t (x(t), y(t)), a t b (x(a), y(a)) t ( ) ( ) dy s + dt dt dt [a, b] a a t < t 1 < < t n b {(x(t i ), y(t i ))} n i ( s(t) ds ) ( ) dy dt + dt dt ( ) d

Size: px
Start display at page:

Download "t (x(t), y(t)), a t b (x(a), y(a)) t ( ) ( ) dy s + dt dt dt [a, b] a a t < t 1 < < t n b {(x(t i ), y(t i ))} n i ( s(t) ds ) ( ) dy dt + dt dt ( ) d"

Transcription

1 1 13 Fall Semester N. Yamada Version: Chapter. Preliminalies (1 3) Chapter 1. (4 16) Chapter. (17 9) Chapter 3. (3 49) Chapter 4. (5 63) Chapter 5. (64 7) Chapter 6. (71 8) 11, ISBN The latest version is available from the URL: nyamada/ 1 Q: A: Q: A: Q: A: peanuts, peanuts Q: ( )

2 t (x(t), y(t)), a t b (x(a), y(a)) t ( ) ( ) dy s + dt dt dt [a, b] a a t < t 1 < < t n b {(x(t i ), y(t i ))} n i ( s(t) ds ) ( ) dy dt + dt dt ( ) ds + dt ( ) dy dt dt ( t ) (line element) s s ds ( ) + ( dy) ds t a ( ) + dt ( dy dt ) dt s (x(t), y(t)) ( ) ( T (t) dt, dy ) dt 1

3 T (t) ( ) + dt ( ) dy dt ( ) ds dt ( ) ( ) dy T (s) + ds ds ( ( ) ( ) ) ( ) dy dt + dt dt ds ( ) ( ) ds dt 1 dt ds.1 y f(x) ds 1 + ( dy) (r(t), θ(t)) x(t) r(t) cos θ(t), y(t) r(t) sin θ(t) dt dr dθ cos θ r sin θ dt dt dy dt dr dθ sin θ + r cos θ dt dt dr cos θ r sin θ dθ dy dr sin θ + r cos θ dθ

4 ( ds) ( ) + ( dy) ( dr) cos θ r dr sin θ cos θ dθ + r sin θ( dθ) + ( dr) sin θ + r dr sin θ cos θ dθ + r cos θ( dθ) ( dr) + r ( dθ) ds ( dr) + r ( dθ). ρ r(θ) ρ, θ θ θ ds + ρ ρ s π ds π ρ dθ πρ.1 ds. x + y 1 ( x 1).3 x-y m (3 ) Pythagoras (circa 75 B. C.??) 3

5 1 θ x a + y b 1 x a cos θ, y b sin θ 1.1 x + y 1 y + z (1,, ), (, 1, 1) (, 1, 1) 1 P 1/ 1/ 1/ 1/ e 1, e, e 3 x cos θ, y sin θ, z sin θ P ( ) cos θ cos θ P sin θ sin θ sin θ x-y 4

6 a S 4 b 1 x a (x u, a du) a 1 4ab 1 u du (u sin θ, du cos θ dθ) 4ab 4ab abπ π/ π/ cos θ dθ 1 + cos θ dθ y ( ) φ x a sin φ, y b cos φ (a > b) 5

7 s φ ( ) ( ) dy s + dφ dφ dφ φ a cos φ + b sin φ dφ φ a (1 sin φ) + b sin φ dφ φ a 1 a b sin φ dφ a φ ( ) a a 1 k sin b φ dφ k E(k, φ) φ 1 k sin φ dφ k (modulus) (imcomplete elliptic integral of the second kind) ( E k, π ) π E(k) 1 k sin φ dφ (complete elliptic integral of the second kind) sin φ z φ arcsin z, dφ a dz 1 z E(k, φ) E(k) sin φ sin φ 1 dz 1 k z 1 z 1 k z dz 1 z 1 k z 1 z dz 6

8 1. k E() 1.3 x 9 + y x + y x + xy + y 1.6 E(k) z z 1 1. ( ) y b sin x a (b : πa : 1 a : ) P (x, y ) s x ( ) dy s 1 + x ( b 1 + a cos x ) a x 1 + b a b x a sin a ϕ x a, ϕ x a, dϕ 1 a a 1 + b a b a sin ϕ dϕ a + b b sin ϕ dϕ ) (k b ϕ ϕ a + b ϕ a + b 1 k sin ϕ dϕ a + b E(k, ϕ ) 7

9 x a π ϕ π s a + b E(k) (imcomplete elliptic integral of the first kind) F (k, φ) φ dφ 1 k sin φ ( k < 1) k (modulus) φ π K(k) F ( k, π ) π/ dφ 1 k sin φ 1 (complete elliptic integral of the first kind) dz cos φ dφ φ dφ z 1 k sin φ z sin φ 1 1 k z dz cos φ ( z φ z (sin φ) ) z 1 1 k z 1 1 sin φ dz F (k, φ) z dz (1 z )(1 k z ) 8

10 1.8 K() π/ dφ 1 dz 1 z π 1.9 < k < 1 K(k) z 1.3 ( ) (lemniscate) r a cos θ x-y x r cos θ, y r sin θ x + y r x y r (cos θ sin θ) r cos θ x + y a r (x y ) (x + y ) a (x y ) x + y a cos ϕ, x y a cos 4 ϕ 9

11 ( ) x a (cos ϕ + cos 4 ϕ) a cos ϕ(1 + cos ϕ) a cos ϕ( sin ϕ) ( a cos ϕ 1 1 ) sin ϕ y a (cos ϕ cos 4 ϕ) a cos ϕ(1 cos ϕ) 1 a cos ϕ sin ϕ x a cos ϕ 1 1 sin ϕ y a sin ϕ cos ϕ ϕ (a, ), ϕ π (, ) a sin ϕ 1 1 sin ϕ cos ϕ sin ϕ + a cos ϕ dϕ 1 1 sin ϕ ( ( a sin ϕ 1 1 ) 1 1 sin ϕ sin ϕ 1 ) cos ϕ dϕ ( a sin ϕ 3 ) 1 1 sin ϕ + sin ϕ dϕ dy a (cos ϕ sin ϕ) dϕ a (1 sin ϕ) dϕ 1

12 ( ds) ( ) + ( dy) { ( a sin ϕ ) } sin ϕ + sin ϕ + a (1 sin ϕ) ( dϕ) a { } 1 1 sin ϕ ( sin ϕ 3 ) + sin ϕ + 1 ( (1 sin ϕ) 1 1 ) sin ϕ ( ) 9 sin ϕ 4 3 sin ϕ + sin 4 ϕ + 1 ( (1 4 sin ϕ + 4 sin 4 ϕ) 1 1 ) sin ϕ ( ) 9 sin ϕ 4 3 sin ϕ + sin 4 ϕ + 1 ( 1 9 ) sin ϕ + 6 sin 4 ϕ sin 6 ϕ 1 ( ) ( ds) a sin ϕ ( dϕ) ds a dϕ 1 1 sin ϕ s a ϕ dϕ 1 1 sin ϕ a ( ) 1 F, ϕ 11

13 ϕ π 4 s( ) 4 a π/ dϕ 1 1 sin ϕ 4 a ( ) 1 K : ( x a z + z3 z3, y az 1 + z4 1 + z 4 x + y a (z + z 6 ) (1 + z 4 ) a z 1 + z 4 x y 4a z 4 (1 + z 4 ) (x + y ) a (x y ) ( ds) ( ) + ( dy) a (1 + z 4 ) (1 + 3z 3z 4 z 6 ) dz a dy (1 + z 4 ) (1 3z 3z 4 + z 6 ) dz a (1 + z 4 ) 4 {(1 + 3z 3z 4 z 6 ) + (1 3z 3z 4 + z 6 ) }( dz) a (1 + z 4 ) 3 (1 + z 4 ) 4 ( dz) a 1 + z 4 (dz) x + y a cos ϕ, x y a cos 4 ϕ 1

14 ϕ : π z : 1 dϕ dz a ds a dϕ 1 1 sin ϕ dϕ 1 1 sin ϕ a dz 1 + z 4 dϕ dz 1 1 sin ϕ 1 + z 4 s a ϕ dϕ 1 1 sin ϕ a ( ) 1 F, ϕ ϕ dϕ 1 1 sin ϕ z 1 ( ) dz 1 F, ϕ 1 + z 4 1 dz 1 ( ) z 4 F, ϕ z ϕ z x + y z cos ϕ 1 + z 4 x a z + z3 z3, y az 1 + z4 1 + z 4 z x + y a 1 + z 4 x + y a z 1 + z 4 13

15 ( x + y az(x + y) x az ) ( + y az ) ( ) az ( (x, y) az, az ) c 1 x 4 x 1 η x η dη x : 1, η : 1 c 1 c 1 x c η x dη η (1 1 η ) dη (1 η )(1 1 η ) (η sin φ, dη cos φ dφ) 1 π/ dφ (c cos ϕ) ϕ 1 1 sin φ 1 ( 1 {F, π ) ( )} 1 F, ϕ 14

16 1 1 x 4 c 1 1 x 4 1 F 1 F ( ) 1, ϕ ( 1, π ) 1 ( ) 1 K z ( ) F, ϕ c 1 1 x 4 z dz 1 + z 4 c cos ϕ z 1 + z 4 dz 1 + z 4 dz 1 z 4 f(x) x 1 x r a cos θ ds ( ( dr) + (r dθ) 1 + r dθ ) dr dr 3 F. Gauss ( ) 15

17 r dr dθ a sin θ r dr dθ a 1 cos θ a 1 r4 a 4 r dr dθ a 4 z 4 1 dθ r dr 1 a 4 r 4 r dθ dr r a 4 r 4 ds 1 + r4 a 4 r 4 dr a 4 a 4 r 4 dr a a 4 r dr 4 s r a a 4 r 4 dr s( ) 4 a a a 4 r 4 dr (r aσ, dr a dσ, r : a, σ : 1) 4 4a 1 1 a 4a K ( 1 a 1 σ a dσ 4 dσ 1 σ 4 ) 16

18 1 3.1 dz (1 z )(1 k z ) ( 1 ) 1 k z 1 z dz ( ) dz (1 + nz ) (1 z )(1 k z ) ( 3 ) k ( k < 1) n 4-5 z sin φ dz cos φ dφ, dz 1 z dφ F (k, φ) E(k, φ) π(k, n, φ) φ φ φ dφ 1 k sin φ 1 k sin φ dφ dφ (1 + n sin φ) 1 k sin φ [, π ] K(k) E(k) π/ π/ 4 A. M. Legendre ( ) 5 C. G. J. Jacobi ( ) dφ 1 k sin φ ( 1 ) 1 k sin φ dφ ( ) 17

19 ( ) F (x, y, z) (f 1 (x, y, z), f (x, y, z), f 3 (x, y, z)) ( ) U(x, y, z) F U f 1 U x, f U y, f 3 U z U F (potential) ( ) 1 r P (x, y, z) U(x, y, z) 1 r 1 x + y + z P ( x U r, y 3 r, z ) 3 r 3 P U 1 r.1 ( ( ) ) ( ) 1 y-z r Q Q(, r cos θ, r sin θ) x-y P (a, b, ) () 18

20 Q P 1 P Q P Q a + (b r cos θ) + r sin θ a + b + r br cos θ a + (b + r) br(1 + cos θ) a + (b + r) 4br cos θ r r dθ U π r P Q dθ π r P Q dθ U r π dθ a + (b + r) 4br cos θ k 4br a + (b + r) θ π φ dθ dφ cos θ 19

21 cos( π φ) sin φ U r π a + (b + r) dθ 1 k cos θ 4r π/ dφ a + (b + r) 1 k sin φ π/ k 4r br r k b K(k) dφ 1 k sin φ. ( ) ( ) 6 Newton 7 ( ) x x(t) x (t) : x (t) : 6 R. Hooke ( ) 7 I. Newton ( )

22 x (t) αx(t) α () x 1 (t) cos αt, x (t) sin αt x(t) Ax 1 (t) + Bx (t) f(x) f(x) (αx + βx 3 ) f(x) f() f(x) x 8 1 f(x) αx f(x) αx βx 3 x ( ) x m d x dt (αx + βx3 ) β ( ) α α m t s dt m ds 1 8 B. Taylor ( ) ds dt dt ds d x ds m d x α dt m α dt 1

23 d x ds m α ( αm x βm x3 ) x β α x3 β α x ξ d ξ β ( x ds βα ) α x3 (ξ + ξ 3 ) ξ x, s t d x dt (x + x3 ) t, x d x dt (x + x3 ) /dt dt dt dt (x + x 3 ) dt dt

24 1 ( ) E x dt x4 4 E 1 ( ) : dt U x + x4 4 : E /dt ( ) dt ± E x x4 x t ± E x x4 dt U E ( ) x a 3

25 E a a4 a 4 + a 4E a E E x 1 x4 1 (a x )( + a + x ) ( a, ) x a cos φ a sin φ dφ x a cos φ a (1 sin φ) a x a sin φ + a + x + a + a (1 sin φ) (1 + a ) a sin φ t ± φ ± φ a sin φ a sin φ (1 + a ) a sin φ dφ dφ (1 + a ) a sin φ 1 ± 1 + a φ dφ 1 k sin φ where k a (1 + a ) ± kf (k, φ) a 4

26 T φ π 4 π T 4 a kk(k) α m t s, β α x ξ dt E (x) m t(ξ ) α dξ ds β ds dt ( dt α mβ dξ ds ) α β E (ξ) m α E(ξ ) β α t, x, E t (x), x (x), E (x), s t (s), ξ x (ξ), E (ξ) a x a x ( ) ( x a cos φ φ ) a x a (x) α β a (ξ) a (x) α β 1 β ( β α E (x) ( ) α + α + 4βE (x) ) 5

27 k (ξ) β a (ξ) (1 + a (ξ) ) a α (x) (1 + β a α (x) ) βa (x) (α + βa (x) ) k (x) m m t (x) ± α t 1 (ξ) ± α 1 + a(ξ) m 1 ± α 1 + β a α (x) m φ ± α + βa(x) T T 4 m α + βa φ φ dφ 1 k (x) sin φ dφ 1 k (x) sin φ π/ 4 m α + βa K(k) dφ 1 k sin φ β cos β > T m d x dt α m, αx sin m T π α α m 4 m α + βa K(k) ( k 6 ) βa (α + βa ) dφ 1 k (ξ) sin φ

28 β k K() π T 4 m π m α π α.3 ( ) ( ) α θ(t) l, m : m ( l dθ ) dt : mgl(1 cos θ) ( m l dθ ) + mgl(1 cos θ) E dt (g ) θ α E mgl(1 cos α) m ( l dθ ) + mgl(1 cos θ) mgl(1 cos α) dt 7

29 ( cos θ cos α sin α θ ) sin dθ g dt k l sin θ k sin α sin θ k sin θ k sin φ dθ g dt l k cos φ θ arcsin(k sin φ) dθ k cos φ 1 k sin φ dφ t t dt 1 1 dt dθ dθ l g l g l g dθ k cos φ 1 k cos φ k cos φ 1 k sin φ dφ dφ 1 k sin φ φ dφ g t g 1 k sin φ dt l l t 8

30 l t F (k, φ) g T θ θ α φ φ π 4 l T 4 g K(k) α k k l T π g g 9.8 (m/s ) π 3.14 l.5 (m) T 1.3 (s) 5 cm.1 T m/s 1 9

31 3 (sin) y x 1 x ( 1 x 1) y arcsin x x sin y u sn 1 x u x (1 x )(1 k x ) sn 1 x u sn 1 (x, k) ( k < 1) sn x sn u sn (u, k) sn u sn 1 x 1 x 1 K(k) u K(k) (sn u) sn u 3. sn 1 sn 3

32 3.3 sn( K(k), k), sn (, k), sn (K(k), k) x sn u K(k) u K(k) K(k) u 3K(k) sn(u, k) sn(k u, k) u K u K(k) sn(3k, k) sn(k, k) 1 sn( K, k) R sn(u, k) 3. cn u cn(u, k), dn u dn(u, k) cn u 1 sn u or cn u 1 sn u dn u 1 k sn u or dn u 1 k sn u ( K(k) u K(k)) ( K(k) u K(k)) k K() π sn(u, ) sin u cn(u, ) cos u dn(u, ) k 3.5 cn( K(k), k), cn (, k), cn (K(k), k), dn( K(k), k), dn (, k), dn (K(k), k) 31

33 () ( ) sn u cn u, dn u cn(u, k) cn(u K, k) (K u 3K) dn(u, k) dn(u K, k) (K u 3K) sn(u, k), cn(u, k) 4K(k), dn(u, k) K(k) 3.6 cn u, dn u 3 R ( K, K ) ( ) k 1 sinh x ex e x, cosh x ex + e x cosh x sinh 1 (sinh x) cosh x, (cosh x) sinh x tanh x sinh x cosh x, sech x 1 cosh x 3

34 (tanh x) 1 cosh x sech x, 1 tanh x 1 cosh x tanh x d tanh 1 x cosh y 1 1 tanh y 1 1 x k 1 u x u k1 (1 x )(1 k x ) x 1 x tanh 1 x sn(u, 1) tanh u cn(u, 1) dn(u, 1) 1 cosh u 3.7 sn(u, k), cn(u, k), dn(u, k) Maple Maxima Maple 33

35 plot(jacobisn(u,.5), u-5..5); plot(jacobicn(u,.5), u-5..5); plot(jacobidn(u,.5), u-5..5, y..1); sn(u,.5) cn(u,.5) dn(u,.5) 34

36 K(.5) ( Maple ) u x cos ϕ dϕ u sin ϕ sin ϕ (1 x )(1 k x ) sn 1 x x sin ϕ cos ϕ dϕ (1 sin ϕ)(1 k sin ϕ) dϕ 1 k sin ϕ sn 1 (sin ϕ) u ϕ am u (amplitude) u sin ϕ ϕ am u am(u, k) dϕ 1 k sin ϕ am 1 ϕ x sin ϕ sin(am u) u am 1 ϕ sn 1 (sin ϕ) sn u sin ϕ sin(am u) cn u 1 sn u 1 sin (am u) cos(am u) x sn u 1 x 1, K u K 4K π ϕ π xsin ϕ 1 x 1 sn u K u K 35

37 sn u sin ϕ sin(am u) ϕ am u K u K π ϕ π K u 3K π ϕ 3π sin ϕ sin(π ϕ) sn u sn(u K) sin(am(u K)) sin( am(u K)) π ϕ am(u K) ϕ π + am(u K) am u 9 am(k) π am(3k) π + am K 3π am(nk(k)) nπ (n Z) x sn u u x (1 x )(1 k x ) du 1 (1 x )(1 k x ) du (1 x )(1 k x ) (1 sn u)(1 k sn u) cn u dn u 9 Maple K u K 36

38 d sn u du cn u dn u k (sin u) cos u d cn u du d dn u du d (1 sn u) du sn u dn u sn u 1 sn u d sn u du (k (cos u) sin u) d 1 k sn du u k sn u 1 k sn u k sn u cn u d sn u du 3.1 ( ) (sn u) cn u dn u (cn u) sn u dn u (dn u) k sn u cn u 3.8 sn u, cn u, dn u u, u ±K 3.9 d sn u du, d cn u, d dn u du du sn 1 x x (1 x )(1 k x ) cn 1 x dn 1 x 37

39 (cn u) sn u dn u 1 x 1 k (1 x ) (here, x cn u) du 1 x 1 k (1 x ) 1 x k + k x (where k 1 k ) u u x 1 (1 x )(k + k x ) ( x 1 u x cn 1 ) 1 x (1 x )(k + k x ) dn u 1 k sn u sn u 1 k (1 dn u) cn u 1 sn u 1 k (dn u + k 1) 1 k (dn u k ) (here, k 1 k ) (dn u) k sn u cn u du 1 1 k 1 x x k k k (1 x )(x k ) (where x dn u) u x 1 (1 x )(x k ) ( x 1 u x dn 1 ) 1 x (1 x )(x k ) 38

40 3. ( ) 1 sn 1 (x, k) cn 1 (x, k) dn 1 (x, k) x 1 x 1 F (k, φ) x (1 x )(1 k x ) (1 x )(k + k x ) (1 x )(x k ) (where k 1 k ) φ dφ 1 k sin φ u sin φ du cos φ dφ F (k, φ) u du 1 u 1 k u F (k, φ) sn 1 u 1 sn u 3 ϕ E(k, ϕ) 1 k sin ϕ dϕ sin ϕ sn u cos ϕ cn u cos ϕ dϕ cn u dn u du dϕ dn u du 1 k sin ϕ 1 k sn u dn u 39

41 E(k, ϕ) u dn u du u sin ϕ sn u ε(u) u ( ) 3 π(k, n, φ) φ dn u du dz (1 + nz ) (1 z )(1 k z ) z sn u dz cn u dn u du 1 z dn u du 1 (1 + nz ) (1 z )(1 k z ) 1 (1 + n sn u) 1 z dn u π(k, n, φ) u du 1 + n sn u 3.3 () F (k, ϕ) sn 1 u E(k, ϕ) π(k, n, φ) u u dn u du du 1 + n sn u ( 1 ) ( ) ( 3 ) x a + y 1 (a b > ) b 4

42 ϕ x a sin ϕ, y b cos ϕ ϕ am(u, k) x a sin(am(u, k)) a sn(u, k) y b cos(am(u, k)) b cn(u, k) k < 1 k a b a cn u dn u du dy b sn u dn u du ds a cn u + b sn u dn u du a ds a s a 1 a b a ds a dn u du u dn u du aε(u) ae(k, ϕ) sn u dn u du 41

43 x a sn(u, k), y b cn(u, k) k a b a k a b a e (eccentricity) e < 1 e e S(ae, ), S ( ae, ) (focus) BS a e + b a a OS ae P (x, y) P S (OS x) + y (ae a sn u) + b cn u a (e sn u) + b (1 sn u) a (e sn u) + a (1 e )(1 sn u) a (1 e sn u + e sn u) 4

44 P S a (1 e sn u) P S (OS + x) + y a (e + sn u) + a (1 e )(1 sn u) a (1 + e sn u + e sn u) P S a (1 + e sn u) P S + P S a(1 e sn u) + a(1 + e sn u) a : l S OA OB b a 1 e ae (where e k 1 e ) P S OS P (x, y) x a sn u OS ae e sn u l P S l y b cn u ae 1 sn u ae 43

45 () P (x, y) P x a sn u, y b cn u dy dy/du /du b a sn u cn u e sn u cn u y b cn u e sn u (x a sn u) cn u b cn u + ae sn u cn u ae cn u (cn u + sn u) ae cn u y e sn u cn u x + ae cn u y px + q q 1 + p l a e cn u 1 + e sn u cn u a e cn u + e sn u a e 1 (1 e ) sn u a e 1 e sn u a e dn u (note that k e ) 44

46 l l ae dn u P (x, y) x a sn u, y b cn u dy e sn u cn u : cn u e sn u tan θ SP : b cn u a sn u ae e cn u sn u e tan θ + S P : b cn u a sn u + ae e cn u sn u + e tan θ tan(θ θ ± ) tan θ tan θ ± 1 + tan θ tan θ ± sn u cn u e cn u e sn u cn u e sn u(sn u e) + e cn u (1 e ) sn u cn u e cn u e ee sn u e cn u(e sn u 1) e e (1 e sn u) e cn u 45

47 S, S P S P ( ) S (vice versa) : ( 3.1 ( 1 ) R (R, θ, φ) x R sin θ cos φ y R sin θ sin φ z R cos θ s(t) s(t) (R sin θ(t) cos φ(t), R sin θ(t) sin φ(t), R cos θ(t)) ( ds) ( ) + ( dy) + ( dz) (R cos θ cos φ dθ R sin θ sin φ dφ) + (R cos θ sin φ dθ + R sin θ cos φ dφ) + R sin θ( dθ) R sin θ(dφ) + R ( dθ) r R sin θ 1 Seifert (????) 46

48 r ( ds) (r dφ) + (R dθ) dr R cos θ dθ ( dr) R cos θ( dθ) R (1 sin θ)( dθ) (R r )( dθ) ( ds) (r dφ) + (R dr) R r s(t) φ k Rφ(t) ks(t) k φ(t) x k 1 s(t) k < k > k k 1 dφ k R ds (ds) k r R ( (R ds) dr) + R r (ds) R 4 (R r )(R k r ) (dr) ( ds) R R ( dr) (R r )(R k r ) x r R 47

49 ( dr) R ( ) ( ds) R θ ( ) s R x R 4 ( ) (R r )(R k r ) ( ) (1 x )(1 k x ) (1 x )(1 k x ) ( x ) r R sn s R z R r R cn s R z R cn s R dn s R 1 k sn s R 1 k r R k r R k ( ) dr R 4 (R r )(R k r ) ds R dr ds R R r k r R dr dθ R r R dr R ds k r dr R dθ R dθ R ds k r R R dθ R θ ds 48 R

50 α R dθ ds cos α sn s R r R cn s R z R dn s R cos α 49

51 4 y b sn x c b, c y b sn x c dy b c cn x c dn x c s ( ds) ( ) + ( dy) ( ) ds ( dy ( b c b ) ) cn x c dn x c (cn u 1 sn u 1 1 k (1 dn u)) (dn u 1 k sn u 1 k ) ( ) b 1 k 1 c k dn x ( ) b c + 1 x c k dn4 c k 1 k c ( ) ds 1 4 x ( ) 1 k dn c + 1 k dn 4 x c s x x k ds x 1 k dn c 1 ds ( x ) 1 k dn c 1 x dn x c x (where k 1 k ) ε(u) u dn u du 5

52 1 k sin ϕ dn u sin ϕ sn u cos ϕ dϕ cn u dn u du cos ϕ 1 sin ϕ 1 sn u cn u dϕ dn u du ε(u) u ϕ dn u du u dn u dn u du 1 k sin ϕ dϕ E(k, ϕ) ( ) x x ξ c u, ξ : x, dn ξ c dξ c x/c dn ξ c dξ dξ c du u : x c s c ( x ) k ε x c ( x ) dn u du cε c x x x a x a sn u u K(k) 51

53 x c a c K(k) K(k) 1 1 c K(k) a x : a u : K(k) E E(k) ϕ : π (by sin ϕ sn u) π/ K 1 k sin ϕ dϕ dn u du ε(k) x x a l c k E a 4aE Kk a x(t) r cos ωt y(t) r sin ωt x + y r : ω : ω : r, : T π ω, : ν ω π v(t) (x (t), y (t)) ( rω sin ωt, rω cos ωt) rω( sin ωt, cos ωt) 5

54 rω f v (t) rω (cos ωt, sin ωt) rω ( ) f ( ) rω x x x, x a ρ ω T : : y x y 53

55 x s (x(s), y(s)) (x (s), y (s)) T cos ψ T ds T x x d ds (T cos ψ) d ( T ) ds ds T cos ψ T ds T y ds ρ ds, y f ρω y ds 54

56 ds y T dy ds Taylor 1 T dy ds + d ds ( T dy ) ds ds ρω y ds d ds ( T dy ) ds ds ( d T dy ) + ρω y ds ds y T ( d T dy ) + ρω y ds ds T ds T d ds ( dy ) + ρω T y p dy dp ds dp dy dy ds dp dy dy ds pdp dy ds p dp dy ds + ρω y T 55

57 ( ) ds ( ) + ( dy) () 1 + p p dp 1 + p dy ρω y T p(y) p dp ρω y dy 1 + p T 1 + p ρω T y + C C p dy (y ) b C 1 + ρω T b 1 + p 1 + ρω T (b y ) p 1 + p 1 + ρω T (b y ) + ρ ω 4 4T (b y ) } p ρω (b y ) {1 + ρω (b y ) T 4T b c (1 η )(1 k η ) ( dy ) y 56

58 ρω b 4T k (then < k < 1) 1 + ρω b 4T 1 c ρω (1 + ρω b ) η y b c T 4T dη (1 η )(1 k η ) 1 c k k b 4.1 (k 1 k ) ( x ) η sn c ( x ) y b sn c b k k c a c 1 c K(k) a ( ) Kx y sn a ( a k ) sn l 4a E k K a ( ) 57

59 l a l 4a E(k) k K(k) a l k b b k k c, 1 c K(k) a 1 c ρω T (1 + ρω b ) 4T x T ω π T ( ) x ψ tan ψ dy b x c cn x c dn x c b x c x T cos ψ T T T b ( ) k T cos ψ c + 1 T + 1 l a () ( ) k (variational principle) i.e., ds 1 + ( ) dy : y ds y 1 + ( ) dy 58

60 y f ω y u 1 ω y f du ρ, ds, dy l l U ρω y ds ρy ds x (x, y) R g(x, y) f(x, y) g(x, y) (x, y) x-y C z f(x, y) C C g(x, y) f(x, y) C (a, b) (a, b) (a, b) C y φ(x) g y (a, b) () y φ(x) f(x, y) 1 f(x, φ(x)) (a, b) x a f x (a, b) + f y (a, b)φ (a) g(x, φ(x)) g x (a, b) + g y (a, b)φ (a) 59

61 φ (a) f x (a, b) g x(a, b) g y (a, b) f y(a, b) f x (a, b) g x (a, b) f y(a, b) g y (a, b) λ f x (a, b) λg x (a, b), f y (a, b) λg y (a, b) 4.1 ( 11 ) g(x, y) f(x, y) (a, b) g x (a, b) g y (a, b) f x (a, b) λg x (a, b), f y (a, b) λg y (a, b) λ λ F (x, y) f(x, y) λg(x, y) R F x (x, y), F y (x, y) f x λg x, f y λg y x, y λ g(x, y) λ λ x, y 4.1 x + y x + y F (x, y) x + y λ(x + y ) F x x λ, F y y λ 11 J. L. Lagrange ( ) 6

62 F x F y x λ, y λ x + y λ (x, y) (1, 1) ( ) y x z x + y z (x 1) + y y b ( I(y) F y, dy ) a y(a) y(b) I(y) y ϕ(a) ϕ(b) ϕ f(ε) I(y + εϕ) : R R ε df dε ε df dε d b ( F y + εϕ, d(y ) + εϕ) dε a b ( F a y ϕ + F ) dϕ p b ( F y d ) F ϕ p ϕ y a F y d ( ) F p 1 ( ) dy 1 + : ( ) dy y 1 + : 1 L. Euler ( ) 61

63 α I(y) a (y α) 1 + ( ) dy F (y, p) (y α) 1 + p dy ( ) d (y α) dy y ( dy ) (y α) d dy 1 + ( dy ) + y 1 d dy 1 + ( dy ) (y α) dy ( 1 log 1 + d y 1 ( ) 3/ 1 + ( dy ) ( dy ) 1 + ( dy ) 1 + ( dy ) 1 d y ( ) 3/ 1 + ( dy ) 1 ( ) 1/ 1 + ( dy ) dy d y 1 + ( dy ) d y 1 + ( dy ) ( 1 + ( dy ) ) 3/ ( ) dy d y y ( ) 1/ 1 + ( dy ) y α 1 + ( dy ) d y y y dy y α ( ) ) dy log y α + Const. 6

64 1 + ( ) dy β(α y ) β : Const. dy y b β 1 (α b ) ( ) dy β(α y ) 1 1 (α b ) (α y ) 1 1 { (α y (α b ) ) (α b ) } 1 (α b ) (b y )(α y b ) ( ) dy α b (b y ) (1 + b y ) (α b ) α b ρω T p ρω (b y ) (1 + (b y )ρω ) T 4T ( ) 63

65 5 k k k 1 sn(u, k) sin u (k ) cn(u, k) cos u (k ) dn(u, k) 1 (k ) 5.1 sin u tanh u sn(u, k) tanh u (k 1) cn(u, k) sech u (k 1) dn(u, k) sech u (k 1) sin(u + v) sin u cos v + cos u sin v tanh(u + v) tanh u + tanh v 1 + tanh u tanh v 5. sinh u, cosh u, tanh u, sech u 5.1 (sn ) sn(u + v) sn u cn v dn v + sn v cn u dn u 1 k sn u sn v u + v c F (u) sn u cn(c u) dn(c u) + sn(c u) cn u dn u 1 k sn u sn (c u) F (u) sn c c u F F u 64

66 sn u s 1, cn u c 1, dn u d 1 sn(c u) s, cn(c u) c, dn(c u) d N F (u) s 1 c d + s c 1 d 1 D F (u) 1 k s 1 s F u N D DN D F/ u N D DN s 1 c 1 d 1, c 1 s 1 d 1, d 1 k s 1 c 1 s c d, c s d, d k s c N s 1 c d + s c 1 d 1 s 1 s + s s 1 N s 1 s s 1 s + s s 1 + s s 1 s s 1 s 1 s s 1 c 1 d 1 + c 1 d 1 s 1 d 1 k s 1 c 1 s 1 (1 k s 1 ) k s 1 (1 s 1 ) (1 + k )s 1 + k s 1 3 s c d c d s d k s c (1 + k )s + k s 3 N (1 + k )s 1 s + k s 1 3 s + (1 + k )s 1 s k s 3 s 1 k s 1 s (s 1 s ) D 1 k s 1 s D k s 1 s s 1 k s 1 s s k s 1 s (s c 1 d 1 s 1 c d ) 65

67 ND (s 1 c d + s c 1 d 1 )k s 1 s (s 1 c d s c 1 d 1 ) k s 1 s (s 1 c d s c 1 d 1 ) k s 1 s (s 1 (1 s )(1 k s ) s (1 s 1 )(1 k s 1 )) (s 1, s (s 1 s ) ) k s 1 s (s 1 s )(1 k s 1 s ) N D F (u) F (u) u F (u) F () sn c sn u Jacobi k sn(u, k) sin u, k 1 sn(u, k) tanh u k, k 1 sin(u + v) sin u cos v + cos u sin v sin u d sin v + d sin u dv du sin v tanh u + tanh v tanh(u + v) 1 + tanh u tanh v d tanh u du 1 cosh u 1 tanh u tanh u d tanh v + d tanh u tanh v dv du tanh u(1 tanh v) + (1 tanh u) tanh v tanh u + tanh v tanh u tanh v(tanh u + tanh v) (tanh u + tanh v)(1 tanh u tanh v) 66

68 tanh(u + v) tanh u tanh v d tanh v tanh u dv tanh u d tanh v dv + d tanh u du 1 tanh u tanh v tanh v 1 tanh u tanh v tanh v + d tanh u du sn u k k 1 sn(u + v) d sn v sn u dv + d sn u du sn v 1 k sn u sn v 5.3 sn u k, k 1 sin u, tanh u 5.4 sin u ((sin u) cos u ) 5. (cn ) cn(u + v) cn u cn v sn u sn v dn u dn v 1 k sn u sn v sn(u + v) sn u cn v dn v + sn v cn u dn u 1 k sn u sn v s 1c d + s c 1 d 1 1 k s 1 s cn (u + v) 1 sn (u + v) (1 k s 1 s ) (s 1 c d + s c 1 d 1 ) (1 k s 1 s ) (1 k s 1 s ) (s 1 + c 1 k s 1 s )(s + c k s 1 s ) (c 1 + s 1 d )(c + s d 1 ) 67

69 (c 1 + s 1 d )(c + s d 1 ) (s 1 c d + s c 1 d 1 ) c 1 c + c 1 s d 1 + c s 1 d + s 1 s d 1 d (s 1 c d + s 1 s c 1 c d 1 d + s c 1 d 1 ) (c 1 c s 1 s d 1 d ) cn (u + v) (c 1c s 1 s d 1 d ) (1 k s 1 s ) u v cn 1 cn(u + v) c 1c s 1 s d 1 d 1 k s 1 s cn u cn v sn u sn v dn u dn v 1 k sn u sn v 5.5 cn u k, k 1 cos u, sech u 5.3 (dn ) dn(u + v) dn u dn v k sn u sn v cn u cn v 1 k sn u sn v dn(u + v) 1 k sn (u + v) 1 k ( s1 c d + s c 1 d 1 1 k s 1 s ) (1 k s 1 s ) k (s 1 c d + s c 1 d 1 ) (1 k s 1 s ) (1 k s 1 s ) k (s 1 c d + s c 1 d 1 ) (d 1 + k s 1 c )(d + k s c 1 ) k (s 1 c d + s c 1 d 1 ) d 1 d + k s 1 c d + k s c 1 d 1 + k 4 s 1 s c 1 c k {s 1 c d + s 1 s c 1 c d 1 d + s c 1 d 1 } d 1 d k s 1 s c 1 c d 1 d + k 4 s 1 s c 1 c (d 1 d k s 1 s c 1 c ) 68

70 dn (u + v) (d 1d k s 1 s c 1 c ) (1 k s 1 s ) u v dn sn u sn v sn(u + v) sn u cn v dn v sn v cn u dn u sn u cn u dn v sn v cn v dn u cn(u + v) sn u cn v dn v sn v cn u dn u sn u cn v dn u sn v cn u dn v dn(u + v) sn u cn v dn v sn v cn u dn u sn(u + v) s 1c d + s c 1 d 1 1 k s s 1c d s c 1 d 1 1 s s 1 c d s c 1 d 1 s 1 c d s c 1 d 1 (1 k s 1 s )(s 1 c d s c 1 d 1 ) s 1 (1 s )(1 k s ) s (1 s 1 )(1 k s 1 ) s 1 (1 s k s + k s 4 ) s (1 s 1 k s 1 + k s 1 4 ) s 1 s + k s 1 s (s s 1 ) (s 1 s )(1 k s 1 s ) sn(u + v) (s 1 s )(1 k s 1 s ) (1 k s 1 s )(s 1 c d s c 1 d 1 ) s 1 s s 1 c d s c 1 d 1 cn (u + v) 1 sn (u + v) 1 (s 1 s ) (s 1 c d s c 1 d 1 ) (s 1c d s c 1 d 1 ) (s 1 s ) (s 1 c d s c 1 d 1 ) 69

71 s 1 c d s 1 s c 1 c d 1 d + s c 1 d 1 (s 1 s ) (s 1 c 1 d s c d 1 ) + s 1 c d s 1 c 1 d + s c 1 d 1 s c d 1 (s 1 s ) (s 1 c 1 d s c d 1 ) + s 1 d (c c 1 ) + s d 1 (c 1 c ) (s 1 s ) (s 1 c 1 d s c d 1 ) + (s 1 s ){s 1 d s d 1 (s 1 s )} (s 1 c 1 d s c d 1 ) cn(u + v) s 1c 1 d s c d 1 s 1 c d s c 1 d 1 v s, c 1, d 1 dn (u + v) 1 k sn (u + v) 1 k (s 1 s ) (s 1 c d s c 1 d 1 ) (s 1c d s c 1 d 1 ) k (s 1 s ) (s 1 c d s c 1 d 1 ) (s 1 c d 1 s c 1 d ) + s 1 c (d d 1 ) + s c 1 (d 1 d ) k (s 1 s ) ( ) + k s 1 c (s 1 s ) + k s c 1 (s s 1 ) k (s 1 s ) ( ) + k (s 1 s ){s 1 (1 s ) s (1 s 1 ) (s 1 s )} (s 1 c d 1 s c 1 d ) dn(u + v) s 1c d 1 s c 1 d s 1 c d s c 1 d 1 v s, c 1, d s 1 sn u, s sn v (s 1 c d s c 1 d 1 ) (s 1 s ) (s 1 c 1 d s c d 1 ) (s 1 c d s c 1 d 1 ) k (s 1 s ) (s 1 c d 1 s c 1 d ) 7

72 6 sn u u x (1 x )(1 k x ) K u K 1 x 1 K u 3K sn u sn(k u) ( 1 x 1) K u 3K sn(u + v) sn u cn v dn v + sn v cn u dn u 1 k sn u sn v sn K 1, cn K, dn K 1 k sn u cn K dn K + sn K cn u dn u sn(u + K) 1 k sn u sn K cn u dn u 1 k sn u cn u dn u K u K sn(u + K) sn(k u) sn K cn u dn u sn u cn K dn K 1 k sn u sn K cn u dn u 71

73 cn(u + v) cn u cn v sn u sn v dn u dn v 1 k sn u sn v cn(u + K) 1 k sn u dn u 1 k sn u k sn u dn u (k 1 k ) cn(u + K) cn(k u) cn K cn u + sn K sn u dn K dn u 1 k sn K sn u k sn u dn u ( cn u ) ( + k sn u ) cn u + (1 k ) sn u dn u dn u dn u 1 k sn u dn 1 u sn (u + K) + cn (u + K) 1 dn (u + K) dn (u + v) dn u dn v k sn u sn v cn u cn v 1 k sn u sn v dn(u + K) 1 k dn u 1 k sn u k dn u [K, K] sn((u + K) + K) cn (u + K) dn (u + K) sn(u + K) sn u sn (K), cn K 1, dn (K) 1 7

74 sn(u + K) sn u cn (K) dn (K) + sn (K) cn u dn u 1 k sn u sn (K) sn u sn(u + K) cn u dn u cn(u + K) k k dn(u + K) dn u k dn(u + K) cn u k dn (u + K) dn u dn u K [K, 3K] sn(u + 4K) sn(u + K + K) sn u cn(u + 4K) cn(u + K + K) cn u sn u, cn u 4K sn(iv) k sin u x dt u arcsin x 1 t x iy iy t iη i y iv dt 1 t arcsin(iy) dη 1 + η i sinh 1 y iv iy sin(iv) i sinh v 73

75 6.1 sinh 1 y ( ) sinh v ev e v y sinh v v v sinh 1 y log (y + ) 1 + y (sinh 1 y) 1 sin θ eiθ e iθ 1 + y i sin(iv) e v e v i 1 e v e v i i sinh v w x dt (1 t )(1 k t ) sn 1 (x, k) t sin θ t x x x sin ϕ t : x θ : ϕ dt cos θ dθ 1 t dθ w ϕ dθ 1 k sin θ sn 1 (sin ϕ, k) iy t iη i iy y dt (1 t )(1 k t ) sn 1 (iy, k) dη (1 + η )(1 + k η ) iv iv η tan ψ, y tan φ dη dψ 1 + tan cos ψ ψ 1 + η dψ cos ψ cos ψ dψ 1 + k η 1 + k tan ψ cos ψ + k sin ψ cos ψ 1 k sin ψ cos ψ (k 1 k ) 74

76 v y φ y tan φ dη φ (1 + η )(1 + k η ) 1 1 k sin ψ dψ sn 1 (sin φ, k ) sin φ sn(v, k ) cos φ 1 sin φ cn(v, k ) y tan φ sin φ cos φ sn(v, k ) cn(v, k ) v sn 1 (iy, k) iv sn K K(k ) iy sn(iv, k) sn(iv, k) i sn(v, k ) cn(v, k ) < y < π < φ < π K < v < K π/ η (1 + η )(1 + k η ) cos ψ dψ dϕ 1 k sin ϕ K < v < K sn(iv, k) i sn(v, k ) cn(v, k ) < v < cn(iv, k) 1 sn (iv, k) 1 + sn (v, k ) cn (v, k ) 1 cn(v, k ) 75

77 (cn 1 ) dn(iv, k) 1 k sn (iv, k) cn (v, k ) + k sn (v, k ) cn (v, k ) 1 (1 k ) sn (v, k ) cn(v, k ) dn(v, k ) cn(v, k ) < v < sn 5 sn(iv, k) i sn(v, k ) cn(v, k ) sn(i(v + w), k) i sn(v + w, k ) cn(v + w, k ) i sn(v,k ) cn(w,k ) dn(w,k )+sn(w,k ) cn(v,k ) dn(v,k ) 1 k sn (v,k ) sn (w,k ) cn(v,k ) cn(w,k ) sn(v,k ) sn(w,k ) dn(v,k ) dn(w,k ) 1 k sn (v,k ) sn (w,k ) i sn(v, k ) cn(w, k ) dn(w, k ) + sn(w, k ) cn(v, k ) dn(v, k ) cn(v, k ) cn(w, k ) sn(v, k ) sn(w, k ) dn(v, k ) dn(w, k ) sn(iv + iw, k) sn(iv, k) cn(iw, k) dn(iw, k) + sn(iw, k) cn(iv, k) dn(iv, k) 1 k sn (iv, k) sn (iw, k) i sn(v,k ) cn(v,k ) 1 cn(w,k ) k k dn(w,k ) cn(w,k ) + i sn(w,k ) cn(w,k ) 1 cn(v,k ) 1 k i sn (v,k ) cn (v,k ) i sn (w,k ) cn (w,k ) dn(v,k ) cn(v,k ) i sn(v, k ) cn(v, k ) dn(w, k ) + sn(w, k ) cn(w, k ) dn(v, k ) cn (v, k ) cn (w, k ) k sn (v, k ) sn (w, k ) s 1 sn(v, k), c 1 cn(v, k), d 1 dn(v, k) s sn(w, k), c cn(w, k), d dn(w, k) s 1 c d + s c 1 d 1 c 1 c s 1 s d 1 d s 1c 1 d + s c d 1 c 1 c k s 1 s 76

78 (s 1 c d + s c 1 d 1 )(c 1 c k s1 s ) (c 1 c s 1 s d 1 d )(s 1 c 1 d + s c d 1 ) (s 1 c d + s c 1 d 1 )(c 1 c (1 k )s 1 s ) (c 1 c s 1 s d 1 d )(s 1 c 1 d + s c d 1 ) s 1 c 1 c 3 d (1 k )s 1 3 s c d + s c 1 3 c d 1 (1 k )s 1 s 3 c 1 d 1 s 1 c 1 c d s c 1 c d 1 + s 1 s c 1 d 1 d + s 1 s c d 1 d (use c 3 (1 s )c, c 1 3 (1 s 1 )c 1 ) (use d 1 1 k s 1, d 1 k s ) s 1 (1 s )c 1 c d s 1 3 s c d + k s 1 3 s c d + s (1 s 1 )c 1 c d 1 s 1 s 3 c 1 d 1 + k s 1 s 3 c 1 d 1 s 1 c 1 c d s c 1 c d 1 + s 1 s c 1 d 1 (1 k s ) + s 1 s c (1 k s 1 )d s 1 s c 1 c d s 1 3 s c d s 1 s c 1 c d 1 s 1 s 3 c 1 d 1 + s 1 s c 1 d 1 + s 1 s c d s 1 s c 1 d 1 (1 c s ) + s 1 s c d (1 c 1 s 1 ) sn(u + iv) sn(u) cn(iv) dn(iv) + cn(u) dn(u) sn(iv) 1 k sn (u) sn (iv) s sn(u, k), c cn(u, k), d dn(u, k) s 1 sn(v, k ), c 1 cn(v, k ), d 1 dn(v, k ) sn(u + iv) sn(u) cn(iv) dn(iv) + cn(u) dn(u) sn(iv) 1 k sn (u) sn (iv) s 1 c 1 d 1 c 1 + cd(i s 1 c 1 ) 1 k s (i s 1 c 1 ) sd 1 + icds 1 c 1 c 1 + k s s 1 sd 1 + icds 1 c 1 1 (1 k s )s 1 sd 1 + icds 1 c 1 1 d s 1 77

79 cn(u + iv) cn(u) cn(iv) sn(u) sn(iv) dn(u) dn(iv) 1 k sn (u) sn (iv) c 1 c 1 s(i s 1 c 1 )d d 1 c 1 1 k s (i s 1 c 1 ) cc 1 isds 1 d 1 1 d s 1 dn(u + iv) dn(u) dn(iv) k sn(u) sn(iv) cn(u) cn(iv) 1 k sn (u) sn (iv) d d 1 c 1 k s(i s 1 c 1 )c 1 c 1 1 k s (i s 1 c 1 ) dc 1d 1 ik scs 1 1 d s cn(u + iv), dn(u + iv) sn u R sn(u + K) sn u sn(u + 4K) sn u 4K ( ) sn, sn K 1, sn (K) sn (3K) 1, sn (4K), cn 1, cn K, cn (K) 1 cn (3K), cn (4K) 1, dn 1, dn K 1 k, dn (K) 1 dn (3K) 1 k, dn (4K) 1 ( ) u C sn(u + K) sn(u + 4K) sn u cn (K) dn (K) + sn (K) cn u dn u 1 k sn u sn (K) sn u sn u cn (4K) dn (4K) + sn (4K) cn u dn u 1 k sn u sn (4K) sn u 78

80 sn u C 4K sn(iv, k) i sn(v, k ) cn(v, k ) sn sn(iv + ik, k) i sn(v + K, k ) cn(v + K, k ) i sn(v, k ) cn(v, k sn(iv, k) ) sn ik sn(ik, k) i sn(k, k ) cn(k, k ) u C v K sn(u + iv, k) sd 1 + icds 1 c 1 1 d s 1 s sn u, c cn u, d dn u s 1 sn(k, k ) 1, c 1 cn(k, k ) d 1 dn(k, k ) 1 k k sn(u + ik, k) i.e., sn(u + ik, k) v K ks 1 d ks 1 (1 k s ) 1 ks 1 k sn(u, k) s 1 sn(k, k ), c 1 cn(k, k ) 1, d 1 dn(k, k ) 1 sn(u + ik, k) sn u sn C ik 6.1 (sn ) u C sn(u + 4K + ik, k) sn (u, k) sn(u + 4mK + nik, k) sn (u, k) (m, n Z) 4K, ik 79

81 [, 4K] [, ik ] sn u R sn u sn u R sn(u + ik ) 1 k sn(u, k) R, R + ik, R + ik v R sn(iv) sn(iv + K) sn(iv) sn(iv + K) sn(iv) cn(k) dn(k) + sn(k) cn(iv) dn(iv) 1 k sn (iv) sn (K) 1 c 1 d 1 c k s 1 c 1 d 1 c 1 + k s 1 d 1 1 (1 k )s 1 1 d 1 R sn(iv + 3K) sn(iv + K) 1 d 1 R sn(ik ), sn(k + ik, k) 1 k, sn(k + ik ) sn(3k + K, k) 1 k 8

82 cn cn(u + K) cn(u) cn(u + 4K) cn u cn(u + ik ) cn(u) 1 cn(k,k ) sn(u) sn(ik ) dn(u) dn(ik ) 1 k sn (u) sn (ik ) cn(u) cn(u + K + ik ) cn(u) 4K, K + ik cn(k + ik ) cn(ik 1 ) cn(k, k ) cn(4k + ik ) cn(k + ik ) 1 sn (K + ik ) (1 k ) i k k k 1 1 k dn dn(u + K) dn(u) dn(ik ) dn(k, k ) cn(k, k ) 1 dn(4ik ) dn(4k, k ) cn(4k, k ) 1 dn(u + 4iK ) dn(u) dn(4ik ) k sn(u) sn(4ik ) cn(u) cn(4ik ) 1 k sn (u) sn (4iK ) dn(u) 81

83 K, 4iK dn(k) 1 k k dn(k + ik ) dn(k) dn(ik ) k sn(k) sn(ik ) cn(k) cn(ik ) 1 k sn (K) sn (ik ) dn(k) k dn(ik ) dn(k, k ) cn(k, k ), dn(k + ik ) 1 k sn (K + ik ) dn(k + 3iK ) dn(k + ik ) dn(3ik ) dn(ik ) 8

4 5.............................................. 5............................................ 6.............................................. 7......................................... 8.3.................................................4.........................................4..............................................4................................................4.3...............................................

More information

(yx4) 1887-1945 741936 50 1995 1 31 http://kenboushoten.web.fc.com/ OCR TeX 50 yx4 e-mail: yx4.aydx5@gmail.com i Jacobi 1751 1 3 Euler Fagnano 187 9 0 Abel iii 1 1...................................

More information

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............

More information

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. 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....

More information

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 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

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

Z: Q: R: C: 3. Green Cauchy

Z: Q: R: C: 3. Green Cauchy 7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................

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

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,, 01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,

More information

(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

(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)

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

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

( ) ( )

( ) ( ) 20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

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 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345

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

Gmech08.dvi

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

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

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.

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,

More information

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.   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)

More information

(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

(1.2) 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

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

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

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

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

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

KENZOU

KENZOU KENZOU 2008 8 2 3 2 3 2 2 4 2 4............................................... 2 4.2............................... 3 4.2........................................... 4 4.3..............................

More information

2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t) = x 0 + v 0 t 1 2 gt2 v(t) = v 0 gt t x = x 0 + v2 0 2g v2 2g 1.1 (x, v) θ

2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t) = x 0 + v 0 t 1 2 gt2 v(t) = v 0 gt t x = x 0 + v2 0 2g v2 2g 1.1 (x, v) θ 1 1 1.1 (Isaac Newton, 1642 1727) 1. : 2. ( ) F = ma 3. ; F a 2 t x(t) v(t) = x (t) v (t) = x (t) F 3 3 3 3 3 3 6 1 2 6 12 1 3 1 2 m 2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t)

More information

30

30 3 ............................................2 2...........................................2....................................2.2...................................2.3..............................

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

II 2 II

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

More information

( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =

( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) = 1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =

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

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

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 (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 (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

More information

b3e2003.dvi

b3e2003.dvi 15 II 5 5.1 (1) p, q p = (x + 2y, xy, 1), q = (x 2 + 3y 2, xyz, ) (i) p rotq (ii) p gradq D (2) a, b rot(a b) div [11, p.75] (3) (i) f f grad f = 1 2 grad( f 2) (ii) f f gradf 1 2 grad ( f 2) rotf 5.2

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

Z: Q: R: C:

Z: Q: R: C: 0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x

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

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

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 ),

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

chap03.dvi

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)

More information

. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n

. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n 003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........

More information

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou (Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.

More information

meiji_resume_1.PDF

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

More information

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g( 06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,

More information

p = mv p x > h/4π λ = h p m v Ψ 2 Ψ

p = mv p x > h/4π λ = h p m v Ψ 2 Ψ II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π

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

II 1 II 2012 II Gauss-Bonnet II

II 1 II 2012 II Gauss-Bonnet II II 1 II 212 II Gauss-Bonnet II 1 1 1.1......................................... 1 1.2............................................ 2 1.3.................................. 3 1.4.............................................

More information

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1 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

More information

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

More information

ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +

ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + 2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

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

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy z fz fz x, y, u, v, r, θ r > z = x + iy, f = u + iv γ D fz fz D fz fz z, Rm z, z. z = x + iy = re iθ = r cos θ + i sin θ z = x iy = re iθ = r cos θ i sin θ x = z + z = Re z, y = z z = Im z i r = z = z

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

#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 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/

More information

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 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

More information

医系の統計入門第 2 版 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, 第 2 版 1 刷発行時のものです.

医系の統計入門第 2 版 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, 第 2 版 1 刷発行時のものです. 医系の統計入門第 2 版 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/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987

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

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

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

振動と波動

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

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

chap1.dvi

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

More information

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s [ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =

More information

2 4 202 9 202 9 6 3................................................... 3.2................................................ 4.3......................................... 6.4.......................................

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

I

I I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............

More information

X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2

More information

m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)

m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120) 2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ

More information

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10% 1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n

More information

, 1.,,,.,., (Lin, 1955).,.,.,.,. f, 2,. main.tex 2011/08/13( )

, 1.,,,.,., (Lin, 1955).,.,.,.,. f, 2,. main.tex 2011/08/13( ) 81 4 2 4.1, 1.,,,.,., (Lin, 1955).,.,.,.,. f, 2,. 82 4.2. ζ t + V (ζ + βy) = 0 (4.2.1), V = 0 (4.2.2). (4.2.1), (3.3.66) R 1 Φ / Z, Γ., F 1 ( 3.2 ). 7,., ( )., (4.2.1) 500 hpa., 500 hpa (4.2.1) 1949,.,

More information

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.

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. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

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

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

O x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0

O x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0 9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )

More information

i 6 3 ii 3 7 8 9 3 6 iii 5 8 5 3 7 8 v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35

More information

1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8..............................

More information

i

i i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,

More information

1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC

1   nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC 1 http://www.gem.aoyama.ac.jp/ nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC r 1 A B B C C A (1),(2),, (8) A, B, C A,B,C 2 1 ABC

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

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

1 B64653 1 1 3.1....................................... 3.......................... 3..1.............................. 4................................ 4..3.............................. 5..4..............................

More information

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59

More information

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

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

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

i 18 2H 2 + O 2 2H 2 + ( ) 3K

i 18 2H 2 + O 2 2H 2 + ( ) 3K i 18 2H 2 + O 2 2H 2 + ( ) 3K ii 1 1 1.1.................................. 1 1.2........................................ 3 1.3......................................... 3 1.4....................................

More information

sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.

sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even. 08 No. : No. : No.3 : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No.0 : No. : sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin

More information

数学の基礎訓練I

数学の基礎訓練I I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............

More information

all.dvi

all.dvi 5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0

More information

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even. I 0 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No.0 : I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd

More information

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35

More information

webkaitou.dvi

webkaitou.dvi ( c Akir KANEKO) ).. m. l s = lθ m d s dt = mg sin θ d θ dt = g l sinθ θ l θ mg. d s dt xy t ( d x dt, d y dt ) t ( mg sin θ cos θ, sin θ sin θ). (.) m t ( d x dt, d y dt ) = t ( mg sin θ cos θ, mg sin

More information

A 2008 10 (2010 4 ) 1 1 1.1................................. 1 1.2..................................... 1 1.3............................ 3 1.3.1............................. 3 1.3.2..................................

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

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)

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

More information