ) 1 a = ln * k 14-13
38 (5) 111
39 (5) 11 ν ks ( ) = f n * ks ( ) 1 = ln Ar * k ks Ar ks Ar ( ) = 5.75log * k S ks
40 (5) 113 k S * ν * ks /ν k S δ S k S / δ S * ks /ν δ S / k S * ks /ν δ S / k S * ks /ν δ S / k S * ks /ν ( ) 1 * ks = ln A * k ks ν A * ks /ν A = A ln( * k S / ν ) / n ( ) = max a 5 Re 1 n 5 Re > 1 n n = log1 Re 14-1 n
41 (5) 114 * * = 1 ln As x ν * = 1 ln Ar x ks = * * S s a δ S A (Schlichting, McGraw-Hill)
42 (5) 115 L ( p1 p )( π ) τ ( π L) = a a ( p1 p )( π ) ( τ π L) p1 p a d p τ = = 14-1 L d x τ D = a a d p D d p τ = = 14- d x 4 d x f = Q /( π ) D p1 p L = f D 1 ρ a 14-3 d p 1 1 f = / ρ d x D 14-4 (14-1) τ * f = 8 = 8 ρ 14-5 max ( ) 1 a = ln ( ) = π ( a ) d * k πa ( max ) π a a = ln ( a )d * k πa = k = 3.75πa ( 1 ξ ) a 1 ln dξ ξ ( ξ = / a) 14-6 = max * 14-7
43 (5) 116 p 1 p max 1 * a = ln A S * k ν max 1 * a = ln k ν * ( ) 1 (14-1) (14-15) = ln Ar * k ks a = A max *.5ln r ks 14-9 / * a =.5ln * k S 14-3 f 1 a =.log1 f f ν 1 ks =.log f D 14-3 f k S / D f =.3164 Re / 4 Re = f =.3.1Re Re = 14-34
44 { log ( Re ). } f = 1/ f Re = 1 8 (5) /Re, (7-1) ( ) (14-1) f n (Re), (14-31) (14-16) f n (k s /D), (14-3) ( ) ( ) ( ) ( ) ( )
45 (5) 118 Re C 4 Re C 4
46 (5) 119
47 (5) 1 x p p = = x (15-1) τ t x v ρ (15- ) v v = x (15-3) τ ρv = ρl ε = l d d d d d d (15-4) x b τ ρε = (15-5) ( ε = kb 1 max min) (15-6) ε R ε b ( ) max min R ε = ε 1 = = const (15-7) k 1
48 - (5) 11 min v =ε x (15-8) ε = kb 1 max (15-9) x = s S b S x max x max ( x) = S f1( x/ s) m bx ( ) = bs f ( x/ s) = bs( x/ s) (15-1)
49 (5) 1
50 (5) 13 η φ( x, ) η = bx ( ) (15-11) Ψ( x, ) = f( x) ψ ( η ) P f ( x) = bss( x/ s) (15-1) dp / dx = J = ρ = ρ = ρb S = const ( x, )d b( x) S ( x / s) ( x, η)dη P m { ψ ( η) } dη (15-13) ε S s ( ψ ) ψψ ψ = b S S (15-14) = : / =, v = : η = : ψ =, ψ = 1 (15-15) : ψ b S b ε σ = = S S 4k1 = s s S (15-16) η ψ ( η) = tanhη (15-17)
51 (5) 14 z z z e e z e e z z z z z cosh sinh tanh cosh sinh = = =
52 S, v 3 J S = ρσs 3 J = ( 1 tanh η) ρσx 3 Jσ v = { η( 1 tanh η) tanhη} 4 ρx = 1 η σ x (5) 15 (15-18) σ σ = 1/ 767. (15-19) b 1 / b b tanh / = b b = 115. b 1 / (15- ) ε = kb 115. σ = b1 4 =. 37b 1 max / max 1 / max (15-1)
53 (5) 16 (form drag), pressre drag (frictional drag)
54 (5) 17 (a) (b) (c) L R D
55 (5) 18 D D f D p D = D f D p ρ ρ ρ (16-1) = C f A C p A = CD A C f C p C D A da p wda da pda pdacos da wda wda sin D p = p cosθda (16- ) A D τ sinθda (16-3) f = A w C D
56 (5) 19 da w da w da sin pda 16-1 (Re = ) (Re = ) C D [ (1976) 4-4 p.8 3] 16- (Re = ) C D C D sting spport.47 separation cbe.8 cbe )
57 (5) 13 d Red/ν (16-4) Re Re Re Re áá Re áá Re Re Re
58 (5) 131
59 (5) 13 p D C D = (16-5) 1 ρ d D Re 1 Re 1 Re 1 5 C D 11. D 5 Re 1 6 5
60 (5) Re < < Re < Re > 3 1 6
61 (5) 134 x x x x x t l l tt Y(T) l (t) Y ( T ) = T l ( t) dt 17-1) t tt (17- ) Y ( T ) = T l ( T ξ) dξ 17-3) T dy l ( T ) = dt 17-4) C l Y CY(C/) l Y(C/) F C = ( ly ) 17-5) = l Y 17-6) l Y T ( T ) Y ( T ) = ( T ) ( T ξ) dξ 17-7) l l l l (T) ( T ) Y ( T ) ( T ) ( T ξ) dξ 17-8) l = T l l
62 (5) 135 v l (t) Y (t) v l (o)
63 (5) 136 l (T)Y(T) l Y dy 1 dy ly = Y = dt dt 17-9) (ensemble mean) l ( T ) Y( T ) 1 dy T = l Y = = l ( T ) l ( T ξ) dξ dt 17-1) 1 d Y T = l Rl ( ξ ) dξ dt 17-11) R ξ ) = ( T ) ( T ξ) / 17-1) l ( l l l R l () Y T η Y = R ( ξ ) dξ dη 17-13) l l one particle analsis
64 (5) 137 C C C Y Y C C
65 (5) 138 R l, R l 1 T R 1 dy K = = l T, dt ( T ) 17-14) Y = l T, ( T ) 17-15) T T Rl ( ξ ) dξ = const = T* 17-16) 1 dy K = = l T* ( = const), ( T >> T* ) dt 17-17) Y = l T T, ( T >> T ) 17-18) * * T * ( ξ ) = exp ξ / T 17-19) ( ) R l * 1 dy T K = = l T* 1 exp dt T* 17- ) T T Y = T l * 1 exp T* T* R l () T TT * T x Tx/ TT * xt * x l T * T *
66 (5) D=T 1. D T a b c
67 (5) 14 ( ) Re 9/4 = = = z w w v x w w z p F z w w w v x w t w z w v v x v v p F z v w v v x v t v z w v x x p F z w v x t z x ρ µ ρ ρ ρ µ ρ ρ ρ µ ρ ρ 18-1) τ ρv = d /d ρν v T /d d ρν ρ = (18- ) T ν T ν K ε ν µ K C T = (18-3) µ C T ν K K- µ C
68 (5) 141 LES(large edd simlation) LES DNS(direct nmerical simlation) DNS
69 (5) 14 present work;, R 8, R 65, R 39, R 5 slope -5/3 others;, R (Grant, et al., 196), R 1(, 1995), R 318(Karakin et al., 1991) NASA, R 15 (Saddoghi & Veeravalli, 1994), R 85 (Coantic & Favre, 1974), R 41 (Sanborn & Marshall, 1965), R 38 (beroi & Fremth, 1969), R 13 (Champagne, 197), R 54 (Kistler & Vrebalovich, 1966), R 7, R 37 (Comte-Bellot & Corrsin, 1971) Saddoghi & Veeravalli(1994) (Flohr, 1999)
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,
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211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
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[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
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t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4
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201711grade1ouyou.pdf
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W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
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9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
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
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Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
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Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
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II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
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量子力学 問題
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構造と連続体の力学基礎
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5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E
5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E 2, S 1 N 1 = S 2 N 2 2 (chemical potential) µ S N
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
p = mv p x > h/4π λ = h p m v Ψ 2 Ψ
II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
1. A0 A B A0 A : A1,...,A5 B : B1,...,B
1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5. A0 A, B Z Z m, n Z m n m, n A m, n B m=n (1) A, B (2) A B = A B = Z/ π : Z Z/ (3) A B Z/ (4) Z/ A, B (5) f : Z Z f(n) = n f = g π g : Z/ Z A, B (6)
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
K E N Z OU
K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
: , 2.0, 3.0, 2.0, (%) ( 2.
2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................
20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
2 1 x 2 x 2 = RT 3πηaN A t (1.2) R/N A N A N A = N A m n(z) = n exp ( ) m gz k B T (1.3) z n z = m = m ρgv k B = erg K 1 R =
1 1 1.1 1827 *1 195 *2 x 2 t x 2 = 2Dt D RT D = RT N A 1 6πaη (1.1) D N A a η 198 *3 ( a =.212µ) *1 Robert Brown (1773-1858. *2 Albert Einstein (1879-1955 *3 Jean Baptiste Perrin (187-1942 2 1 x 2 x 2
I 1
I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
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
IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
![18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb](/thumbs/93/114079295.jpg "18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb")
r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)
液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k
![4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k](/thumbs/94/119235351.jpg "4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k")
4.6 (E i = ε, ε + ) T Z F Z = e ε + e (ε+ ) = e ε ( + e ) F = kt log Z = kt loge ε ( + e ) = ε kt ln( + e ) (4.8) F (T ) S = T = k = k ln( + e ) + kt e + e kt 2 + e ln( + e ) + kt (4.20) /kt T 0 = /k (4.20)
S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
QMII_10.dvi
65 1 1.1 1.1.1 1.1 H H () = E (), (1.1) H ν () = E ν () ν (). (1.) () () = δ, (1.3) μ () ν () = δ(μ ν). (1.4) E E ν () E () H 1.1: H α(t) = c (t) () + dνc ν (t) ν (), (1.5) H () () + dν ν () ν () = 1 (1.6)
chap9.dvi
9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =
. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
LLG-R8.Nisus.pdf
d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
振動と波動
Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3