Formation of hot jupiters by slingshot model Naoya Okazawa Department of Earth Sciences, Undergraduate school of Science, Hokkaido University

Size: px
Start display at page:

Download "Formation of hot jupiters by slingshot model Naoya Okazawa Department of Earth Sciences, Undergraduate school of Science, Hokkaido University"

Transcription

1 Formation of hot jupiters by slingshot model Naoya Okazawa Department of Earth Sciences, Undergraduate school of Science, Hokkaido University Planetary and Space Group

2 1,.,,.,.,.,..,.,.,,.,.,,,.,.,.,.,,,..

3 i APPENDIX

4 (Hayashi et al. 1985)...,..,. 10,..,,.,., Peg b, 0.052AU (Marcy et al. 1997) (Schneider 2011),. (e. g. Schilling 1996)., 1., 3,,,.,, (Nagasawa et al. 2008).

5 , 2,..,,. 1.3, ,

6 ,..,,.,.,.,.,, , 2, 2. 1.,.,,. 1: (National Space Science Data Center) [m ] [AU]

7 2 4 1, 2,.,.., 0.1AU.,,. 2,.. 1, (a < 0.1 AU),. 3. 1: , 481 (Schneider 2011 )

8 2 5 2: , 514 (Schneider 2011 ) 2.3 (Weidenschilling & Marzari 1996)..,,,., (Lin et al. 1996) (Rasio & Ford 1996), (Bodenheimer et al. 2000).,,.,.,,...,, (Lin et al. 1996). 3.

9 ,. 1,,.,.,., 4,.,, (Nagasawa et al. 2008). 3.1 Chambers et al. (1996), Marzari & Weidenschilling (2002), Chatterjee et al (2008) , (Gladman 1993). 2 3r H(1,2) (3.1.1) 1 a 1, m 1, 2 a 2, m 2, M, r H(1,2). r H(1,2) = a 1 + a 2 2 ( ) m1 + m 1/3 2 (3.1.2), 3,., 1, 2 (Marzari & Weidenschilling 2002). 3M, a i+1 = a i + Kr H(i,i+1) (3.1.3) K, (Chatterjee et al 2008).,

10 3 7,, (Chatterjee et al 2008).. 3.2,.,.,,,.,,..,... ( ), ( ).,,. 2, 1, 1... m 1 m , E L. E = Gµ(m 1 + m 2 ) (3.2.1) 2a L = µ (1 e 2 )(m 1 + m 2 )ag (3.2.2) G, a, µ µ m 1 m 2 /(m 1 + m 2 ). (3.2.1), ȧ = Gm 1m 2 2E 2 Ė 2a 2 = Ė (3.2.3) Gm 1 m 2. Ė, Ė < 0. ȧ < 0.

11 4 8 (3.2.2), L = 0. ė = 1 e2 2ae ȧ (3.2.4) (3.2.3), ė = 1 (1 e)(1 + e)a Ė (3.2.5) Gm 1 m 2 e q = a(1 e) e 1 (3.2.5). ė = Ė < 0 ė < 0. 2q Gm 1 m 2 Ė (3.2.6) a fin a ini e ini, (3.2.2) a fin = a ini (1 e 2 ini) = q ini (1 + e ini ) (3.2.7). a fin = q fin, e ini 1, q fin 2q ini, 2. q ini (1 + e ini )/(1 e ini ) 2q ini. 1,.,, , 1, 2, 3, 4. j,. m j d 2 r j dt 2 = i j Gm i m j r(i,j) 3 r ( i, j) (4.1.1) r j j, m i i, G, r (i,j) i j..

12 , ,,. x. dx dt = f(x, t) (4.2.1) t, x, f x t. t, x(t + t). x(t + t) = x(t) + dx(t) t + 1 d 2 x(t) dt 2 dt 2 t 2 + (4.2.2) ( t) 2, t i+1 = t + t x i+1. x i+1 = x i + f(x i, t i ) t (4.2.3)., t = t i, t = t i + t.. k 1 = f(x i, t i ) (4.2.4) k 2 = f(x i + k 1 t/2, t i + t/2) (4.2.5) k 3 = f(x i + k 2 t/2, t i + t/2) (4.2.6) k 4 = f(x i + k 3, t i + t) (4.2.7) x i+1 = x i + (k 1 + 2k 2 + 2k 3 + k 4 ) t (4.2.8) 6 k 1 t = t i, k 2 k 1 t/ d 2 x = g(x, t) (4.2.9) dt2

13 x 2 (4.2.9). g. t = t i a i 1, x p, v p. x pi = x i + v i t + t2 2 a i + t3 6 v pi = v i + a i t + t2 2 da i dt da i dt (4.2.10) (4.2.11) x i, v i t = t i. da i /dt g(x, t),, a i = Gm k r k r 3 k da i dt = Gm k [ vk r 3 k 3(r ] k v k )r k rk 5 (4.2.12) (4.2.13) m k k, r k, v k.,, (4.2.12), (4.2.13). t = t i + t a i+1 da i+1 /dt. a i, da i /dt, a i+1, da i+1 /dt d 2 a i /dt 2, d 3 a i /dt 3 (7.5 ). d 2 a i dt 2 = d 3 a i dt 3 = x i+1, v i ( ) 6(a i a i+1 ) t 4 da i dt + 2 da i+1 dt t 2 (4.2.14) ( 12(a i a i+1 ) + 6 t dai dt + da i+1 dt t 3 (4.2.15) x i+1 = x pi + t4 24 v i+1 = v pi + t3 6 ) d 2 a i dt 2 + t5 d 3 a i 120 dt 3 (4.2.16) d 2 a i dt 2 + t4 d 3 a i 24 dt 3 (4.2.17) 2 xy 3,.,.,, 2.

14 4 11, v 1 = v i m i /m 1, r 1 = r i m i /m 1. E( ) E i = 1 ( 2 m i 1 + m ) i (v i v cen ) 2 Gm 1m i (4.3.1) m 1 r (1,i). v cen 2.,., L, ( L i = m i 1 + m ) i (r i (v i v cen )) (4.3.2) m 1. 2,. 2, µ = m 1m 2 m 1 +m 2 a e (7.3.2 ). a i = Gµ(m 1 + m i ) (4.3.3) 2E i L 2 i e i = 1 µ 2 (4.3.4) (m 1 + m i )ag, 10 3 M k AU (k = 1, 5), 2.5 AU k 10 3 M (k = 1, 5), 3 30., M, 2 AU. 1, 1.. (Marzari & Weidenschilling 2002). 1.

15 semi-major axis [AU] time [year] 3: 3, 4. 4,. Chatterjee et al :, 5. 5,.

16 4 13 5:, AU, (a 5 AU).,. 6:,, q q < 0.05 AU (Rasio & Ford 1996). 0.5 AU,. 1, 2,.,,.

17 ,.. E = Gµ(m 1 + m 2 ) + 1 2a 2 I ω 2 (5.1.1) L = µ (1 e 2 )(m 1 + m 2 )ag + I ω (5.1.2) I, ω. (5.1.1), Ω p, Ė = Gµ(m 1 + m 2 ) 2a 2 ȧ + I ω ω., (5.1.2), = 1 2 µω2 paȧ + I ω ω (5.1.3) L = 1 2 µ G(m1 + m 2 ) a 3 aȧ + I ω = 1 2 µω paȧ + I ω (5.1.4). L = 0 (5.1.4). (5.1.5) (5.1.3), I ω = 1 2 µω paȧ (5.1.5) Ė = 1 2 µω paȧ(ω Ω p ) (5.1.6). Ė < 0 ȧ (ω Ω p ).,., (ω < Ω p ) (Ω p < ω ). 7.,,...

18 5 15 7: ( 2003 ).,.. 2.3,.,,.,. 5.2,.,.,., ( a < 0.1 ).. m, a 0 a 1, e 0 e 1

19 5 16. M L, ( ) L = m GMa fin (1 e 2 fin ) GMa ini (1 e 2 ini ) (5.2.1) (7.3.1 ). ω, ω = L I (5.2.2) I. 10. a ini = 2.5 AU( ), e ini = 0.98, a fin = 0.02 AU, e fin = 0, P rot,. ( ) I ( ) MR M 1/2 ( ) m 1 P rot (5.2.3) ,,. 8,. 1. M m J 8: ( Mamajek & Hillenbrand 2008, Baliunas 1996, Noyes et al ), (Lanza 2010 )

20 ,,.,.,,... T- 8 (Bouvier et al. 1993).. 8,.,. r, M, T, Ω p Ω p = GM/r 3, Ω Ω = 2π/T r c. r c = ( ) T 2/3 (GM) 1/3 (5.3.1) 2π.. 5.4,,. 9. a < 0.04 AU , a 0.04 AU 4.92, AU., τ a τ a (m 1/2 /m p ) (Murray & Dermott 1999) AU, ( ) 1/2 / ( ) 1/2 /.,,. 9

21 5 18.,.,,.,. 9: (Schneider 2011 ) 10: a < 0.04 AU, ( ) 1/2 / (Schneider 2011 )

22 6 19 6,,.,.,. 3,, 1.,,.,.,.,.,.,,,.,.,.,.,.,.,.,.,,,.

23 7 APPENDIX 20 7 APPENDIX 7.1,. r, R, M, M p, Ω K. a m.,. GM m (R a) 2 = GM pm a 2 + m(r a)ω 2 K (7.1.1) Ω 2 K = GM r 3, (r a) 2, GM m (r a) 2 = GM pm a 2 + Gm(r a)m r 3 (7.1.2) 1 (r a) 2 1 r 2 + 2a r 3 (7.1.3), M r 2 + 2aM r 3 a = r = M p + M a 2 ( Mp 3M r 2 ) 1 3 am r 3 (7.1.4) (7.1.5)

24 7 APPENDIX : ( 2007 II ( ) ).,.. F P r, FF P r = 2a r, FF C (7.2.2) (7.2.1) r 2 = r 2 + (2ae) 2 + 2(2ae)r cos ϕ (7.2.1) r(1 + e cos ϕ) = a(1 e 2 ) (7.2.2) a(1 e 2 ) = l (7.2.3) r = l 1 + e cos ϕ (7.2.4),., l, e, (a, e),.,.

25 7 APPENDIX 22,. E. E = m 2 (v2 r + vθ 2 ) GMm r = m ( ) dr 2 ( + r dθ ) 2 GMm 2 dt dt r (7.3.1) m, v r r, v θ θ. L = mr 2 dθ dt, E = m 2 ( ) dr 2 + L2 dt 2mr 2 GMm r (7.3.2) d dt = dθ dt d dθ = L mr 2 d dθ, u 1/r, E = m 2 ( ) L dr 2 mr 2 + L2 dθ 2mr 2 GMm r (7.3.3) cos l ( ) E = L2 du 2 + L2 u 2 GMmu (7.3.4) 2m dθ 2m ( ) 2mE du 2 ) 2 L 2 = + (u GMm2 dθ L 2 G2 M 2 m 4 L 4 (7.3.5) du dθ = ± ( ) (7.3.6) 2 2mE u GMm2 L 2 L + G 2 M 2 m 4 2 L 4 dx (a 2 x 2 ) = arccos ( x a) θ = ± arccos cos θ = r = u GMm2 L 2 (7.3.7) 2mE + G2 M 2 m 4 L 2 L 4 u GMm2 L 2 (7.3.8) 2mE + G2 M 2 m 4 L 2 L 4 L 2 GMm EL2 cos θ G 2 M 2 m 3 L2, e 1 + 2EL2 GMm 2 G 2 M 2 m 3 (7.3.9) r = l 1 + e cos θ (7.3.10)

26 7 APPENDIX 23.,. l, e,., a = l l, e 1 e 2, e 1 + 2EL2 G 2 M 2 m 3 E = GMm 2a (7.3.11) L = m (1 e 2 )agm (7.3.12) 7.3.2,.,., m 1, r 1,, m 2, r 2, R, r. R = m 1r 1 + m 2 r 2 m 1 + m 2 (7.3.13) r = r 1 r 2 (7.3.14),. r 1, r 2 (7.3.13), (7.3.14) L m 2 r 1 = r 1 R = r (7.3.15) m 1 + m 2 r 2 m 1 = r 2 R = r (7.3.16) m 1 + m 2 L = r 1 dr 1 m 1 dt + dr r 2 2 m 2 dt m 1 m 2 = r dr m 1 + m 2 dt = µr dr dt E. (7.3.13), (7.3.14) (7.3.17) m 2 r 1 = R + r (7.3.18) m 1 + m 2 m 1 r 2 = R r (7.3.19) m 1 + m 2 E, V E = 1 2 (m 1ṙ m 2 ṙ 2 2) + V (r 1 r 2 ) (7.3.20)

27 7 APPENDIX 24 (7.3.18), (7.3.19) ṙ1 2 = Ṙ2 + 2m 2 m 2 2 Ṙṙ + m 1 + m 2 (m 1 + m 2 ) 2 ṙ2 (7.3.21) ṙ2 2 = Ṙ2 + 2m 1 m 2 1 Ṙṙ + m 1 + m 2 (m 1 + m 2 ) 2 ṙ2 (7.3.22) E = m 1 + m 2 Ṙ 2 + m 1m 2 )ṙ2 Gm 1m 2 2 (m 1 + m 2 r (7.3.23) Ṙ = 0. µ m 1m 2 /(m 1 + m 2 ) E = µṙ 2 Gm 1m 2 r = µ 2 (v2 r + vθ 2 ) Gm 1m 2 r = µ ( ) dr 2 ( + r dθ ) 2 Gm 1m 2 2 dt dt r (7.3.24). v r r, v θ θ. E L a, e.,,.. E = Gµ(m 1 + m 2 ) (7.3.25) 2a L = µ (1 e 2 )(m 1 + m 2 )ag (7.3.26) 7.4 x(t + t), x(t + t) = x(t) + t dx(t) dt t n = n! n=0 + t2 2 d 2 x(t) dt 2 + t3 6 d 3 x(t) dt 3 + (7.4.1) d n x(t) dt n (7.4.2),.., t 2 O( t 2 ) O( t). 1., 4 4., 5, 4.

28 7 APPENDIX ,. k 1, k 2, k 3, k 4 t, x, f, k 1 = f(x) (7.4.3) ( k 2 = f x + 1 ) 2 k 1 t = f (x + 12 ) f(x) t ( 1 = f(x) + f (x) + 1 ( 1 6 f (x) 2 f(x) t ) 2 f(x) t + 1 ( 1 2 f (x) 2 f(x) t ) 3 + O( t 4 ) = f(x) f(x)f (x) t f 2 (x)f (x) t f 3 (x)f (x) t 3 + O( t 4 ) (7.4.4) ( k 3 = f x + 1 ) 2 k 2 t ( ) 1 = f(x) + f (x) 2 k 2 t + 1 ( ) f (x) 2 k 2 t + 1 ( ) f (x) 2 k 2 t + O( t 4 ) = f(x) f (x) (f(x) + 12 f(x)f (x) t + 18 ) f 2 (x)f (x) t 2 t + 1 ( 8 f (x) f(x) + 1 ) 2 2 f(x)f (x) t t = f(x) + 1 [ 1 2 f(x)f (x) t + + [ 3 16 f (x)f (x)(f(x)) k 4 = f(x) + f(x)f (x) t + + ) 2 48 f (x)(f(x)) 3 t 3 + O( t 4 ) 4 f(x)(f (x)) ] 8 f 2 (x)f (x) t 2 ] 48 f 3 (x)f (x) t 3 + O( t 4 ) (7.4.5) [ 1 2 f(x)(f (x)) ] 2 f 2 (x)f (x) t 2 [ 1 4 f(x)(f (x)) f 2 (x)f (x)f (x) f 3 (x)f (x) x(t + t). x(t + t) = x(t) + (k 1 + 2k 2 + 2k 3 + k 4 ) t 6 ] t 3 + O( t(7.4.6) 4 ) = x(t) + f(x) t f(x)f (x) t [(f(x))2 f (x) + f(x)(f (x)) 2 ] t [(f(x))3 f (x) + 4(f(x)) 2 f (x)f (x) + f(x)(f (x)) 3 ] t 4 +O( t 5 ) (7.4.7)

29 7 APPENDIX 26. d dt f(x(t)) = f dx dt = ff d 2 dt 2 f(x(t)) = d dt (ff ) = f(f ) 2 + f 2 f d 3 dt 3 f(x(t)) = d dt (f(f ) 2 + f 2 f ) = f(f ) 3 + 4f 2 f f + f 3 f (7.4.8) (7.4.8) (7.4.7), x(t + t) = x(t) + f(x) t + 1 df(x) t 2 2 dt + 1 d 2 f(x) 6 dt 2 t d 3 f(x) 24 dt 3 t 4 + O( t 5 ) (7.4.9). t f(x). S i (x) = A 0 + A 1 x + A 2 x A n x n (7.5.1) [a, b] [x i 1, x i ] (i = 1, 2, n), x 0, x 1,, x n. f(x), A 0, A 1, A n. f(x). f(x) f (x), 2n+2, A 2n a i (t i ) 3 S i. S i (t i ) = A 0 + A 1 t i + A 2 t 2 i + A 3 t 3 i (7.5.2) t i = i t. (7.5.2) a i (t i ). (7.5.2), (7.5.3) i = 0, 1, S i (t i ) = a i (t i ) (7.5.3) Ṡ i (t i ) = ȧ i (t i ) (7.5.4) a 0 = A 0 (7.5.5) a 1 = A 0 + A 1 t + A 2 t 2 + A 3 t 3 (7.5.6)

30 7 APPENDIX 27 (7.5.2), (7.5.4) i = 0, 1, ȧ 0 = A 1 (7.5.7) ȧ 1 = A 1 + 2A 2 t + 3A 3 t 2 (7.5.8). (7.5.5), (7.5.6), (7.5.7), (7.5.8) A 0, A 1, A 2, A 3,. A 0 = a 0 (7.5.9) A 1 = ȧ 0 (7.5.10) A 2 = 3(a 0 a 1 ) t(2ȧ 0 + ȧ 1 ) t 2 (7.5.11) A 3 = 2(a 0 a 1 ) + t(ȧ 0 + ȧ 1 ) t 2 (7.5.12), a(t + t) O( t 3 ), t = 0,, a(t + t) a(t) + da(t) t + 1 d 2 a(t) dt 2 dt 2 t d 3 a(t) 6 dt 3 t 3 (7.5.13) a( t) a 0 + da 0 dt t + 1 d 2 a 0 2 dt 2 t2 + 1 d 3 a 0 6 dt 3 t3 (7.5.14) d 2 a 0 dt 2 = 2A 2 = 6(a 0 a 1 ) t(4ȧ 0 + 2ȧ 1 ) t 2 (7.5.15) d 3 a 0 dt 3 = 6A 3 = 12(a 0 a 1 ) + 6 t(ȧ 0 + ȧ 1 ) t 3 (7.5.16)

31 7 APPENDIX 28,.,.,. 3,,.,,.,.

32 7 APPENDIX 29 Baliunas, S., D. Sokoloff, and W. Soon Magnetic Field and Rotation in Lower Main-Sequence Stars: an Empirical Time-dependent Magnetic Bode s Relation?. Astrophysical Journal Letters L99 Bouvier, J., S. Cabrit, M. Fernandez, E. L.Martin, and J. M.Matthews Coyotes-I - the Photometric Variability and Rotational Evolution of T-Tauri Stars. Astronomy and Astrophysics Chambers, J. E., G. W. Wetherill, and A. P.Boss The Stability of Multi- Planet Systems. Icarus Chatterjee, S., E. B. Ford, S. Matsumura, and F. A. Rasio Dynamical Outcomes of Planet-Planet Scattering. The Astrophysical Journal Lanza, A. F Hot Jupiters and the evolution of stellar angular momentum. Astronomy and Astrophysics A77 Lin, D. N. C, P. Bodenheimer, and D. C. Richardson Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature Mamajek, E. E., and L. A. Hillenbrand Improved Age Estimation for Solar- Type Dwarfs Using Activity-Rotation Diagnostics. The Astrophysical Journal Marcy, G. W., R. P. Butler, E. Williams, L. Bildsten, J. R. Graham, A. M. Ghez, and J. G. Jernigan The Planet around 51 Pegasi. Astrophysical Journal Marzari, F., and S. J. Weidenschilling Eccentric Extrasolar Planets: The Jumping Jupiter Model. Icarus Nagasawa, M., S. Ida, and T. Bessho Formation Of Hot Planets By A Combination Of Planet Scattering, Tidal Circularization, And The Kozai Mechanism. The Astrophysical Journal Noyes, R. W., L. W. Hartmann, S. L. Baliunas, D. K. Duncan, and A. H. Vaughan Rotation, convection, and magnetic activity in lower main-sequence stars. Astrophysical Journal Rasio, F. A., and E. B. Ford Dynamical instabilities and the formation of extrasolar planetary systems. Science

33 7 APPENDIX 30 Schilling, G Hot Jupiters Leave Theorists in the Cold. Science Weidenschilling, S. J., and F. Marzari Gravitational scattering as a possible origin for giant planets at small stellar distances. Nature ,, HARP 1. - HPC). Vol No 72 Jean Schneider The Extrasolar Planets Encyclopaedia. National Space Science Data Center II ( ) 10 ebisawa/teaching/2007univtokyo.html

Formation process of regular satellites on the circumplanetary disk Hidetaka Okada Department of Earth Sciences, Undergraduate school of Scie

Formation process of regular satellites on the circumplanetary disk Hidetaka Okada Department of Earth Sciences, Undergraduate school of Scie Formation process of regular satellites on the circumplanetary disk Hidetaka Okada 22060172 Department of Earth Sciences, Undergraduate school of Science, Hokkaido University Planetary and Space Group

More information

Gmech08.dvi

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

More information

A 99% MS-Free Presentation

A 99% MS-Free Presentation A 99% MS-Free Presentation 2 Galactic Dynamics (Binney & Tremaine 1987, 2008) Dynamics of Galaxies (Bertin 2000) Dynamical Evolution of Globular Clusters (Spitzer 1987) The Gravitational Million-Body Problem

More information

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

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 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

More information

DVIOUT

DVIOUT 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)

More information

( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (

( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) ( 6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b

More information

A

A A05-132 2010 2 11 1 1 3 1.1.......................................... 3 1.2..................................... 3 1.3..................................... 3 2 4 2.1............................... 4 2.2

More information

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 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

More information

K E N Z OU

K E N Z OU K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................

More information

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

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

More information

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 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)

More information

Contents 1 Jeans (

Contents 1 Jeans ( Contents 1 Jeans 2 1.1....................................... 2 1.2................................. 2 1.3............................... 3 2 3 2.1 ( )................................ 4 2.2 WKB........................

More information

mugensho.dvi

mugensho.dvi 1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

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

More information

2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( )

2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( ) http://astr-www.kj.yamagata-u.ac.jp/~shibata f4a f4b 2 f4cone f4eki f4end 4 f5meanfp f6coin () f6a f7a f7b f7d f8a f8b f9a f9b f9c f9kep f0a f0bt version feqmo fvec4 fvec fvec6 fvec2 fvec3 f3a (-D) f3b

More information

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

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

More information

sec13.dvi

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:

More information

x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v

x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v 12 -- 1 4 2009 9 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 c 2011 1/(13) 4--1 2009 9 3 x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2

More information

Standard Model for Formation of the Solar System ADACHI Toshitaka Department of Earth Sciences, Undergraduate school of Science, Hokkaido University P

Standard Model for Formation of the Solar System ADACHI Toshitaka Department of Earth Sciences, Undergraduate school of Science, Hokkaido University P Standard Model for Formation of the Solar System ADACHI Toshitaka Department of Earth Sciences, Undergraduate school of Science, Hokkaido University Planetary Physics Laboratory 2009 4 24 , Hayashi et

More information

II 1 3 2 5 3 7 4 8 5 11 6 13 7 16 8 18 2 1 1. x 2 + xy x y (1 lim (x,y (1,1 x 1 x 3 + y 3 (2 lim (x,y (, x 2 + y 2 x 2 (3 lim (x,y (, x 2 + y 2 xy (4 lim (x,y (, x 2 + y 2 x y (5 lim (x,y (, x + y x 3y

More information

i

i 009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3

More information

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

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)

More information

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin 2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ

More information

ライトカーブ観測から何がわかるか

ライトカーブ観測から何がわかるか 2004.7.2 M 2.5log F F 10 0.4*M M F S= (abc)[sin 2 A(sin 2 ( )/a 2 +cos 2 ( )/b 2 )+cos 2 (A)/c 2 ] 1/2 a,b,c A aspect angle S M = (abc)[sin 2 (A)/b 2 +cos 2 (A)/c 2 ] 1/2 S m = (abc)[sin 2 (A)/a 2 +cos

More information

Untitled

Untitled 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

More information

untitled

untitled - k k k = y. k = ky. y du dx = ε ux ( ) ux ( ) = ax+ b x u() = ; u( ) = AE u() = b= u () = a= ; a= d x du ε x = = = dx dx N = σ da = E ε da = EA ε A x A x x - σ x σ x = Eε x N = EAε x = EA = N = EA k =

More information

2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................

More information

untitled

untitled 20010916 22;1017;23;20020108;15;20; 1 N = {1, 2, } Z + = {0, 1, 2, } Z = {0, ±1, ±2, } Q = { p p Z, q N} R = { lim a q n n a n Q, n N; sup a n < } R + = {x R x 0} n = {a + b 1 a, b R} u, v 1 R 2 2 R 3

More information

OHP.dvi

OHP.dvi t 0, X X t x t 0 t u u = x X (1) t t 0 u X x O 1 1 t 0 =0 X X +dx t x(x,t) x(x +dx,t). dx dx = x(x +dx,t) x(x,t) (2) dx, dx = F dx (3). F (deformation gradient tensor) t F t 0 dx dx X x O 2 2 F. (det F

More information

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0, .1.1 Y K L Y = K 1 3 L 3 L K K (K + ) 1 1 3 L 3 K 3 L 3 K 0 (K + K) 1 3 L 3 K 1 3 L 3 lim K 0 K = L (K + K) 1 3 K 1 3 3 lim K 0 K = 1 3 K 3 L 3 z = f(x, y) x y z x-y-z.1 z = e x +xy y 3 x-y ( ) z 0 f(x,

More information

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 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

More information

gr09.dvi

gr09.dvi .1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {

More information

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v = 1. 2. 3 3. 4. 5. 6. 7. 8. 9. I http://risu.lowtem.hokudai.ac.jp/ hidekazu/class.html 1 1.1 1 a = g, (1) v = g t + v 0, (2) z = 1 2 g t2 + v 0 t + z 0. (3) 1.2 v-t. z-t. z 1 z 0 = dz = v, t1 dv v(t), v

More information

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

More information

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b) 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h 0 f(a + h, b) f(a, b) h............................................................... ( ) f(x, y) (a, b) x A (a, b) x

More information

215 11 13 1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4

More information

2011de.dvi

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

More information

all.dvi

all.dvi 72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G

More information

dynamics-solution2.dvi

dynamics-solution2.dvi 1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj

More information

: , 2.0, 3.0, 2.0, (%) ( 2.

: , 2.0, 3.0, 2.0, (%) ( 2. 2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................

More information

太陽系外惑星探査

太陽系外惑星探査 http://hubblesite.org/newscenter/archive/2001/38/ Terra MODIS http://modarch.gsfc.nasa.gov/ http://www.nasa.gov/home/index.html / 2 Are we alone? Origins Where are they? (Fermi 1950) 3 4 0.5 arcsec 10pc

More information

note1.dvi

note1.dvi (1) 1996 11 7 1 (1) 1. 1 dx dy d x τ xx x x, stress x + dx x τ xx x+dx dyd x x τ xx x dyd y τ xx x τ xx x+dx d dx y x dy 1. dx dy d x τ xy x τ x ρdxdyd x dx dy d ρdxdyd u x t = τ xx x+dx dyd τ xx x dyd

More information

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N

More information

2000年度『数学展望 I』講義録

2000年度『数学展望 I』講義録 2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53

More information

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i

More information

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1

More information

D 24 D D D

D 24 D D D 5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6

More information

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) 4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7

More information

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ

More information

Korteweg-de Vries

Korteweg-de Vries Korteweg-de Vries 2011 03 29 ,.,.,.,, Korteweg-de Vries,. 1 1 3 1.1 K-dV........................ 3 1.2.............................. 4 2 K-dV 5 2.1............................. 5 2.2..............................

More information

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

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 23 2 2.1 n n r x, y, z ˆx ŷ ẑ 1 a a x ˆx + a y ŷ + a z ẑ 2.1.1 3 a iˆx i. 2.1.2 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, 2.1.3 n a b a i b i a i b i a x b x + a y b y + a z b

More information

( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n

More information

ver.1 / c /(13)

ver.1 / c /(13) 1 -- 11 1 c 2010 1/(13) 1 -- 11 -- 1 1--1 1--1--1 2009 3 t R x R n 1 ẋ = f(t, x) f = ( f 1,, f n ) f x(t) = ϕ(x 0, t) x(0) = x 0 n f f t 1--1--2 2009 3 q = (q 1,..., q m ), p = (p 1,..., p m ) x = (q,

More information

1 1 u m (t) u m () exp [ (cπm + (πm κ)t (5). u m (), U(x, ) f(x) m,, (4) U(x, t) Re u k () u m () [ u k () exp(πkx), u k () exp(πkx). f(x) exp[ πmxdx

1 1 u m (t) u m () exp [ (cπm + (πm κ)t (5). u m (), U(x, ) f(x) m,, (4) U(x, t) Re u k () u m () [ u k () exp(πkx), u k () exp(πkx). f(x) exp[ πmxdx 1 1 1 1 1. U(x, t) U(x, t) + c t x c, κ. (1). κ U(x, t) x. (1) 1, f(x).. U(x, t) U(x, t) + c κ U(x, t), t x x : U(, t) U(1, t) ( x 1), () : U(x, ) f(x). (3) U(x, t). [ U(x, t) Re u k (t) exp(πkx). (4)

More information

() 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 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc 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 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n ( 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

More information

通信容量制約を考慮したフィードバック制御 - 電子情報通信学会 情報理論研究会(IT) 若手研究者のための講演会

通信容量制約を考慮したフィードバック制御 -  電子情報通信学会 情報理論研究会(IT)  若手研究者のための講演会 IT 1 2 1 2 27 11 24 15:20 16:05 ( ) 27 11 24 1 / 49 1 1940 Witsenhausen 2 3 ( ) 27 11 24 2 / 49 1940 2 gun director Warren Weaver, NDRC (National Defence Research Committee) Final report D-2 project #2,

More information

f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14

f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14 B p.1/14 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) f(x 1,...,x n ) (x 1 x 0,...,x n 0), (x 1,...,x n ) i x i f xi

More information

B ver B

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.......................

More information

x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n

x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n 1, R f : R R,.,, b R < b, f(x) [, b] f(x)dx,, [, b] f(x) x ( ) ( 1 ). y y f(x) f(x)dx b x 1: f(x)dx, [, b] f(x) x ( ).,,,,,., f(x)dx,,,, f(x)dx. 1.1 Riemnn,, [, b] f(x) x., x 0 < x 1 < x 2 < < x n 1

More information

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { ( 5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx

More information

I 1

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

More information

4 19

4 19 I / 19 8 1 4 19 : : f(e, J), f(e) Phase mixing Landau Damping, violent relaxation : 2 2 : ( ) http://antwrp.gsfc.nasa.gov/apod/ap950917.html ( ) http://www-astro.physics.ox.ac.uk/~wjs/apm_grey.gif

More information

, 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,,,,

, 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,

More information

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) 2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x

More information

1 a b cc b * 1 Helioseismology * * r/r r/r a 1.3 FTD 9 11 Ω B ϕ α B p FTD 2 b Ω * 1 r, θ, ϕ ϕ * 2 *

1 a b cc b * 1 Helioseismology * * r/r r/r a 1.3 FTD 9 11 Ω B ϕ α B p FTD 2 b Ω * 1 r, θ, ϕ ϕ * 2 * 448 8542 1 e-mail: ymasada@auecc.aichi-edu.ac.jp 1. 400 400 1.1 10 1 1 5 1 11 2 3 4 656 2015 10 1 a b cc b 22 5 1.2 * 1 Helioseismology * 2 6 8 * 3 1 0.7 r/r 1.0 2 r/r 0.7 3 4 2a 1.3 FTD 9 11 Ω B ϕ α B

More information

F S S S S S S S 32 S S S 32: S S rot F ds = F d l (63) S S S 0 F rot F ds = 0 S (63) S rot F S S S S S rot F F (63)

F S S S S S S S 32 S S S 32: S S rot F ds = F d l (63) S S S 0 F rot F ds = 0 S (63) S rot F S S S S S rot F F (63) 211 12 1 19 2.9 F 32 32: rot F d = F d l (63) F rot F d = 2.9.1 (63) rot F rot F F (63) 12 2 F F F (63) 33 33: (63) rot 2.9.2 (63) I = [, 1] [, 1] 12 3 34: = 1 2 1 2 1 1 = C 1 + C C 2 2 2 = C 2 + ( C )

More information

II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (

II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x ( II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )

More information

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

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 II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =

More information

2 1958 10 2 2 60 60020 20 10 1 10 2, 3 2, 3 5 6 108 6 357

2 1958 10 2 2 60 60020 20 10 1 10 2, 3 2, 3 5 6 108 6 357 3 860 8555 2 39 1 e-mail: keitaro@sci.kumamoto-u.ac.jp 1 3 1958 1 195710 1957 7 * 1 12 Fred Lawrence Whipple * 1 1957 7 1 1958 12 31 356 2015 6 2 1958 10 2 2 60 60020 20 10 1 10 2, 3 2, 3 5 6 108 6 357

More information

Note.tex 2008/09/19( )

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..................................

More information

A

A A04-164 2008 2 13 1 4 1.1.......................................... 4 1.2..................................... 4 1.3..................................... 4 1.4..................................... 5 2

More information

š ( š ) 2,973,655 3,774,545 4,719,254 1,594,319 3,011,432 1,517,982 1,493, ,503 2,591, , , , , ,000 f21 500,000 24

š ( š ) 2,973,655 3,774,545 4,719,254 1,594,319 3,011,432 1,517,982 1,493, ,503 2,591, , , , , ,000 f21 500,000 24 š ( š ) 812,488 8,633,171 390,374,410 324,279,452 9,953,269 17,329,976 2,944,796 2,944,796 6,866,917 341,279,452 12,000,000 12,000,000 2,000,000 2,000,000 1,000,000 1,000,000 500,000 600,000 I 1,000,000

More information

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

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

More information

phs.dvi

phs.dvi 483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....

More information

6.1 (P (P (P (P (P (P (, P (, P.101

6.1 (P (P (P (P (P (P (, P (, P.101 (008 0 3 7 ( ( ( 00 1 (P.3 1 1.1 (P.3.................. 1 1. (P.4............... 1 (P.15.1 (P.15................. (P.18............3 (P.17......... 3.4 (P................ 4 3 (P.7 4 3.1 ( P.7...........

More information

08-Note2-web

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)

More information

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C602E646F63>

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C602E646F63> スピントロニクスの基礎 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. http://www.morikita.co.jp/books/mid/077461 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 1 2 ii 3 5 4 AMR (anisotropic magnetoresistance effect) GMR (giant magnetoresistance

More information

2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+

2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+ R 3 R n C n V??,?? k, l K x, y, z K n, i x + y + z x + y + z iv x V, x + x o x V v kx + y kx + ky vi k + lx kx + lx vii klx klx viii x x ii x + y y + x, V iii o K n, x K n, x + o x iv x K n, x + x o x

More information

23 7 28 i i 1 1 1.1................................... 2 1.2............................... 3 1.2.1.................................... 3 1.2.2............................... 4 1.2.3 SI..............................

More information

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),

More information

pdf

pdf http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg

More information

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F

More information

B 1 B.1.......................... 1 B.1.1................. 1 B.1.2................. 2 B.2........................... 5 B.2.1.......................... 5 B.2.2.................. 6 B.2.3..................

More information

6.1 (P (P (P (P (P (P (, P (, P.

6.1 (P (P (P (P (P (P (, P (, P. (011 30 7 0 ( ( 3 ( 010 1 (P.3 1 1.1 (P.4.................. 1 1. (P.4............... 1 (P.15.1 (P.16................. (P.0............3 (P.18 3.4 (P.3............... 4 3 (P.9 4 3.1 (P.30........... 4 3.

More information

6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2

6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2 1 6 6.1 (??) (P = ρ rad /3) ρ rad T 4 d(ρv ) + PdV = 0 (6.1) dρ rad ρ rad + 4 da a = 0 (6.2) dt T + da a = 0 T 1 a (6.3) ( ) n ρ m = n (m + 12 ) m v2 = n (m + 32 ) T, P = nt (6.4) (6.1) d [(nm + 32 ] )a

More information

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)

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)

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,. 24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)

More information

,, 2. Matlab Simulink 2018 PC Matlab Scilab 2

,, 2. Matlab Simulink 2018 PC Matlab Scilab 2 (2018 ) ( -1) TA Email : ohki@i.kyoto-u.ac.jp, ske.ta@bode.amp.i.kyoto-u.ac.jp : 411 : 10 308 1 1 2 2 2.1............................................ 2 2.2..................................................

More information

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A .. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.

More information

() 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)

() 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 ()

More information

Chap11.dvi

Chap11.dvi . () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x

More information

Gmech08.dvi

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.)

More information

曲面のパラメタ表示と接線ベクトル

曲面のパラメタ表示と接線ベクトル L11(2011-07-06 Wed) :Time-stamp: 2011-07-06 Wed 13:08 JST hig 1,,. 2. http://hig3.net () (L11) 2011-07-06 Wed 1 / 18 ( ) 1 V = (xy2 ) x + (2y) y = y 2 + 2. 2 V = 4y., D V ds = 2 2 ( ) 4 x 2 4y dy dx =

More information

総研大恒星進化概要.dvi

総研大恒星進化概要.dvi The Structure and Evolution of Stars I. Basic Equations. M r r =4πr2 ρ () P r = GM rρ. r 2 (2) r: M r : P and ρ: G: M r Lagrange r = M r 4πr 2 rho ( ) P = GM r M r 4πr. 4 (2 ) s(ρ, P ) s(ρ, P ) r L r T

More information

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a = II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [

More information

1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D

1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA  appointment Cafe D 1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00

More information

untitled

untitled ( ) c a sin b c b c a cos a c b c a tan b a b cos sin a c b c a ccos b csin (4) Ma k Mg a (Gal) g(98gal) (Gal) a max (K-E) kh Zck.85.6. 4 Ma g a k a g k D τ f c + σ tanφ σ 3 3 /A τ f3 S S τ A σ /A σ /A

More information

1 c Koichi Suga, ISBN

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

More information

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 - M3............................................................................................ 3.3................................................... 3 6........................................... 6..........................................

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information