δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b
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- れいな やまがた
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1 n n r x, y, z ˆx ŷ ẑ 1 a a x ˆx + a y ŷ + a z ẑ a iˆx i i1 i j k e x e y e z 3 a b a i b i i 1, 2, 3 x y z ˆx i ˆx j δ ij, n a b a i b i a i b i a x b x + a y b y + a z b z i1 Einstein 2 a b θ ab cos θ 1 ˆx ˆx 1 ŷ ˆx 2 ẑ ˆx 3 2!
2 δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b y ŷ + b z ẑ a y b z a z b y ˆx + a z b x a x b z ŷ + a x b y a y b x ẑ ˆx ŷ ẑ a x a y a z b x b y b z c a b a b a b θ ab sin θ a b c ˆx ŷ ẑ i c i ϵ ijk a j b k ϵ ijk ϵ xyz ϵ yzx ϵ zxy 1 ϵ xzy ϵ zyx ϵ yxz 1 others ϵ ijk ϵ lmk ϵ ijk ϵ klm ϵ ijk ϵ mkl δ il δ jm δ im δ jl
3 a b b a, a b + c a b + a c, ca b a cb ca b, a b c b c a c a b, a b c ba c ca b a b c a b c 6 ϵ ijk a b c ϵ ijk a j ϵ klm b l c m δ il δ jm δ im δ jl a j b l c m a j b i c j a j b j c i ba c ca b [ ]. a b a i b j a x b x a x b y a x b z a y b x a y b y a y b z a z b x a z b y a z b z tensor 2.2 a t da dt lim t 0 at + t at t da dt dax dt, da y dt, da z dt
4 r r dr/dt a dv/dt d 2 r/dt 2 t ct at bt d dt d dt d dt ca dc dt a + cda dt, da db a b b + a dt dt, da db a b b + a dt dt [ ]. 3 s rs ˆt dr ds r lim s 0 s v dr dt dr ds ds dt ˆtv θ ˆn dˆt dθ dˆt ds ds dθ R dˆt ds 1 dˆt κ ds R ds/dθ κ 1/R ˆn ˆt a dv dt dv dt ˆt + v dˆt dθ ds dθ ds dt dv dt ˆt + v ˆn 1 dv v. R dt ˆt + v2 ˆn R r x, y, z r 2
5 f f x, f y, f z 3 m r f/m d 2 r dt 2 f m ṙ [ ] r field scalar field vector field ϕr ϕx, y, z 1 ϕ/x ϕ/y ϕ/z 3 /x /y /z ϕx, y, z ϕ ϕ x, y, ˆx z x + ŷ y + ẑ z Ar A x r, A y r, A z r A A ϕr ϕx, y, z grad ϕr ϕr, x ϕx, y, z, y ϕx, y, z z 3 f x f y f z f f
6 ϕr ϕr ϕr dϕr ϕr dr ϕr 2 ϕr 1 ϕr dr r 1 r 2 r 1 r 2 r 1 r 2 r m ϕr mgz fr ϕr 0, 0, mg mgẑ mg M r m ϕr G Mm r fr ϕr GMm x r 3, GMm y r 3, GMm z r 3 G Mm r r 2 r q r ϕr 1 4πϵ 0 q r Er ϕr 1 4πϵ 0 q r 2 r r divergence Ar Ar : divar Ar ˆx x + ŷ y + ẑ A x ˆx + A y ŷ + A z ẑ z A x x + A y y + A z z
7 ˆx ŷ ẑ ρr vr x, y, z x + x, y + y, z + z : M M jr ρrvr Mx, y, z, t + t Mx, y, z, t [ j x x, y y, z z + j xx + x, y y, z z y z + j y x x, y, z z + j yx x, y + y, z z z x + j z x x, y y, z + j zx x, y + 1 ] 2 y, z + z x y t jx x, y, z + j yx, y, z + j zx, y, z x y z t x y z x y z M/ x y z ρ ρx, y, z t + jx, y, z jr 1 fr 0, 0, g g, m g m fr GM r r 2 r GM ˆr, r2 fr m Er Er GM r 0. for r r3 q r 4πϵ 0 r 3, q r 0 4πϵ 0 r3 for r
8 [ ] vr 2 3 vr r, vr r r rotation Ar Ar rotation curl : rotar Ar ˆx x + ŷ y + ẑ A x ˆx + A y ŷ + A z ẑ z Az y A y Ax ˆx + z z A z Ay ŷ + x x A x ẑ y ˆx ŷ ẑ x y z A x A y A z [ ] Ar r, Ar By, 0, 0, Ar 0, Bx, 0, Ar 1 2 By, 1 2 Bx, 0, y vr x 2 + y, x y 2 x 2 + y, 0 2 ρ, x ρ, 0, vr 1 y x 2 + y 2 x 2 + y, 2 x x 2 + y 2, 0 y ρ 2, x ρ 2, x 0, y 0.z 0 x 0 + x, y 0.z 0 x 0 + x, y 0 + y.z 0 x 0, y 0 + y.z 0 x 0, y 0.z 0
9 r 0 r 0 + x ˆx r 0 + x ˆx + y ŷ r 0 + y ŷ r Ar r I Ar dr I Ar x ˆx x ˆx + Ar 0 + x ˆx + 1 y ŷ y ŷ 2 +Ar x ˆx + y ŷ x ˆx + Ar y ŷ y ŷ 2 Ay x A x x y y A ẑ x y A r 0 1 A z v vorticity z ϕ ω r vr v r sin θ sin ϕ ϕ, r sin θ cos ϕ ϕ, 0 ωy, ωx, 0 ωẑ r ω r ω ωẑ vr ˆx ŷ ẑ x y z ωy ωx 0 2ωẑ 2ω vr z z! z : z I Br µ 0I y 2π ρ, x ρ,
10 ρ x 2 + y 2 ρ 0 B 0 Laplacian Laplacian ϕ ϕ 2 ϕ ϕ ˆx x + ŷ y + ẑ ˆx z x + ŷ y + ẑ ϕ z 2 ϕ x ϕ y ϕ z x y z A x y z 2 A 2 A x x A x y A x z 2, 2 A y x A y y A y z 2, 2 A z x A z y A z z [ ] fr r x 2 + y 2, fr ln r ln x 2 + y ??? 4
11 ϕψ ϕψ + ϕ ψ, ϕa ϕ A + ϕ A, ϕa ϕ A + ϕ A, A B B A + A B, A B B A B A A B + A B, A B B A + A B + B A + A B, ϕ 0, A 0, A A 2 A [ ] x r cos ϕ, y r sin ϕ r x 2 + y 2, ϕ Arctan y x ˆr ϕ ˆϕ ˆr cos ϕ ˆx + sin ϕ ŷ, ˆϕ sin ϕ ˆx + cos ϕ ŷ ar 6 5 ϕ ϕr 6 ar a r r ˆr + a ϕ r ˆϕ 2.4.5
12 4.1: ar a x r ˆx + a y r ŷ a r ˆr a ˆr a x ˆx + a y ŷ a x cos ϕ + a y sin ϕ, a ϕ ˆϕ a ˆϕ a x ˆx + a y ŷ a x sin ϕ + a y cos ϕ f r ˆr + 1 r ϕ ˆϕ ˆr r + ˆϕ 1 f r ϕ f ˆx x + ŷ y ˆr cos ϕ ˆϕ r sin ϕ x r + ϕ x ϕ ˆr cos ϕ ˆϕ x sin ϕ r r sin ϕ r ϕ ˆr x cos ϕ + y sin ϕ r r + ˆϕ sin2 ϕ + cos 2 ϕ r + ˆr sin ϕ + ˆϕ r cos ϕ y + ˆr sin ϕ + ˆϕ cos ϕ ϕ ϕ/x sin ϕ/r ϕ/y cos ϕ/r r + ϕ y ϕ y r r + cos ϕ r ϕ
13 : ˆr r 0, A ˆr r + ˆϕ 1 r ϕ A r r + A r r + 1 r 1 r ˆϕ r 0, A rˆr + A ϕ ˆϕ A ϕ ϕ [ r ra r + A ϕ ϕ ˆr ϕ ˆϕ, ] ˆϕ ϕ 2 f f ˆr r + ˆϕ 1 r ϕ r ˆr + 1 r ϕ ˆϕ 2 f r r r f r 2 ϕ 2 1 r r r r ˆr r 2 2 f ϕ x ρ cos ϕ, y ρ sin ϕ, z z ˆρ ϕ ˆϕ z ẑ ˆr cos ϕ ˆx + sin ϕ ŷ, ˆϕ sin ϕ ˆx + cos ϕ ŷ, ẑ ẑ x r sin θ cos ϕ, y r sin θ sin ϕ, z r cos θ ˆρ ϕ ˆϕ z ẑ ˆr sin θ cos ϕ ˆx + sin θ sin ϕ ŷ + cos θ ẑ, ˆθ cos θ cos ϕ ˆx + cos θ sin ϕ ŷ sin θ ẑ, ˆϕ sin ϕ ˆx + cos ϕ ŷ
14 [ ] z ω vr ωẑ r v x, v y, v z ωy, ωx, v ρ, v ϕ, v z 0, ωρ, v r, v θ, v ϕ 0, 0, ωr sin θ ω ω x, ω y, ω z 0, 0, ω ω ρ, ω ϕ, ω z 0, 0, ω ω r, ω θ, ω ϕ ω cos θ, ω sin θ, f r ˆr + 1 r ˆr r + ˆθ 1 r θ ˆθ + 1 r sin θ ϕ ˆϕ f ϕ θ + ˆϕ 1 r sin θ A ˆr r + ˆθ 1 r θ + ˆϕ 1 r sin θ A r r + A r r + 1 A θ r θ + A θ A r r + 2A r 1 r 2 sin θ : ˆr r 0 ˆθ r 0 ˆr θ ˆθ ˆθ θ ˆr ˆr ϕ ˆϕ sin θ A rˆr + A θˆθ + Aϕ ˆϕ ϕ A r r sin θ sin θ + A θ r sin θ cos θ + 1 A ϕ ϕ r sin θ r + 1 r θ + cot θ r A θ + 1 A ϕ r sin θ ϕ [ r 2 sin θa r + r θ r sin θa θ + ] ϕ ra ϕ ˆθ ϕ ˆϕ cos θ ˆϕ r ˆϕ θ ˆϕ ϕ ˆr sin θ ˆθ cos θ A ˆr r + ˆθ 1 r θ + ˆϕ 1 r sin θ A rˆr + A θˆθ + Aϕ ˆϕ ϕ
15 A θ r ˆϕ A ϕ r ˆθ 1 A r r θ ˆϕ + 1 A ϕ r θ ˆr + A θ r ˆϕ + 1 A r r sin θ ϕ ˆθ 1 A θ r sin θ ϕ ˆr A ϕ ˆθ sin θ + ˆr cos θ r sin θ 1 A ϕ r θ + cos θa ϕ r sin θ 1 A θ 1 A r ˆr + r sin θ ϕ r sin θ ϕ A ϕ r A ϕ ˆθ r Aθ + r 1 A r r θ + A θ ˆϕ r [ 1 r sin θ θ sin θa ϕ 1 ] [ A θ 1 A r ˆr + r sin θ ϕ r sin θ ϕ 1 ] r r ra ϕ ˆθ [ 1 + r r ra θ 1 ] A r ˆϕ r θ 2 f f ˆr r + ˆθ 1 r θ + ˆϕ 1 r sin θ 2 f r r r f r 2 θ ϕ θ ˆθ + 1 r ˆr + 1 r r sin θ + 1 r 2 sin 2 θ r sin θ 1 2 f + r 2 sin 2 θ ϕ 2 2 f r r r f r 2 θ 2 + cos θ r 2 sin 2 θ θ f r 2 sin 2 θ ϕ [ 2 1 r 2 r 2 sin θ + sin θ + sin θ r r θ θ ϕ ϕ ˆϕ r sin θ θ cos θ 1 sin θ ]. ϕ f ρ ˆρ + 1 ρ ϕ ˆϕ + z ẑ ˆρ ρ + ˆϕ 1 ρ ϕ + ẑ f z A ˆρ ρ + ˆϕ 1 ρ ϕ + ẑ z 1 ρ ρ ρa ρ + 1 ρ 1 ρ : A ϕ ϕ + A z z [ ρ ρa ρ + A ϕ ϕ + z ρa z A ρˆρ + A ϕ ˆϕ + Az ẑ ] ˆρ ρ 0 ˆϕ ρ 0 ẑ ρ
16 ˆρ ϕ ˆϕ ˆϕ ϕ ˆρ ˆρ z 0 ˆϕ z 0 ẑ ϕ ẑ z A ˆρ ρ + ˆϕ 1 ρ ϕ + ẑ A ρˆρ + A ϕ ˆϕ + Az ẑ z 1 A z ρ ϕ A ϕ Aρ ˆρ + z z A z ˆϕ ρ 1 + ρ ρ ρa ϕ 1 A ρ ẑ ρ ϕ 2 f f ˆρ ρ + ˆϕ 1 ρ ϕ + ẑ z ρ ˆρ + 1 ρ ϕ ˆϕ + 2 f ρ ρ ρ f ρ 2 ϕ f z 2 1 ρ ρ ρ ρ z ẑ + 1 ρ 2 2 f ϕ f z [ ]
notekiso1_09.dvi
39 3 3.1 2 Ax 1,y 1 Bx 2,y 2 x y fx, y z fx, y x 1,y 1, 0 x 1,y 1,fx 1,y 1 x 2,y 2, 0 x 2,y 2,fx 2,y 2 A s I fx, yds lim fx i,y i Δs. 3.1.1 Δs 0 x i,y i N Δs 1 I lim Δx 2 +Δy 2 0 x 1 fx i,y i Δx i 2 +Δy
II 2 II
II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................
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)
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
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
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
sec13.dvi
13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r)
( : December 27, 215 CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x f (x y f(x x ϕ(r (gradient ϕ(r (gradϕ(r ( ϕ(r r ϕ r xi + yj + zk ϕ(r ϕ(r x i + ϕ(r y j + ϕ(r z k (1.1 ϕ(r ϕ(r i
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ
![R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ](/thumbs/80/82073891.jpg "R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ")
.3.2 3.3.2 Spherical Coorinates.5: Laplace 2 V = r 2 r 2 x = r cos φ sin θ, y = r sin φ sin θ, z = r cos θ.93 r 2 sin θ sin θ θ θ r 2 sin 2 θ 2 V =.94 2.94 z V φ Laplace r 2 r 2 r 2 sin θ.96.95 V r 2 R
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30
2.4 ( ) 2.4.1 ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) I(2011), Sec. 2. 4 p. 1/30 (2) Γ f dr lim f i r i. r i 0 i f i i f r i i i+1 (1) n i r i (3) F dr = lim F i n i r i. Γ r i 0 i n i
x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)
2011 I 2 II III 17, 18, 19 7 7 1 2 2 2 1 2 1 1 1.1.............................. 2 1.2 : 1.................... 4 1.2.1 2............................... 5 1.3 : 2.................... 5 1.3.1 2.....................................
Gmech08.dvi
51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
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
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
2.5 (Gauss) (flux) v(r)( ) S n S v n v n (1) v n S = v n S = v S, n S S. n n S v S v Minoru TANAKA (Osaka Univ.) I(2012), Sec p. 1/30
2.5 (Gauss) 2.5.1 (flux) v(r)( ) n v n v n (1) v n = v n = v, n. n n v v I(2012), ec. 2. 5 p. 1/30 i (2) lim v(r i ) i = v(r) d. i 0 i (flux) I(2012), ec. 2. 5 p. 2/30 2.5.2 ( ) ( ) q 1 r 2 E 2 q r 1 E
B ver B
B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
![II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re](/thumbs/94/118770263.jpg "II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re")
II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
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 ()
7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a
9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,
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
d dt P = d ( ) dv G M vg = F M = F (4.1) dt dt M v G P = M v G F (4.1) d dt H G = M G (4.2) H G M G Z K O I z R R O J x k i O P r! j Y y O -
44 4 4.1 d P = d dv M v = F M = F 4.1 M v P = M v F 4.1 d H = M 4.2 H M Z K I z R R J x k i P r! j Y y - XY Z I, J, K -xyz i, j, k P R = R + r 4.3 X Fig. 4.1 Fig. 4.1 ω P [ ] d d = + ω 4.4 [ ] 4 45 4.3
量子力学 問題
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79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
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SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ
SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )
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