1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h
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1 IB IIA 1 1 r, θ, φ 1 (r, θ, φ)., r, θ, φ 0 r <, 0 θ π, 0 φ < 2π., θ, φ, φ, r, r, θ r, θ, φ,., r, θ, φ,,,. r, θ, φ r, θ, φ, 3 (r, θ, φ)., r, θ, φ e r, e θ, e φ., e r, e θ, e φ e r, e θ, e φ.,., (r, θ, φ) e r (r, θ, φ), e θ (r, θ, φ), e φ (r, θ, φ), (r, θ, φ) e r, e θ, e φ., 2 (r, θ, φ) dr (r, θ, φ) (r dr, θ dθ, φ dφ), dr h r dr e r h θ dθ e θ h φ dφ e φ (1). h r, h θ, h φ h r 1, h θ r, h φ r sin θ (2). r, θ, φ, h r (r, θ, φ), h θ (r, θ, φ), h φ (r, θ, φ), h r, h θ, h φ. 1
2 1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h φ dφ 1 e φ dr 2 h r dr 2 e r h θ dθ 2 e θ h φ dφ 2 e φ (3) 2
3 z r sinθ dϕ eϕ r dθ eθ ds dθ θ r x 0 ϕ r sinθ dϕ r sinθ dϕ y 2 ds r dθ eθ x r sinθ dϕ eϕ r sinθ dθ dϕ er 3 ( ). z dr er r sinθ dϕ eϕ θ dθ r r dθ eθ x 0 ϕ dϕ y 2 dv r sinθ dr dθ dϕ 4. ds dr 1 dr 2 h θ h φ dθ 1 dφ 1 dθ 2 dφ 2 e dφ r h φ h 1 dr 1 r dφ 2 dr 2 e dr θ h r h 1 dθ 1 θ dr 2 dθ 2 e φ (4).. 3
4 ,., r. dθ, dφ 3 2 dr 1 h θ dθ e θ r dθ e θ dr 2 h φ dφ e φ r sin θ dφ e φ (5). 2 ds h θ dθ e θ h φ dφ e φ h θ h φ dθ dφ e r r 2 sin θ dθ dφ e r (6)., ds n ds n e r, ds h θ h φ dθ dφ r 2 sin θ dθ dφ.,, θ, ds h φ dφ e φ h r dr e r h φ h r dφ dr e θ r sin θ dφ dr e θ (7),,, φ, ds h r dr e r h θ dθ e θ h r h θ dr dθ e φ r dr dθ e φ (8)., dr, dθ, dφ. 4, dr, dθ, dφ (r, θ, φ) 3, dr 1 h r dr e r dr e r dr 2 h θ dθ e θ r dθ e θ dr 3 h φ dφ e φ r sin θ dφ e φ (9) dv dr 1 (dr 2 dr 3 ) h r dr e r (h θ dθ e θ h φ dφ e φ ) h r h θ h φ dr dθ dφ r 2 sin θ dr dθ dφ (10). 4
5 2., r. V V V r (r, θ, φ)e r V θ (r, θ, φ)e θ V φ (r, θ, φ)e φ (11)., Γ {(r, θ, φ) a r b, θ θ 0, φ φ 0 } (12)., Γ (a, θ 0, φ 0 ) (b, θ 0, φ 0 ). Γ θ, φ,, dθ 0, dφ 0, Γ dr h r dr e r (13)., Γ (r, θ 0, φ 0 ) V dr V dr V r (r, θ 0, φ 0 )h r (r, θ 0, φ 0 ) dr (14), Γ V Γ V dr b a b a V r (r, θ 0, φ 0 )h r (r, θ 0, φ 0 ) dr V r (r, θ 0, φ 0 ) dr (15)., V r cos θ cos φ e r r 2 sin θ sin φ e θ r 3 tan θ tan φ e φ (16) Γ V dr b a r cos θ 0 cos φ 0 dr 1 2 (b2 a 2 ) cos θ 0 cos φ 0 (17). r, θ Γ {(r, θ, φ) r r 0, a θ b, φ φ 0 } (18) 5
6 V dr Γ b a b a V θ (r 0, θ, φ 0 )h θ (r 0, θ, φ 0 ) dθ V θ (r 0, θ, φ 0 )r 0 dθ (19)., φ Γ {(r, θ, φ) r r 0, θ θ 0, a φ b} (20) V dr Γ b a b a V φ (r 0, θ 0, φ)h φ (r 0, θ 0, φ) dφ V φ (r 0, θ 0, φ)r 0 sin θ 0 dφ (21).., Γ., p r r(p), θ θ(p), φ φ(p) (22) Γ (r(p), θ(p), φ(p)). V V V r (p)e r V θ (p)e θ V φ (p)e φ (23)., V r (r(p), θ(p), φ(p)) V r (p)., dr dr dp dp, dθ dθ dp dp, dφ dφ dp (24) dp, Γ { dr h r (p) dr dp e r h θ (p) dθ dp e θ h φ (p) dφ } dp e φ dp (25) 6
7 ., h r (r(p), θ(p), φ(p)) h r (p)., Γ (r(p), θ(p), φ(p)) V dr { V dr V r (p)h r (p) dr dp V θ(p)h θ (p) dθ dp V φ(p)h φ (p) dφ } dp (26) dp, Γ V { V dr V r (p)h r (p) dr Γ Γ dp V θ(p)h θ (p) dθ dp V φ(p)h φ (p) dφ } dp dp { V r (p) dr dp V θ(p)r(p) dθ dp V φ(p)r(p) sin θ(p) dφ } dp dp Γ (27)... V V V r (r, θ, φ)e r V θ (r, θ, φ)e θ V φ (r, θ, φ)e φ (28)., Σ {(r, θ, φ) r r 0, a θ b, c φ d} (29)., Σ r. 1 (6), ds h θ h φ dθ dφ e r (30)., Σ (r 0, θ, φ) V ds V ds V r (r 0, θ, φ)h θ (r 0, θ, φ)h φ (r 0, θ, φ) dθ dφ (31), Σ V Σ V ds d b c a d b c a V r (r 0, θ, φ)h θ (r 0, θ, φ)h φ (r 0, θ, φ) dθ dφ V r (r 0, θ, φ)r 2 0 sin θ dθ dφ (32) 7
8 ., V r cos θ cos φ e r r 2 sin θ sin φ e θ r 3 tan θ tan φ e φ (33) Σ V ds d b c a r 3 0 cos θ sin θ cos φ dθ dφ 1 2 r3 0(cos 2 a cos 2 b)(sin d sin c) (34)., Σ {(r, θ, φ) c r d, θ θ 0, a φ b} (35) V ds Σ d b c a d b c a V θ (r, θ 0, φ)h φ (r, θ 0, φ)h r (r, θ 0, φ) dφ dr V θ (r, θ 0, φ)r sin θ 0 dφ dr (36)., Σ {(r, θ, φ) a r b, c θ d, φ φ 0 } (37) V ds Σ d b c a d b c a V φ (r, θ, φ 0 )h r (r, θ, φ 0 )h θ (r, θ, φ 0 ) dr dθ V φ (r, θ, φ 0 )r dr dθ (38).., Σ., p, q r r(p, q), θ θ(p, q), φ φ(p, q) (39) 8
9 Σ (r(p, q), θ(p, q), φ(p, q)). V V V r (p, q)e r V θ (p, q)e θ V φ (p, q)e φ (40)., V r (r(p, q), θ(p, q), φ(p, q)) V r (p, q)., 1 (4) θ dr 1 dp, dθ 1 dp, dφ 1 p q p q p θ dr 2 dq, dθ 2 dq, dφ 2 q q q p p q p dp dq (41), Σ { (θ, φ) ds h θ (p, q)h φ (p, q) (p, q) e (φ, r) r h φ (p, q)h r (p, q) (p, q) e θ (r, θ) h r (p, q)h θ (p, q) (p, q) e φ } dp dq (42)., h r (r(p, q), θ(p, q), φ(p, q)) h r (p, q)., α (α, β) (p, q) p β p q q α q β q., Σ (r(p, q), θ(p, q), φ(p, q)) V ds { (θ, φ) V ds V r (p, q)h θ (p, q)h φ (p, q) (p, q) (φ, r) V θ (p, q)h φ (p, q)h r (p, q) (p, q) } (r, θ) V φ (p, q)h r (p, q)h θ (p, q) dp dq (p, q) p p (43) (44) 9
10 , V Σ { (θ, φ) V ds V r (p, q)h θ (p, q)h φ (p, q) Σ Σ (p, q) (φ, r) V θ (p, q)h φ (p, q)h r (p, q) (p, q) } (r, θ) V φ (p, q)h r (p, q)h θ (p, q) dp dq (p, q) { V r (p, q)r 2 (θ, φ) (p, q) sin θ(p, q) Σ (p, q) (φ, r) V θ (p, q)r(p, q) sin θ(p, q) (p, q) } (r, θ) V φ (p, q)r(p, q) dp dq (p, q). (45) Ω f. 1, (10)., f dv Ω. Ω Ω f(r, θ, φ) h r (r, θ, φ)h θ (r, θ, φ)h φ (r, θ, φ) dr dθ dφ (46) f(r, θ, φ) r 2 sin θ dr dθ dφ 3 f f f dr df (47)., dr. dr t ds f t df ds 10 (48)
11 . (48) f t. (47), (r, θ, φ) f f 1 f h r f e r 1 f e θ 1 f e φ (49) h θ θ h φ f e r 1 f e θ 1 f e φ (50) r θ r sin θ. (49). f r, θ, φ r e r h r, θ e θ h θ, φ e φ h φ (51)., f f f r θ θ f φ (52)., f (49). (r, θ, φ) f dr h r dr f e r h θ dθ f e θ h φ dφ f e φ (53). f f df dr f f dθ dφ (54) θ,, r f e r 1 h r ( f ) (55)., θ, φ, f (49). V V ( V ) S 11 σ V dr (56)
12 ., σ, S. S n S ( V ) n 1 S σ V dr (57). (57) V n. (56), (r, θ, φ) V { V 1 ( Vφ ) h φ h θ h φ θ { 1 ( Vr ) h r h φ h r { 1 ( Vθ ) h θ h r h θ } Vθ h θ } Vφ h φ } Vr h r θ e r e θ e φ (58) { ( Vφ ) } 1 r sin θ Vθ r V r 2 sin θ θ { 1 ( Vr ) } Vφ r sin θ e θ r sin θ { 1 ( Vθ ) } r Vr e φ r θ e r (59). (58). V e r h θ h φ e θ h φ h r θ e φ h r h θ V r h r V θ h θ V φ h φ (60). f (49) ( f) 0 (61) 12
13 .,. V, f f(r, θ, φ) r r 0 V r (r, θ 0, φ 0 )h r (r, θ 0, φ 0 ) dr θ φ V θ (r, θ, φ 0 )h θ (r, θ, φ 0 ) dθ V φ (r, θ, φ )h φ (r, θ, φ ) dφ θ 0 φ 0 (62), V V f., V (58)., (56) σ 5 γ 1 : A B, γ 2 : B C, γ 3 : C D, γ 4 : D A (63) 4., σ σ γ 1 γ 4., γ 1, γ 3., γ 1 t e θ, γ 3 t e θ V dr V dr γ 1 γ 3 θ0 θ θ 0 {v θ (r 0, θ, φ 0 ) v θ (r 0, θ, φ 0 φ)} dθ., v θ V θ h θ., v θ (r 0, θ, φ 0 φ) φ 2 vθ v θ (r 0, θ, φ 0 φ) v θ (r 0, θ, φ 0 ) φ (65) 0 r 0 θ (64) θ0 θ vθ V dr V dr φ dθ γ 1 γ 3 θ 0 0 r 0 θ vθ Vθ h θ θ φ θ φ 0 r 0 θ 0 0 r 0 θ 0 (66)., 2 θ θ θ 0., γ 2, γ 4., γ 2 t e φ, γ 4 t e φ 13
14 V dr V dr γ 2 γ 4 φ0 φ φ 0 {v φ (r 0, θ 0 θ, φ) v φ (r 0, θ 0, φ)} dφ (67)., v φ V φ h φ., v φ (r 0, θ 0 θ, φ) θ 2 vφ v φ (r 0, θ 0 θ, φ) v φ (r 0, θ 0, φ) θ (68) θ 0 0 φ0 φ vφ V dr V dr θ dφ γ 2 γ 4 φ 0 θ 0 0 vφ Vφ h φ θ φ θ φ θ 0 φ 0 r 0 θ 0 φ 0 r 0 (69)., 2 φ φ φ 0., σ σ σ { ( Vφ ) h φ V dr θ 0 φ 0 r 0 Vθ h θ 0 r 0 θ 0 } θ φ (70)., σ n e r, S h θ h φ θ φ, (57), ( V ) e r 1 h θ h φ { ( Vφ ) h φ θ 0 φ 0 r 0 Vθ h θ., ( V ) e θ, ( V ) e φ, r 0, θ 0, φ 0 r, θ, φ, V (58). 0 r 0 θ 0 } (71) V V V V 14 ω V ds (72)
15 z 2 S r sinθ θ ϕ er A D B x S C θ 0 ϕ0 A( r0, θ0, ϕ0 ) B( r0, θ0 θ, ϕ0 ) C( r0, θ0 θ, ϕ0 ϕ) D( r0, θ0, ϕ0 ϕ) θ0 ϕ r0 y 5 V r. z θ θ0 r0 r σ2 x σ1 0 ϕ0 σ4 σ3 ϕ σ6 σ5 y 6 V. 15
16 ., ω V, ω ω. (72), (r, θ, φ) V { ( Vr ) } 1 h θ h φ Vθ h φ h r Vφ h r h θ V (73) h r h θ h φ θ V 1 r 2 sin θ { ( Vr r 2 ) sin θ Vθ r sin θ θ } Vφ r (74). (73). V (58) ( V ) 0 (75).,., V (73)., (72) ω 6 σ 1 : (r 0 r, θ, φ ) σ 2 : (r 0, θ, φ ) σ 3 : ( r, θ 0 θ, φ ) σ 4 : ( r, θ 0, φ ) σ 5 : ( r, θ, φ 0 φ) σ 6 : ( r, θ, φ 0 ) r 0 r r 0 r, θ 0 θ θ 0 θ, φ 0 φ φ 0 φ (76) 6., ω ω σ 1 σ 6., σ 1 σ 2., σ 1 n e r, σ 2 n e r V ds V ds σ 1 σ 2 φ0 φ θ0 θ φ 0 θ 0 {v r (r 0 r, θ, φ) v r (r 0, θ, φ)} dθ dφ., v r V r h θ h φ., v r (r 0 r, θ, φ) r 2 vr v r (r 0 r, θ, φ) v r (r 0, θ, φ) r (78) 0 (77) 16
17 φ0 φ θ0 θ vr V ds V ds r dθ dφ σ 1 σ 2 φ 0 θ 0 0 vr r θ φ 0 θ 0 φ ( 0 ) Vr h θ h φ r θ φ 0 θ 0 φ 0 (79)., 2 θ, φ θ θ 0, φ φ 0. σ 3 σ 4, σ 5 σ 6 V ds ω { ( Vr ) } (80) h θ h φ Vθ h φ h r Vφ h r h θ r θ φ 0 θ 0 0 θ 0 φ 0 φ 0 r 0., V h r h θ h φ r θ φ, r 0, θ 0, φ 0 r, θ, φ, V (73). r 0 θ 0 17
18 { 2 1 h θ h φ f f h r h θ h φ h r 1 f r 2 r2 ( h φ h r f θ h θ θ 1 r 2 sin θ ) ( h r h θ h φ ) ( f sin θ θ θ ) } f 1 r 2 sin 2 θ ( 2 ) f 2 (81). 18
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
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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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
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.............................
A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν 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
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν 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](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν 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")
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
1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8..............................
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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
18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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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)
6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P
6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ 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 = 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
(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {
![(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {](/thumbs/94/118399854.jpg "(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {")
7 4.., ], ], ydy, ], 3], y + y dy 3, ], ], + y + ydy 4, ], ], y ydy ydy y y ] 3 3 ] 3 y + y dy y + 3 y3 5 + 9 3 ] 3 + y + ydy 5 6 3 + 9 ] 3 73 6 y + y + y ] 3 + 3 + 3 3 + 3 + 3 ] 4 y y dy y ] 3 y3 83 3
c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l
![c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l](/thumbs/85/91372642.jpg "c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l")
c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2
85 4
85 4 86 Copright c 005 Kumanekosha 4.1 ( ) ( t ) t, t 4.1.1 t Step! (Step 1) (, 0) (Step ) ±V t (, t) I Check! P P V t π 54 t = 0 + V (, t) π θ : = θ : π ) θ = π ± sin ± cos t = 0 (, 0) = sin π V + t +V
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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
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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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
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =
#A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/
知能科学：ニューラルネットワーク
2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,
知能科学：ニューラルネットワーク
2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,
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66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI
65 8. K 8 8 7 8 K 6 7 8 K 6 M Q σ (6.4) M O ρ dθ D N d N 1 P Q B C (1 + ε)d M N N h 2 h 1 ( ) B (+) M 8.1: σ = E ρ (E, 1/ρ ) (8.1) 66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3)
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
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( 9:: 3:6: (k 3 45 k F m tan 45 k 45 k F m tan S S F m tan( 6.8k tan k F m ( + k tan 373 S S + Σ Σ 3 + Σ os( sin( + Σ sin( os( + sin( os( p z ( γ z + K pzdz γ + K γ K + γ + 9 ( 9 (+ sin( sin { 9 ( } 4
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
1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3
1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}
K E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................
2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
1. 1 213 1 6 1 3 1: ( ) 2: 3: SF 1 2 3 1: 3 2 A m 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
( ) e + e ( ) ( ) e + e () ( ) e e Τ ( ) e e ( ) ( ) () () ( ) ( ) ( ) ( )
n n (n) (n) (n) (n) n n ( n) n n n n n en1, en ( n) nen1 + nen nen1, nen ( ) e + e ( ) ( ) e + e () ( ) e e Τ ( ) e e ( ) ( ) () () ( ) ( ) ( ) ( ) ( n) Τ n n n ( n) n + n ( n) (n) n + n n n n n n n n
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
9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
I-2 (100 ) (1) y(x) y dy dx y d2 y dx 2 (a) y + 2y 3y = 9e 2x (b) x 2 y 6y = 5x 4 (2) Bernoulli B n (n = 0, 1, 2,...) x e x 1 = n=0 B 0 B 1 B 2 (3) co
16 I ( ) (1) I-1 I-2 I-3 (2) I-1 ( ) (100 ) 2l x x = 0 y t y(x, t) y(±l, t) = 0 m T g y(x, t) l y(x, t) c = 2 y(x, t) c 2 2 y(x, t) = g (A) t 2 x 2 T/m (1) y 0 (x) y 0 (x) = g c 2 (l2 x 2 ) (B) (2) (1)
, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,
6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,
(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