Korteweg-de Vries
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- えりか いせき
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1 Korteweg-de Vries
2 ,.,.,.,, Korteweg-de Vries,.
3 K-dV K-dV K-dV K-dV
4
5 K-dV.,.,.,, (solitary wave) -on..,,. Korteweg-de Vries (, K-dV ), 1895 D. J. Korteweg G. de Vries,,. K-dV. u t 6u u x + 3 u = 0. (1.1.1) x3 x, t, u. 1965, N. J. Zabusky M. D. Kruskal K-dV *1,., K-dV., K-dV. *1 Zabusky and Kruskal (1965) K-dV u t + u u x + δ2 3 u x 3 = 0, (1.1.1),.
6 , K-dV, K-dV, K-dV., 1,. 3, K-dV 2., K-dV,,. 4.
7 2 K-dV 5 2 K-dV 2.1 K-dV,. u t 6u u x + 3 u x 3 = 0 (2.1.1). u/ t, K-dV (2.1.1) *1., (2.1.2). u t 6u u x = 0 (2.1.2), u t + c u x = 0 c = const. u = g(x ct). g x ct. (2.1.2) u,, (2.1.2) u = f(η) η = x + 6ut (2.1.3) *1 (2.1.2 ) Burgers
8 2 K-dV 6. (2.1.2), ( u u t 6u u x = 6t., t 6u u x f = du dη ) f = 0 (2.1.4). (2.1.4), (2.1.3) (2.1.2). (2.1.3), K-dV. (2.1.2) (2.1.3). η = x + 6ut, 6u.,,. u,. (2.1.2),,.. u t 6u u x = 0 (2.1.2),. 3.0, 128, ,, 32 *2. u, 100.0, (2.1.2) u = f(x + 6ut).,, 2.1.1,,., 2.1.1,,.,.,,. *2 u,.
9 2 K-dV 7 (a) (b) 16 (c) : (2.1.2). x, u.
10 2 K-dV K-dV u t 6u u x + 3 u x 3 = 0 (2.1.1). (2.1.1),. u t + 3 u x = 0 (2.2.1) 3, (2.2.1). t = 0 u = ϕ(x). ϕ(x) 2l. ϕ(x) 2l/n (n = 1, 2, ), ϕ(x) = n=0 α n cos nπx l + n=1 β n sin nπx l (2.2.2). α n, β n. k n k n = nπ/l, (2.2.2) ϕ(x) = α n cos k n x + n=0 β n sin k n x (2.2.3). n ω n, u n u n exp(ik n x ω n t), (2.2.1), n n=1 ω n = k 3 n (2.2.4). t > 0 k n x ω n t, t > 0 (2.2.1) u = α n cos (k n x ω n t) + n=0. β n sin (k n x ω n t), n 1 (2.2.5) n=1 α 0 = 1 2l α n = 1 l β n = 1 l l l l l l l ϕ(x)dx, ϕ(x) cos k n x dx, n 1 ϕ(x) sin k n x dx, n 1 (2.2.6)
11 2 K-dV 9. (2.2.5), (2.2.1)., (2.2.4), k n, v gn v gn = 3k 2 n (2.2.7)., n 2.,,,.,., A n Fourier u = A n = α 2 n + β 2 n, n = 0, 1, 2.,,,.,,.. (2.2.1),. 3.0, 128, , 1000.,,, u = 10 3 sech 2 ( x 3.0/ /10 3 ) * ,, x..,,, (2.2.6),. *3 K-dV. 2.4 K-dV,.
12 2 K-dV 10 (a) (b) 400 (c) : (2.2.1). x, u.
13 2 K-dV K-dV, K-dV. u t 6u u x + 3 u x 3 = 0 (2.1.1) , (a) u., (b).,.,,,.,.,,,.,,,. (a) (b) 2.3.1:. x, u, u.,.
14 2 K-dV 12,,., K-dV,,.
15 2 K-dV , K-dV, K-dV (2.1.1), 1., c (c > 0) u = f(η), η = x ct. (2.1.1), K-dV. cf 6ff + f = 0. (2.4.1) f f η. η, f = f = f = = 0, (2.4.1) f = (c + 3f)f. f, (f ) 2 = (C f)f 2. c/2 = C. c > 0, C < 0., > 0 C f < 0. g = C f, ln g i C g + i C = i C(η x 0 ). i, x 0,. g g = i C tanh{i C/2(η x 0 )}. u = f(η) = C + g 2, C = A { } A u = Asech 2 2 (x 2At x 0) (2.4.2). 1. C < 0 A > 0.
16 2 K-dV 14 K-dV 1 (2.4.2)., A > 0,.. 2A,.,., sech 1, 1/ A/2.,., K-dV,. (2.4.2) 1,.,,.
17 ,,., 2. K-dV u t 6u u x + 3 u x 3 = 0 (2.1.1).,. K-dV (2.1.1). u t 6uu x + u xxx = 0 (3.1.1), u φ u = 2 2 ln φ (3.1.2) x2. x, (3.1.1) ( ) t x ln φ + 6 x ln φ x ln φ = 0 4., K-dV. φ x φ t φφ xt = 3(φ xx ) 2 4φ x φ xxx + φφ xxxx (3.1.3) φ = 1 + e 2θ, θ = kx ωt (3.1.4)
18 3 2 16, (3.1.3) ω = 4k 3 (3.1.5). k, ω (3.1.4). (3.1.5), (3.1.4) K-dV., (3.1.4) (3.1.2), u, u = 2k 2 sech 2 (kx ωt) (3.1.6). (3.1.6) 1, (3.1.4) K-dV., (3.1.4) A,, (3.1.2) A = e 2δ, φ = 1 + Ae 2θ (3.1.7) u = 2k 2 sech 2 (kx 4k 3 t δ), A. (3.1.7),,, i, φ = 1 + A i e 2θ i, θ i = k i x ω i t, i = 1, 2. K-dV.,, A 3 φ = 1 + A 1 e 2θ 1 + A 2 e 2θ 2 + A 3 e 2(θ 1+θ 2 ) (3.1.8), ω i = 4ki 4, i = 1, 2 (3.1.9) ( ) 2 k1 k 2 A 3 = A 1 A 2 (3.1.10) k 1 + k 2, K-dV *1. (3.1.8) (3.1.2), K-dV 2,., φ (3.1.8), 2. *1 (3.1.8) K-dV (3.1.3), e 2θ i, e 2(θ 1+θ 2 ), e 4θ 1+2θ 2, e 2θ 2+4θ 2,, 0 (3.1.9), (3.1.10).
19 , φ 2 φ = 1 + A 1 e 2θ 1 + A 2 e 2θ 2 + A 3 e 2(θ 1+θ 2 ) (3.1.8).., θ i = k i x ω i t ω i = 4ki 4, i = 1, 2 (3.1.9) ( ) 2 k1 k 2 A 3 = A 1 A 2 (3.1.10) k 1 + k 2.,. (3.1.9), (3.1.10), A 1, A 2, k 1, k 2., (3.1.6), k i, (3.1.7) (3.1.2) A i., t ± u *2., φ. 1 0, k 1 x ω 1 t 0. 2 θ 2, θ 2 θ 1, θ 2 = k 2 x ω 2 t = k 2 (θ 1 ) + k ( 2 1 ω 2 k 1 k 1 ω 1 k 1 = k 2 k 1 (θ 1 ) + k 2 k 1 ( 1 k2 2 k 2 1 k 2 ) ω 1 t ) ω 1 t (3.2.1). k 1 > k 2 > 0. 1, 2,. t, θ 2 0, k 1 > k 2 (3.2.1) θ 2 k ( ) 2 1 k2 2 ω k 1 k1 2 1 t t., e 2θ 2 0, φ φ = 1 + A 1 e 2θ 2 (3.2.2) *2.
20 A 1 = exp{ 2δ ( ) 1 }, (3.1.2) u, u = 2k 2 1sech 2 {k 1 x ω 1 t δ ( ) 1 } (3.2.3)., t θ 1 = 0. (3.2.3), k 2 1., t +, θ 2 k ( ) 2 1 k2 2 ω k 1 k1 2 1 t + t +, e 2θ 2 e 2θ 1. φ, ( φ = A A ) 3 e 2θ 1 e 2θ 2 (3.2.4) A 2. t, u, u = 2k 2 1sech 2 {k 1 x ω 1 t δ (+) 1 } (3.2.5)., δ (+) 1 = 1 ( ) 2 ln A3 A 2., θ 2 0 t ± u, t, t +. u = 2k2sech 2 2 {k 2 x ω 2 t δ ( ) 2 }, (3.2.6) δ ( ) 2 = 1 ( ) 2 ln A3, A 1 u = 2k 2 2sech 2 {k 2 x ω 2 t δ (+) 2 }, (3.2.7) δ (+) 2 = 1 2 ln A 2
21 3 2 19, t 2 [ ] u = 2ki 2 sech 2 {k i x ki 3 t δ ( ) i }, (3.2.8) t + i=1 δ ( ) 1 = 1 2 ln A 1, δ ( ) 2 = 1 2 ln ( A3 A 1 ), u = 2 [ 2ki 2 sech 2 {k i x ki 3 t δ (+) i } ], (3.2.9) i=1 δ (+) 1 = 1 ( ) 2 ln A3, A 2 δ (+) 2 = 1 2 ln A 2. (3.2.8) (3.2.9), δ (±) i ± i x i, x i 4k 2 i t, i = 1, 2, t. t x 1 < x 2, t +, x 1 > x 2. +,., 2 2k 2 1, 2k 2 2., t t +., 2k 2 1 1, 1 = 1 k 1 (δ (+) 1 δ ( ) 1 ), = 1 2k 1 ln A 1A 2 A 3., 2k = 1 ln A 3 2k 2 A 1 A 2. (3.1.10) k 1 > k 2, ( ) 2 A 1 A 2 k1 + k 2 = > 1 A 3 k 1 k 2,, 1 < 0, 2 > 0., 1 < 2,.
22 K-dV 2,, 2,,, K-dV. K-dV u t 6uu x + u xxx = 0 (3.1.1) u t = x (3u2 u xx ) (3.3.1). x u, u x, u xx, 0, (3.3.1) x +,, d dt + udx = 0. I 1 = + udx (3.3.2) (3.1.1) u uu t = 6u 2 u x uu xxx ( ) u 2 = (2u 3 uu xx + (u ) x) 2 t 2 x 2. x +,, + ( ) 1 I 2 = 2 u2 dx (3.3.3)., (3.1.1) u 2, u 3, u 4,. K-dV u,, u x, u xx,.
23 K-dV. u t uu x + u xxx = 0 (3.4.1), 1 2,. 2,,. (3.4.1). t, x, j, i, K-dV u j+1 i u j 1 i 2 t = u j i u j i+1 uj i 1 2 x uj i+2 2uj i+1 + 2uj i 1 uj i 2 2( x) 3 (3.4.2)., u, x + 0., u. (3.4.2),,,. (3.4.2) u., u 2. u, u 2., u 2, (3.4.2) u x,. K-dV (3.4.1) u, t ( u 2 2 ) (u j 2) 2 uj i+1 uj i 1 u j u j i+2 2uj i+1 + 2uj i 1 uj i 2 i (3.4.3) 2 x 2( x) 3, u, 0., 0., (3.4.1) u 2. (3.4.1) ) uu x = αuu x + β x α + β = 1 ( 1 2 u2, (3.4.4)
24 , uu x αu i u i 1 u i+1 2 x + β 1 2 u2 i u2 i 1 2 x (3.4.5). (3.4.5), u α, β. u 2. (3.4.5) u, u u i,, 1 3 x (u3 ) u i u i+1 ( αu i + β 2 u i+1 ) u i u i 1 ( αu i + β 2 u i 1 x,.,, ). α = 1 3, β = 2 3 (3.4.6),,., (3.4.6) (3.4.5), (3.4.2)., u j+1 i u j 1 i 2 t = u j i (u j i+1 uj i 1 ) 6 x + (uj i+1 )2 (u j i 1 )2 6 x uj i+2 2uj i+1 + 2uj i 1 uj i 2 2( x) , 256, , ( ) ( ) x X1 x X2 u = U1sech 2 U2sech 2 2α/U1 2α/U2.. α = 6.0, U1 = , U2 = 720.0, X1 = 3.0/4.0, X2 = 3.0/2.0. xu (a) 2
25 (a) (b) 6000 (c) (d) : K-dV 2., (b) x. 3.2, A 2A. u,.,. (c),.,.,,,,.. (d).., 3.2
26 (a) (b) 3.4.2: a: xtu. z u.,.. b: xt u x.,,., (a). x 2.0, t 2.5,. xt 3.4.2(b). (b),,,.,., t t +.
27 K-dV. K-dV,, K-dV,. 2,,.
28 4 26,.,..,.,.,,., DCL.
29 4 27 [1], 1995: ,, 219pp. [2],, 1984: 6,, 183pp. [3], 1985:,, 249pp. [4], 2007: - KdV -, [5] Zabusky, J. N. and Kruskal, M. D., 1965: Interatction of solitons in a collisionless plasma and the recurrence of initial states, P hys.rev.lett., 15,
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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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x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
![d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r](/thumbs/85/92098187.jpg "d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r")
2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t)
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
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................
(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
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 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
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A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
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3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
() 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 ()
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KENZOU 2008 8 2 3 2 3 2 2 4 2 4............................................... 2 4.2............................... 3 4.2........................................... 4 4.3..............................
#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/
微分積分 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, 初版 1 刷発行時のものです.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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
Morse ( ) 2014
Morse ( ) 2014 1 1 Morse 1 1.1 Morse................................ 1 1.2 Morse.............................. 7 2 12 2.1....................... 12 2.2.................. 13 2.3 Smale..............................
1 c Koichi Suga, ISBN
c Koichi Suga, 4 4 6 5 ISBN 978-4-64-6445- 4 ( ) x(t) t u(t) t {u(t)} {x(t)} () T, (), (3), (4) max J = {u(t)} V (x, u)dt ẋ = f(x, u) x() = x x(t ) = x T (), x, u, t ẋ x t u u ẋ = f(x, u) x(t ) = x T x(t
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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
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W
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003 7 14 Black-Scholes [1] Nelson [] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-Wu Nelson e-mail: takatoshi-tasaki@nifty.com kabutaro@mocha.freemail.ne.jp
1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8..............................
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12 -- 1 1 2009 1 17 1-1 1-2 1-3 1-4 2 2 2 1-5 1 1-6 1 1-7 1-1 1-2 1-3 1-4 1-5 1-6 1-7 c 2011 1/(21) 12 -- 1 -- 1 1--1 1--1--1 1 2009 1 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1,
(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
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(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 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)
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
Gmech08.dvi
63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
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
05Mar2001_tune.dvi
2001 3 5 COD 1 1.1 u d2 u + ku =0 (1) dt2 u = a exp(pt) (2) p = ± k (3) k>0k = ω 2 exp(±iωt) (4) k
x E E E e i ω = t + ikx 0 k λ λ 2π k 2π/λ k ω/v v n v c/n k = nω c c ω/2π λ k 2πn/λ 2π/(λ/n) κ n n κ N n iκ k = Nω c iωt + inωx c iωt + i( n+ iκ ) ωx
x E E E e i ω t + ikx k λ λ π k π/λ k ω/v v n v c/n k nω c c ω/π λ k πn/λ π/(λ/n) κ n n κ N n iκ k Nω c iωt + inωx c iωt + i( n+ iκ ) ωx c κω x c iω ( t nx c) E E e E e E e e κ e ωκx/c e iω(t nx/c) I I
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta
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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
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
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ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k.
K E N Z OU 8 9 8. F = kx x 3 678 ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k. D = ±i dt = ±iωx,
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....
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3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
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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
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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
chap03.dvi
99 3 (Coriolis) cm m (free surface wave) 3.1 Φ 2.5 (2.25) Φ 100 3 r =(x, y, z) x y z F (x, y, z, t) =0 ( DF ) Dt = t + Φ F =0 onf =0. (3.1) n = F/ F (3.1) F n Φ = Φ n = 1 F F t Vn on F = 0 (3.2) Φ (3.1)
1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC
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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)
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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
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4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
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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)
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[2642 ] Yuji Chinone 1 1-1 ρ t + j = 1 1-1 V S ds ds Eq.1 ρ t + j dv = ρ t dv = t V V V ρdv = Q t Q V jdv = j ds V ds V I Q t + j ds = ; S S [ Q t ] + I = Eq.1 2 2 Kroneher Levi-Civita 1 i = j δ i j =
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第1章 微分方程式と近似解法
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, 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,,,,
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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
Acrobat Distiller, Job 128
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f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
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) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p
2012 IA 8 I 1 10 10 29 1. [0, 1] n x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 2. 1 x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 1 0 f(x)dx 3. < b < c [, c] b [, c] 4. [, b] f(x) 1 f(x) 1 f(x) [, b] 5.
chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
φ s i = m j=1 f x j ξ j s i (1)? φ i = φ s i f j = f x j x ji = ξ j s i (1) φ 1 φ 2. φ n = m j=1 f jx j1 m j=1 f jx j2. m
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dy + P (x)y = Q(x) (1) dx dy dx = P (x)y + Q(x) P (x), Q(x) dy y dx Q(x) 0 homogeneous dy dx = P (x)y 1 y dy = P (x) dx log y = P (x) dx + C y = C exp
+ P (x)y = Q(x) (1) = P (x)y + Q(x) P (x), Q(x) y Q(x) 0 homogeneous = P (x)y 1 y = P (x) log y = P (x) + C y = C exp{ P (x) } = C e R P (x) 5.1 + P (x)y = 0 (2) y = C exp{ P (x) } = Ce R P (x) (3) αy