30 (11/04 )

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1 30 (11/04 )

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3 i, 1,, II I?,,,,,,,,, ( ),,, ϵ δ,,,,, (, ),,,,,, 5 : (1) ( ) () (,, ) (3) ( ) (4) (5) ( ) (1),, (),,, () (3), (),, (4), (1), (3), ( ), (5),,,,,,,,

4 ii,,,,,,,, Richard P. Feynman, The best teaching can be done only when there is a direct individual relationship between a student and a good teacher a situation in which the student discusses the ideas, think about the things, and talk about the things. It s impossible to learn very much by simply sitting in a lecture, or even by simply doing problems that are assigned. The Feynmann Lectures on Physics, 1 1,,,,,...?,,,...,,,,,, I,?...,,,,, p, s, π?... II,,,, II,?...,,,, (?), xxx, xxx,?...,, :, 003, 1,, TA, : ( 15 ), , ( ),,,

5 iii ( ) : :

6 iv : : : : : ()

7 v ( ) : n ( ) (rotation) (rotation)

8 vi : ( )

9 ,,,,,,,,,,,,, f(x) f (x) = F (f(x), x) (16.1) F (f(x), x), f(x) x,, x 0, f(x + x) f(x) + f (x) x (16.), x f(x), (16.1) f (x) x 0, (16.) f(x + x) f(x), f(x + x) 1 :, (16.1), x x + x, f (x + x) = F (f(x + x), x + x) (16.3), f(x + x), f (x + x), (16.) x x + x f(x + x) f(x + x) + f (x + x) x (16.4), (16.3) f (x+ x) f(x + x) f(x + x), f(x + x), f(x + x) f(x + 3 x), f(x + 3 x), f(x + 4 x),, f, f(x), 14 C C(t), dc dt = αc (16.5) α, t,, x t, f(x) C(t), F (f(x), x) αc ky ; α = 0.1 /ky, A B C 1 t C (t) C(t) 0.0 =-0.1*C =C+B*(A3-A) A t ( t),, 0. ky, 00 0 ky,, C, C(0) 1.0

10 16,,, *1, B, C (t), C C, α, 0.1, B =-0.1*C, C3, C(t + t) (16.) (16.), f(x) C, f (x) B, x A3-A, C3 =C+B*(A3-A) C (t), B B3 B, C3 C4 C C,, t,, C 50 t, ( t, ),,,,,, A C, C(t), D, C(t) = C(0) exp( αt), A D C(t) C D? :, : 1,,,, C(0), C (0),, t = 0 t = t C( t), C ( t),, t = t C( t),, 503 N(t) N(t) (α, β ): dn dt = αn βn (16.6) (1) α = 1.0, β = 0.005, N(0) = 10,,, () (16.6),, : N(t) = N 0 e αt 1 + N 0 β(e αt 1)/α (16.7) α = 1.0, β = 0.005, N 0 = 10, (16.7) (3) α = 1.0, β = 0.0 ( ), N(0) = 10,, (1)() *1, 504 (16.7),

11 N = α β { ( αt tanh + 1 ln N 0 β ) } + 1 α N 0 β (16.8),, 16.3 (16.6), dn dt = αn βn (16.9), (α, β) ( t),, α = 1.0, β = 0.005, t, t,,, N(0) = 10.0 N(0) = 10.1, ( N(t), M(t) ) A B C D E 1 t α β t N (t) N(t) M (t) M(t) A t 0.1 A5 =A4+A$,, A6 A, A, t, A5, B4,, =$B$*C4 - $C$*C4*C4, B5 B C5 =C4+B4*($A5- $A4), C6 C ( t A ) N(t) M(t) D4 D B4, E5 E C5 C5 A,,,,, 505, (1) t 50 N(t) M(t) N(t) M(t) () t, 0.5, 1.0, 1., 1.3,...,,, t?n(t) M(t), (3) t.6, N(t) M(t) (4) t 3.0,,, N(t) M(t), t = 50 (5) t 3.0 M(t), E4, 10 (10.001, M(t) :,,

12 4 16,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 16.4,,,, S(t), W (t) t,, t, α 1 S t α 1, W S, W S, t,, β 1 SW t β 1, t S, S α 1 S t β 1 SW t (16.10),,,, α W t α,,,, β SW t β, t W, W α W t + β SW t (16.11) 506 SW, SW 507 ds dt = α 1S β 1 SW (16.1) dw dt 1 = α W + β SW (16.13)?...,,, α 1 (16.1) (16.13),, t S(t), W (t)

13 * α 1, α, β 1, β,, S(0) =.0, W (0) = 1.0, α 1 = 1., α = 1.1, β 1 = 0.6, β = 0.7 A B C D E, S W,, 1 t α 1 α β 1 β t S (t) W (t) S(t) W (t) ,,,,...,, t t α 1,,, 1,,, 4 (t = 0.0, S, W, A5 =A4+A$, A6 A, A, t, A5, B4, =B$*D4 D$*D4*E4, B5 B C4, C5 C,, D5 D B, E5 E C, D5 =D4+B4*(A5 A4) D, E 508, t = 0 t = 5, S(t), W (t), t 511,, ( t ) γs t ds/dt dw/dt, γ = 0.1, γ?,,,, (), (),,,,,,, *,,,, t

14 6 16,,, (,,,,,, ),,,,,,,,,,,,,,,,,,,,,,,,,,,, (, ),,?...,, βn,,,,,, 51, 3, 16.5,, ( ) A, B, A B (16.14) ) dt A B, dt, [A], [A]? A A,,, A,, dt A B, [A] dt ( ), [A], dt [A] d[a], k ( ), d[a] = k[a] dt (16.15), [A], d[a] dt = k[a] (16.16) (16.16),, 3 A, B, C, A + B C (16.17) (A, B (16.14) A, B ),, A B

15 16.6 7,, [A][B]dt, A B,, [A], d[a] dt = k[a][b] (16.18) ( k, (16.16) k ) (16.18), ), E [E], d[e]/dt = 0, (16.1), : k +1 [E][S] + k 1 [ES] + k [ES] = 0 (16.3) 514 (1) (16.0) (16.3) : k +1 [E 0 ][S] + [ES](k +1 [S] + k 1 + k ) = 0 (16.4) () (16.4) :, ( ) ( ) S (substrate ), ( ) P (product ) E (enzyme ): E + S ES E + P (16.19) [ES] = (3) : d[p] dt = k +1 [E 0 ][S] k 1 + k + k +1 [S] k +1 k [E 0 ][S] k 1 + k + k +1 [S] (16.5) (16.6), E S, ES ; k +1, ES E S ; k 1, ES E P ( k ),,, E E, E [E 0 ],, : [E] + [ES] = [E 0 ] (16.0) 513 : d[e] dt d[p] dt (16.19), = k +1 [E][S] + k 1 [ES] + k [ES] (16.1) = k [ES] (16.), ( P, S (4) k [E 0 ]=V max, (k 1 + k )/k +1 = K m : v = V max[s] K m + [S] (16.7) (16.7) 515 (16.7) (1) [S] = 0 v = 0 () [S], v V max (3) [S] = K m, v = V max / (4), (16.7) [S], v 0, 0 [S] 16.6,, : m d r dt = F (16.8)

16 8 16 *3, t, F, r F,,,, ,, m, x, (x = 0), x,,, r = (x, y, z) (16.30),,, x,, y = z = 0, x, x OK, x 0 ( x ), x (16.9), kx, (16.8) x, m d x = kx (16.31) dt,! () 516 (16.31), (1) (16.31),, x(t), v(t) 16.1,,,,,, x, F = kx (16.9) k, *4, (Hooke),, (16.8) dx dt = v dv dt = k m x (16.3) () m=1.0 kg, k=1.0 N/m, x(0) = 1.0 m, v(0) = 0 m/s,, t 0.0, t 0 15,, *3 d r/dt a,, F = ma *4 (16.9), x x,, ( ), x,, ( ),, (16.31), v(t)), 1 (16.3),, n, n, n

17 , 1,,, (16.31),, ( ), γv 517, (1) (16.31) m d x dt dx = kx γ dt (16.33) () (16.33), dx dt = v dv dt = k m x γ m v (16.34) (3) m=1.0 kg, k=1.0 N/m, γ=0.5 N s/m, x(0) = 1.0 m, v(0) = 0 m/s,,, t 0.0, t 0 15,,, (16.8) (16.7) : ( ), analytic N O 16.3 α=1.0, β=0.005, N 0 =10 α=1.0, β=0.0, N 0 = ( αt exp + 1 ln N 0 β ) N 0 β = α N 0 β α N 0 β eαt/ t (16.35), (16.8), : N = α β = α β { N0β α N 0β eαt/ α N0β N 0β e αt/ N0β α N 0β eαt/ + { N0 β N0 β α N 0 β eαt/ + α N0β N 0β e αt/ + 1 α N 0 β eαt/ α N0 β N 0 β e αt/ } } (16.36)

18 10 16 S, W O S(t) W(t) (α N0 β)(n 0 β)e αt/ (16.37) t S, W O S(t) W(t) β 1 = β = 0.4 t, N = α β = N 0 βe αt (N 0 β)e αt + α N 0 β N 0 e αt 1 + N 0 β(e αt 1)/α (16.38) (16.39) 506, (β 1, β!) 507 S = S(t+ t) S(t), S = S(t + t) S(t) α 1 S t β 1 SW t t t (16.1) (16.13) S, W O 16.6 S(t) W(t) G, S, W 1 grass sheep 10 wolf 8 t 6 510,,,,,,,,,, 4 O t grass ( ), sheep ( ),wolf( )

19 ,,, ds dt = α 1S β 1 SW γs dw dt = α W + β SW 51,, (G),, dg dt = α 0G β 01 GS ds dt = α 1S + β 10 GS β 1 SW dw dt = α W + β 1 SW 16.7, α 0 = α 1 = α = 1.0, β 01 = β 10 = β 1 = β 1 = 0.5, G(0) = 1, S(0) =, W (0) = () E + S ES, E E S [E][S] k +1, [E] k +1 [E][S] ( ), E, [ES] k 1, [E] k 1 [ES], ES E + P, ES E, [ES] k, [E] k [ES], (16.1) P, ES E + P, k [ES] (16.) 514 (1) ( [E] ) () ( (16.4) ) (3) ( (16.) [ES] ) (4) ( (16.6) k +1 ) 16.9 ( )...,...,

20 ?... Alfred J. Lotka Vito Volterra Wikipedia!/,! /,...?,,,, /!?,,...,,,,, ( ),,,????,,,? II,...,,,,,,,,, ,,,,, etc......,,,,,, /...,,,...,

21 13 17 e x,,,? (?),,,,,,, (absorbance),, (17.5), (17.6), κ ln 10 c d = A (17.7) c = A (κ/ ln 10)d (17.8), κ/ ln 10 (17.8) A (ln 10/(κd)), K, c = K A (17.9) 17.1 ( ),, x, x I(x), c, I(x + dx) = I(x) κ c I dx (17.1) (κ, di = κ c I (17.) dx κ, c x,, I(x) = I(0) exp ( κ c x) (17.3) I(x) = I(0) 10 κ c x/ ln 10 (17.4) κ ( I(x) ) ln 10 c x = log 10 I(0) (17.5) x d ( d = 1 cm ), (17.5) ( I(d) ) A := log 10 I(0) (17.6),,,, a,,,, b, (17.9) (17.1), 1,, (17.1) : I(x + dx) = I(x) κ 1 c 1 I dx κ c I dx (17.10), κ 1 κ 1,, c 1, c 1,,, (17.7), κ 1 ln 10 c 1 d + κ ln 10 c d = A (17.11) c 1, c, 1, c 1, c!,

22 14 17,, κ 1, κ, λ 1, λ, A 1, A, (17.11) : κ 11 ln 10 c 1 d + κ 1 ln 10 c d = A 1 (17.1) κ 1 ln 10 c 1 d + κ ln 10 c d = A (17.13) κ ij, λ i, j (i, j 1 ) (17.1), (17.13),, [ ] [ ] [ ] κ11 d/ ln 10 κ 1 d/ ln 10 c1 A1 = κ 1 d/ ln 10 κ d/ ln 10 c A (17.14) (17.1), (17.13), (17.14), c 1, c, (17.14),, (17.14), [ ] 1 [ ] κ11 d/ ln 10 κ 1 d/ ln 10 K11 K = 1 κ 1 d/ ln 10 κ d/ ln 10 K 1 K, [ ] [ ] [ ] c1 K11 K = 1 A1 c K 1 K A,, (17.15) (17.16) c 1 = K 11 A 1 + K 1 A (17.17) c = K 1 A 1 + K A (17.18) (17.16) (17.17), (17.18), (17.9) , a b 1,, λ 1 = nm, λ = nm, (17.16), [ 1.5 µg/ml ].85 µg/ml 4.91 µg/ml 0.31 µg/ml (17.19) *1, A 1 = 0.30, A = 0.13, a b,,, (17.14) (17.16) 519 (17.14) (17.16)?, 1? :,, 50 ( ), (17.3) κc,,, κc, ( 0.48 µm): κc 0.01 m 1 ( 0.55 µm): κc 0.05 m 1 ( 0.68 µm): κc 0.5 m 1 (1) 1 m, () 10 m,, :,,,,,,, :, 3,,,,, *1, 67, ,

23 17. 15? :,? 5 (17.5) 17.,,, t T (t) T T 0, T 1 T 0 T 1, t,, ( ), T 1 T (t), T 1 T (t) K, J, J(t) = K(T 1 T (t)) (17.0), dt, J(t) dt = K(T 1 T (t)) dt (17.1), C,, /C, t + dt, t, K(T 1 T (t)) dt/c, (1) (17.5) : dt T 1 T (t) = K dt (17.6) C () D : ln T 1 T (t) = K C t + D (17.7) (3) : T 1 T (t) = ± exp ( K ) C t D (17.8) (4) t = 0, : T 1 T 0 = ± exp( D) (17.9) (5) (17.8), (17.9), : T 1 T = (T 1 T 0 ) exp ( K ) C t (6) : (17.30) T = (T 0 T 1 ) exp ( K ) C t + T 1 (17.31) (17.31),, C K = τ (17.3) T (t + dt) = T (t) + K(T 1 T (t)) dt C, (17.), (17.31) : ( T (t) = (T 0 T 1 ) exp t ) + T 1 (17.33) τ T (t + dt) T (t) = K(T 1 T (t)) dt C dt, T (t + dt) T (t) dt = K(T 1 T (t)) C (17.3) (17.4) dt, T, dt dt = K(T 1 T (t)) C (17.5), τ (time constant),,,, exp( t/τ),, τ 53, (17.33), (1) T (0) = T 0, T ( ) =

24 16 17 T 1 () T t, T 0 > T 1 T 0 < T 1 (3) τ,, (4) T (τ) T 0 T 1? (5) 30 s ( ), ( 0 ), 1 3? 0.5?,,,,,,,, 54? 55,? 17.3,,,,,, I IV,,, V, I, R,, V = RI (17.34) (resistance) ( ),,, R, (, ), R,, R,, R, (17.34), R = V/I,, SI V/A ( =V/A) 1 V 3 A 4,,??,,, (condenser; ),?,,,,,, V, +Q, Q, C, Q = CV (17.35), V = Q (17.36) C,, C

25 , C, (17.35), C = Q/V,, SI C/V C=A s, A s/v F (F=A s/v) F 3 V 15 C 5 F * 1 F... V R (t) = RI(t) (17.37), V C, (17.36) V C (t) = Q(t) C (17.38), V R (t) + V C (t) V : V = V R (t) + V C (t) (17.39), 17.4, R, C, t = 0 V 17.1; I P.85 5 V = RI(t) + Q(t) C (17.40), (t) (17.40) t, : 0 = R di dt + 1 dq C dt (17.41) V, I,, dt I dt, dq, dq = I dt (17.4) 17.1,,,,,,, (t),, (t) t I(t), Q(t), V R (t), (17.34), dq dt = I (17.43) (17.43) (17.41), 0 = R di dt + I C (17.44) (,, Q(t) ), : di dt = I CR (17.45), (17.45), * 5 F, 10 6 F 10 9 F 56 (17.45), : I(0) = I 0,

26 18 17 : ( I(t) = I 0 exp t ) CR, (17.40), t = 0, V = RI(0) + Q(0) C (17.46) (17.47),, Q(0) = 0,, V = RI(0), I 0 = V/R (17.46), : I(t) = V ( R exp t ) CR (17.37), ( V R (t) = V exp t ) CR (17.39), { ( V C (t) = V V R = V 1 exp t )} CR 57 (17.50) 58 (17.48) (17.49) (17.50), (17.49) (17.49) (17.50) ( t ), CR = τ (17.51), (17.46) : ( I(t) = I 0 exp t ) τ τ 59 (17.5) τ, I(t), 530 R = 1 k, 1000, C = 1 F, F, (1) s () I(t) I 0 1? 531 C,? ,,, (1) 1 m, 0.01 m (100 cm ) 1, 1% m, 1 p m n,, F, F = 1 (1 p) n (17.53) ( ) () 1 m L m, np = L (17.54) F n L p (3) (17.54) L,, n, p 0, F = 1 e L (17.55), 1 1, (4) 1 m 1 m, (5) 1 k P k P k = n C k p k (1 p) n k (17.56) : n k, n k (6) (17.54) L,,

27 n, p 0, : P k = Lk e L k! (17.57) (7),,, θ, 1, p p cos θ, p, n, X, X k, P (X = k), P (X = k) = n C k p k (1 p) n k (17.58), n,, p,, np ( np = λ ), 53, P (X = k) = λk e λ k! (17.59), λ (17.57), L (17.59),...,, 533 (1) 1, 1, 4 () 10 1, 1 3!, ,, ( ) (),, 1, 1, 1 / ?, 16? 535 DNA 4 (A,T,G,C) (1) 4,? 1 4 (A,T,G,C )? () DNA, 30 ( ),? (3)? (4) CD (compact disc)? (5) DVD (digital versatile disc)?, 17.1 (, ),,,,,, (),,, p, I, I = log p (17.60) I p, 17.

28 0 17 I O p I = log p, (),,,,,,,, () H, H = p s log p s p r log p r (17.61) I, yes no 1, yes no 1/,, I = log (1/) = 1, 1, yes no DNA, A, T, G, C 4 1/4,, (),,,, p 1, p, p 3,, p n, n H = p k log p k (17.6) k=1 (, p 1 + p + + p n = 1), () 537 3/4, 1/4,?? 17., 17.1, (, ),,,,?, p s, p r,,, 538, (1) 3/4, 1/4, () 9/10, 1/10, (3) p, 1 p,, H = p log p (1 p) log (1 p) p 0, p p 1 (4), p = 1/ (5),,?

29 DNA, 4,, DNA, A, T, G, C 0.34, 0.4, 0.19, 0.3, 0.8, 0.15, 0.10, 0.47,,?, :, n,, p 1, p, p 3,, p n, n H = p k log p k (17.63) k=1 *3 H, 540, (17.63) H, (1) 3, H () 4, H? (3), n, H n? (4) 3, 0., 0,3, 0.5 H?, 3?, 0.1, 0., 0.7?,,, S S 1 = 1 dq T (17.64) ( T, Q S 1, S 1, )?, (17.64) S,,, : S = k B n k=1 p k ln p k (17.65), n, *4, p k, k B ( ) (17.65),, (17.6) (17.65), (k B ), (17.65),,, (17.6),, (17.63),,, 17.7 : e ix = cos x + i sin x (17.66) *3,, (Claude Elwood Shannon) Norbert Wiener), 0, *4, S 1 S 1

30 17, sin x = eix e ix i cos x = eix + e ix (17.67) (17.68), i,,, ( ) sinh x := ex e x cosh x := ex + e x (17.69) (17.70) tanh x := sinh x cosh x = ex e x e x + e x (17.71) cosh, sinh, tanh,,,, sin h h, sinh,,, 541, cosh x sinh x = 1 (17.7) (cosh x) = sinh x (17.73) (sinh x) = cosh x (17.74) (tanh x) 1 = cosh x, x = 0, sinh 0 = e0 e 0 cosh 0 = e0 + e 0 = 1 1 = (17.75) = 0 (17.76) = 1 (17.77) tanh 0 = sinh 0 cosh 0 = 0 1 = 0 (17.78), y = f(x) = sinh x : f(0) = sinh 0 = 0, f (0) = cosh 0 = 1, x = 0 1 x e x, e x 0, f(x) sinh( x) = e x e x = ex e x = sinh x sinh x, y = sinh x y x y = sinh x! y = sinh x, 0 (y = x 3 )... (sinh x) = cosh x, cosh 0 = 1, x = y = f(x) = cosh x f(0) = cosh 0 = 1, (0, 1) f (0) = sinh 0 = 0, x = 0 0 x e x, e x 0 f(x) cosh( x) = e x + e x = cosh x cosh x y, y = cosh x 17.4 y = f(x) = tanh x f(0) = tanh 0 = 0, f (0) = 1/ cosh 0 = 1 x = 0 1 x y 1

31 y x -1 T T 0 T 1 0 τ τ 3τ 17.6 T 0 > T 1 t y = cosh x tanh( x) = sinh( x) cosh( x) = sinh x cosh x = tanh x tanh x, y = tanh x 17.5 y 3 (1) x y = 1 () xy, (x, y), ( x, y),, cosh t x, sinh t y,,,,, ) 3,...,, x y = tanh x 54, y = sinh x, y = cosh x, y = tanh x 543 t, x = cosh t, y = sinh t ( ) κ 11 κ κ 1 κ 1 = 0 50 (1) : 0.99, : 0.95, : 0.61 () : 0.90, : 0.61, : (1) () 17.6, 17.7 (3), T (t), T (t) ( )

32 4 17 T T 1 T 0 0 τ τ 3τ 17.7 T 0 < T 1 t, I(t) ( ) 530 (1) F= F = (V/A)(C/V)= C/A= s () I(t)/I 0 = exp( t/τ) = 0.01, t/τ = ln , t 4.6 τ s s (4) (17.33) t = τ, T (τ) = (T 0 T 1 )e 1 + T 1 (17.79), T (τ) T 0 T 1 T 0 = (T 0 T 1 )e 1 + T 1 T 0 T 1 T 0 = e = 1 1 = 0.63 (17.80) e, T 0 T 1, 63 (5) T 1, T 1 = T T 0 exp( t/τ) 1 exp( t/τ) (17.81), τ = 30 s, T 0 = 0, t = 60 s, T = 3, T 1 = , 1.9, t, t = τ ln T 0 T 1 T T 1 (17.8) 0.5, T = 33.4, t = 99.8 s, 100, 54 t,, exp( t/τ), τ 55 C/K, K, C,, , I(t) 531,,, 53 (1) 1, 1 1 p n, n, (1 p) n, n, 1 (1 p) n 1 m,, 1 (1 p) n F () np = L, p = L/n, F = 1 (1 L/n) n (3), n, (1+x/n) n e x x = L, F = 1 (1 L/n) n 1 e L, F (L) = 1 e L (4) F (1) = 1 1/e (5) n k nc k,, k 1, 1 p k (1 p) n k, P k = n C k p k (1 p) n k (17.83) : (6) p = L/n, P k = = n! k!(n k)! pk (1 p) n k n! k!(n k)!( L n ) k ( 1 L ) n k n n(n 1) (n k + 1) = n k k! ( = ) ( 1 k 1 ) n n L k ( 1 L ) n k n ( Lk 1 L ) n ( 1 L ) k (17.84) k! n n

33 17.8 5, n, ( ) ( n 1 k 1 n ) 1 (17.85) ( 1 L ) n e L n (17.86) ( 1 L ) k 1 n (17.87), n (17.84), H P k Lk e L k! (17.88) O p (7) 1, p p cos θ, L L cos θ 533 ( ) (1), 0.67?? (), ???? ( ) 534 8, 8 = 56 16, 16 = (1) 4 : A,T,G,C, 4 4 =, () 30, 60 (3) 60 /8=7.5 (4) CD 700 = 7, CD 1 (5) DVD 5 = 50, DVD A, T, G, C, log (1/4) = 537, log (3/4) = 0.4,, log (1/4) = 538 (1) p s = 3/4, p r = 1/4, H = p s log p s p r log p r =0.81 () p s = 9/10, p r = 1/10, H = p s log p s p r log p r =0.47 (3) 17.8 (4) H, dh/dp , dh dp = 1 ( 1 p ) ln ln = 0 (17.89) p, (1 p)/p = 1,, p = 1 p,, p = 1/ (5) p = 1/, H, 539 : H = 1.97 : H = 1.77, 540 (1) H = log (1/3) = 1.58 () H = log (1/4) =.0 (3) n, 1/n, (17.63), p k = 1/n, H = n k=1 = n 1 n log 1 n log 1 n 1 n = log 1 n = log n (4) 0., 0,3, 0.5, H = , H 0.1, 0.,

34 , H = 1.16, 541 ( e cosh x sinh x + e x ) ( e x e x x = = ex + e x + ex + e x = = 1 ( e (cosh x) x + e x ) e x e x = = = sinh x ( e (sinh x) x e x ) e x + e x = = = cosh x ( sinh x ) (tanh x) = cosh x = (sinh x) cosh x sinh x(cosh x) cosh x = cosh x sinh x cosh x (1) (17.7) 1 = cosh x () y 1 0 x y (cosh t, sinh t) y ( cosh t, sinh t) y = ±x )..., 1 /...,,,...!!...,,,,,,,,,,?,,,,,,,,,,,,,

35 7 18 1:,,,,,,,,,,,,,,,,,,,,, (0 ) 18., 0 (!),,,, R, C, Z 544 (1)? () N? (3) Q? 18.1,,,, (0 ),, (0 ),,,,,,,, ( ), 1, 3+4=7, 3 4 7,, 18.3,, X, K,, X K x X, α K, αx X (18.1) x, y X, x + y X (18.),,, R ( K = R ),, C ( K = C ) R, C, K, (18.1) α

36 8 18 1:, R C,,,,, R , 545? *1, (18.1), : X x K α, αx X, (18.), : X x y, x + y X (18.1), (18.), X x, y, (18.1), X, (18.), X,,, 18.1 A = {1,, 3, 4, 5} A,? x, y A, x = 1, y = 5, x + y = 6, A (18.), A 18. S = {,,,, } *4 S S? x S x =, 0.1.5,, +,,,, S,, (18.1) (18.),,, S,, *,,,,, (),, *3,,,?,?,,,,,,, 18.3 R? x R, ( )αx, α,, αx R, (18.1), x, y R,, x + y R, (18.), R,,,, (18.1) (18.),,,, *1,! *, p. 19 *3,,, *4

37 x αx, (, 3) ( 3, 1, 1.5),,, 18. x, y x + y!!!, 3,,,,, 3,,,?,,,, : 18.4 X,? x X, αx, ( 18.1) (18.1), x, y X, x + y, ( 18.) (18.), X 18.5 n n,, n n? n R n *5, (18.3) 4, R 4 *6, R n? x R n, x 1, x,, x n, x 1 x x =. x n (18.4), )α, αx 1 αx αx =. αx n (18.5), R n (18.1) 18.4,,,,,,,,, *5 R n n,,, R n = R R R,, *6,, ( ) [ ],

38 :, R n y 1 y y =. y n, x y x 1 + y 1 x + y x + y =. x n + y n (18.6) (18.7), R n (18.), R n *7 548,,,,,,,,,,, 547?? 18.3, x,,,, x,,,,,,, x, X ( )x, α, x α αx,, xα,,,,,,,,,,,,, *7,,,,, ( ) 18.6 Z Z, x =, α = 0.1, αx = 0., Z, (18.1) Z *8 18.3, R,,, 18.6, Z,,,?, (!) *9 *8, Z,,, Z? Z *9,,,,

39 , R (1) B = {n n 1 } () C, R (3) C, C (4) {0} (5) {1}, Y, (18.11), p exp(qx), (18.11) sinh x, 17.3,, f(x) = p exp(qx), p Y 18.7 P * 10, P = {f(x) = px + qx + r p, q, r R} (18.8) P?, P f(x) = px + qx + r, α α αf(x) = α(px + qx + r) = αpx + αqx + αr 18.9 x, 1 x 1 f(x) X f(x) X, αf(x) 1 x 1, αf(x) X, f 1 (x), f (x) X, f 1 (x) + f (x) 1 x 1, f 1 (x) + f (x) X, X,, 1 x 1,, P (18.1), P g(x) = sx + tx + u s, t, u R, 18.7 P 18.8 Y, 18.9 X,, f(x) + g(x) = (px + qx + r) + (sx + tx + u) = (p + s)x + (q + t)x + (r + u), P (18.), P,,, ,,, * 11,,,, 18.8 Y, f(x) = p exp(qx), x (p, q ),,,, Y = {f(x) = p exp(qx) p, q R} (18.9), Y?, ( ) f 1 (x) = 1 exp(x), f (x) = 1 exp( x) (18.10), f 1 Y, f Y, f 1 (x) + f (x) = exp(x) exp( x) (18.11) *10, x 0, P 0, , f(x) = x 1, f( 1) = f(1) = 0, f(x) 0 0,, 0,,, *11,,,, ,

40 3 18 1:,,, ( ),,, (18.)!), X () 550 α, β, γ, (x, y, z) αx + βy + γz = 0 (18.19) : : x + y + z = 0 (18.1) 1 3,, * 1,, (x, y, z) = ( 1, 1, 1) (18.13) (x, y, z) = (0, 1, ) (18.14),, : X = {(x, y, z) x + y + z = 0} :, (x 1, y 1, z 1 ) X (18.15) (x, y, z ) X (18.16), x 1 + y 1 + z 1 = 0 (18.17) x + y + z = 0 (18.18), α, α(x 1, y 1, z 1 ),, (αx 1, αy 1, αz 1 ), (18.1), αx 1 + αy 1 + αz 1 = α(x 1 + y 1 + z 1 ) = 0 ( (18.17) ), α(x 1, y 1, z 1 ) (18.1), α(x 1, y 1, z 1 ) X ( (18.1)!), (x 1, y 1, z 1 )+(x, y, z ), (x 1 +x, y 1 +y, z 1 +z ) (18.1), x 1 + x + (y 1 + y ) + z 1 + z = (x 1 + y 1 + z 1 ) + (x + y + z ) = 0 ( (18.17) (18.18) ), (x 1, y 1, z 1 ) + (x, y, z ) (18.1), (x 1, y 1, z 1 ) + (x, y, z ) X ( *1 :,, (, 18.3 R, 18.9 ),, 551, (1) 3, () 3, :, X = {ln x 0 < x R} (18.0) α ln x = ln x α X, ln x 1 + ln x = ln x 1 x X, X, R R R,,, X = {f(x) = ln x 0 < x R} (18.0) f(x) = ln x, f(x) = ln x,,,,,,, X = {sin x x R} (18.1)

41 , X = {x x R, 1 x 1} (18.), (18.1),,? ( ) ( ), ( ),,, 18.4,, K X, n (n 1 ): x 1, x,, x n X, n : α 1, α,, α n K, α 1 x 1 + α x + + α n x n (18.3), n α k x k (18.4) k=1,, superposition,,,,,,,,,,,,,,, 3,,,, * 15,, 55 (1) () (3) 553 X, X 554 ( (18.1) (18.)), * 13 * 14 x X, y X, α K, β K, αx + βy X (18.5) 1 X, Y, X, Y, X Y :,, R, R R, R R, 554,,,, *15,, *13 *14, (18.5),

42 : (1) 544 () (1) 0, (), 3, 5 N, 3 5 = / N, * 16, N (3) 0 ), Q, 545, (3) 553 x 1, x,, x n X, α 1, α,, α n K X, α 1 x 1 X n S n = α k x k (18.6), (18.1), 546 α n+1 x n+1 X (18.8) k=1, S 1 = α 1 x 1 X, n N, S n X x n+1 X, α n+1 K (18.7) 547, (18.), 548 S n + α n+1 x n+1 X (18.9) 549 (1) B, 0.1 R, 0.1 = 0. / B (), (3), (4) {0} 0, α R, α0 = 0 {0}, {0} ( 0 ), = 0 {0}, {0} ( 0 ), {0} (5) 1 {1}, R, 1 = / {1} S n+1, S n+1 X,, X X 554 (18.1), α K, β K, x X, y X, αx X βy X (18.), αx + βy X (18.5), (18.5), β = 0 (18.1), α = β = 1 (18.) (18.1) (18.), (18.1) (18.), (18.5),......?..., *16, 1

43 35 19,,,, 556, ( ), x + y + z = 0 (19.3) P ,,,,, ( ) * (19.1),,,, * 555 : 19. x(t) { x + y + z = 0 x + y z = 0 (19.1) dx dt + x = 0 (19.4) *3,,, x(t) = Ae t A,, 19.1 x, y, : xy = 1 (19.)?, (x, y) = (1, 1),, (x, y) = (, ), A,, (19.4) x(t), X,, X = {x(t) = Ae t A R} (19.5),, e t,, A, e t, (19.4),, = 4 1, (, ) ( ), *, *1, dx/dt = x, *3, x(t) (t),,,

44 36 19,, (19.4) : x 1 (t), x (t) X ( (19.4) ), m d x dt dx = kx γ dt : (16.33) (19.1) dx 1 dt + x 1 = 0 (19.6) dx dt + x = 0 (19.7), α, αx 1 (t) X ( (19.4) ), d(αx 1 ) dt ( dx1 ) + (αx 1 ) = α dt + x 1 = 0 (19.8) (19.6), X X, x 1 (t) + x (t), X, d(x 1 + x ) + (x 1 + x ) dt (19.9) ( dx1 ) ( = dt + x dx ) 1 + dt + x = 0 (19.10) (19.6), (19.7), X X, X, (19.4) 19.3 (19.11) (19.1), t,,,,?,, (17.) (16.6),, *4, (19.11) (19.1) *5 19.3, x(t) (19.11) d x dt dx x = 0 (19.13) dt,,,,,! 557 d x dt dx x = 0 (19.11) dt 558, m, k, γ 0 t = 0, x = 5 dx dt =, dx d x, (19.13),,, x(t) *4,, dx dt = x + t,, *5,

45 ,, ( d dt d dt ) x = 0 (19.14), (19.14) (),,, (operator) x(t) ,, ( d dt )( d dt + 1 ) x = 0 (19.15),, (19.14) O x(t) = 4e t + e t 4e t e t t = 0 t, (19.13),, x(t) x(t),,, ( d dt + 1 ) x = 0 (19.16) dx dt + x = 0 (19.17) x(t), (19.15), (19.13), (19.14) ( d dt + 1 )( d dt ) x = 0 (19.18) x(t) = a 1 exp( t) (19.1) () (19.0), a x(t) = a exp(t) (19.) (3), x(t) = a 1 exp( t) + a exp(t) (19.3), (19.13) (4) a 1, a, x = 4e t + e t (19.4) 19.1,, ( d dt ) x = 0 (19.19) dx x = 0 (19.0) dt x(t), (19.13)!, x(t),,,,,, (, ),, 559, (19.13),,, (1) (19.17), a 1,,

46 38 19, : () (19.14) (), (19.15) (19.16),,, (19.16) ( d ) x dt + 1 = 0 (19.5),,, d/dt λ, (19.14), λ λ = 0 (19.6), λ ( ), : (λ + 1)(λ ) = 0 (19.7) λ d/dt, (19.6), d/dt λ, ( d dt + ω 0 ) x = 0 (19.30) (4),,, (19.30), i ( d dt iω 0 )( d dt + iω 0) x = 0 (19.31) (5) a 1, a, x = a 1 e iω0t + a e iω0t (19.3) (6) t = 0 x = x 0, v = v 0 : x 0 = a 1 + a (19.33) v 0 = iω 0 (a 1 a ) (19.34) (7), a 1, a, : ( a 1 = x 0 + v ) 0 / (19.35) iω 0 ( a = x 0 v ) 0 / (19.36) iω 0 (8), (19.3) : x = x 0 cos ω 0 t + v 0 ω 0 sin ω 0 t (19.37) (9), a, δ, x = a sin(ω 0 t + δ) (19.38) 560 (1) ω 0 = 16.1 k m (19.8),,,,, (16.31) d x dt + ω 0x = 0 (19.9) () (19.9) (3),, 561, P.9 517(3) (1),, : d x dt + dx + x = 0 (19.39) dt

47 x(t) t (19.4) 16.9 : t = 0 x = 1, dx/dt = 0 (),, a 1, a x(t) = ( i ) ( 1 15 i ) a 1 exp t + a exp t 4 4 (19.40) (3) : x(t) = e t/4{ ( 15 i ) ( 15 i )} a 1 exp t + a exp t 4 4 (19.41) (4), a 1, a, : x(t) = e t/4 ( cos 15 4 t ) sin 15 4 t (19.4) (5),, 517(3) (), : d [B] dt + (k 1 + k ) d dt [B] + k 1k [B] = 0(19.47) (3), : ( d dt + k 1 )( d dt + k ) [B] = 0 (19.48) (4) a, b, : [B] = ae k 1t + be k t (19.49) (5) t = 0, [A]= A 0, [B]=0, : [B] = k 1A 0 k k 1 (e k 1t e k t ) (19.50),,,,, 19.4,,, (19.4) (19.11),, (16.8), t, * A, B, C,, : 563? A B C (19.43) A B k 1, B C k, (19.4) (19.11) (1) : d[a] dt d[b] dt d[c] dt = k 1 [A] (19.44) = k 1 [A] k [B] (19.45) = k [B] (19.46) *6, r = (x(t), y(t), z(t)), x(t), y(t), z(t) 3 ( ) () 3, t (x, y, z ),,

48 40 19,, ( (x, y, z) (0, 0, 0) ),,, *7,,,,,,,,,,,,,,, *8 f(x, y) : f x + f y = 0 (19.51), 3 f(x, y, z) 3 : f x + f y + f z = 0 (19.5),, 0,?, 567 (19.51) f(x, y) X (1) f 1 (x, y) X f = f 1 (x, y) (19.51), α, f = αf 1 (x, y) (19.51) () f 1 (x, y), f (x, y) X f = f 1 (x, y) f = f (x, y) (19.51), f = f 1 (x, y) + f (x, y) (19.51) (3) X, 564? 19.4 II-1, ( 565 f(x, y),, (19.51) (1) f(x, y) = x y () f(x, y) = ln (x + y ) (x, y) (0, 0) ) ϕ,, : ϕ x + ϕ y = 0 (19.54), 1 P.44 (8), f(x, y, z), 3 (19.5) I, ϕ = ρ s I π ln r 0 r (19.55) f(x, y, z) = 1 x + y + z (19.53), ρ s, r, r 0, (19.54) *7,, *8 (Pierre-Simon Laplace), 18-19, (19.55),, (x, y) r, r = x + y (19.56)

49 , (1) (19.56) (19.55) ϕ = ρ s I 4π ln(x + y ) + ρ s I π ln r 0 (19.57) () (19.57) (19.54) : (19.57) (19.54), ,, (19.3) (19.4) 19.6,,, 19.5, 567,,,, 19.4,,,,, I P.46 (4),, :,,,,,,,,,,,,, 569,,,,,,,,, *9,,,, 570?,, (16.33) (19.8) ω 0, : α = γ m (19.58) *9,

50 4 19 α, (16.33) d x dt + αdx dt + ω 0x = 0 (19.59) (1) x 1 (t), x (t) () α ω 0, x 1 (t), x (t), t 0 () (3) α < ω 0, x 1 (t), x (t), 0 3 ( ) (16.6),,, α = 1.0, β = 0.005, N(0) = 0,,, 503(3), 4 ( ) (19.13), (1) x (t) y(t), (19.13), : { x (t) = y(t) y (t) = y(t) + x(t) (19.60) () (19.60) : d dt [ ] x(t) = y(t) [ ] [ ] x(t) y(t) (19.61) (3) (19.61) A A * 10, (19.13),, (19.6),,, (4) A, λ 1, λ *10,, P, [ ] P 1 λ1 0 AP = 0 λ (19.6) (λ 1, λ, P ) (5), (19.61) : d dt P 1 [ ] [ ] [ ] x(t) λ1 0 = P 1 x(t) y(t) 0 λ y(t) (19.63) :, (19.6) A = ( P, P 1, ) A, (19.61) P 1 (6), [ ] [ ] X(t) = P 1 x(t) Y (t) y(t), { X (t) = λ 1 X(t) Y (t) = λ Y (t) (19.64) (19.65) (7), X(t), Y (t) t, X(t), Y (t) 1 (8) (19.64), x(t), (9) : x(0) = 5 x (0) =,,...,...,!!!...,

51 X... X X, x x,...,?...,

52 (19.1) X (x 1, y 1, z 1 ) X, (x, y, z ) X, x 1 + y 1 + z 1 = 0 (19.66) x 1 + y 1 z 1 = 0 (19.67) x + y + z = 0 (19.68) x + y z = 0 (19.69), α, α(x 1, y 1, z 1 ), (αx 1, αy 1, αz 1 ) (19.1), αx 1 + αy 1 + αz 1 = α(x 1 + y 1 + z 1 ) = 0 αx 1 + αy 1 αz 1 = α(x 1 + y 1 z 1 ) = 0, (19.1), α(x 1, y 1, z 1 ) X, (x 1, y 1, z 1 )+(x, y, z ), (x 1 +x, y 1 + y, z 1 + z ) (19.1), (x 1 + x ) + (y 1 + y ) + (z 1 + z ) = (x 1 + y 1 + z 1 ) + (x + y + z ) = = 0 (x 1 + x ) + (y 1 + y ) (z 1 + z ) = (x 1 + y 1 z 1 ) + (x + y z ) = = 0, (19.1), (x 1, y 1, z 1 ) + (x, y, z ) X, X ( (18.1), (18.)),, (19.1) (19.11), d (αx 1 ) dt d(αx { 1) d x 1 (αx 1 ) = α dt dt dx } 1 dt x 1 (19.70), α 0, 0 αx 1 (19.11), x = x 1 + x (19.11), d (x 1 + x ) dt d(x 1 + x ) (x 1 + x ) dt { d x 1 = dt dx } { 1 d dt x x 1 + dt dx } dt x (19.70), (19.71), 0 + 0, 0 x 1 + x (19.11), X, X (1) () (3) (19.3) (19.13), = 0 (4) t = 0 x = 5, 5 = a 1 + a t = 0 dx/dt =, = a 1 + a, a 1 = 4, a = 1 (19.3) 556,, (, 0, 4) (1, 0, 1),, (3, 0, 5), = 4, 0 (3) (4) 557 (19.11) X x 1 (t), x (t) X (, (t) ), 560 (1) () (5) (19.31), : ( d dt iω 0) x = 0, ( d dt + iω 0) x = 0 d x 1 dt dx 1 dt x 1 = 0 (19.70) d x dt dx dt x = 0 (19.71) α, x = αx 1,, dx x = iω 0dt, x 1 = a 1 e iω 0t,

53 , x = a e iω 0t,, (6) (7) (8) x = a 1 e iω0t + a e iω0t (9) a = x 0 + (v 0/ω 0 ),, sin δ = x 0 /a, cos δ = v 0 /(aω 0 ) δ, (1) A B, A k 1, (19.44) * 11 B C, B ( [B]) C, k, (19.46), B k [B] C ( [B] ),, A k 1 [A] B ( [B] ), (19.45) () : (19.45) t, : d [B] dt = k 1 d[a] dt k d[b] dt, 1 d[a]/dt, (19.44), d [B] dt = k 1[A] k d[b] dt b = a, (19.49), [B] = a(e k 1t e k t ) (19.7) t, d[b] dt = a( k 1 e k 1t + k e k t ) (19.73) t = 0, t = 0 d[b]/dt = a( k 1 + k ), (19.45), t = 0 d[b]/dt = k 1 A 0,, t = 0 d[b]/dt = a( k 1 + k ) = k 1 A 0, a = k 1A 0 k k 1 (19.7), (19.50) (1), f x =, f = (19.74) y f x + f y = 0 (19.75), 1 [A], (19.45), d [B] dt = k 1 ( d[b] dt ) d[b] + k [B] k dt, (19.47) (3) ( (19.47) ) (4) () (5) (19.49), t = 0, [B](0) = a+b, [B](0)=0, a + b = 0, *11 6 (), f x = x + y 4x (x + y ) (19.76) f y = x + y 4y (x + y ) (19.77) f x + f y = 4 x + y 4x + 4y (x + y ) = 0 (19.78)

54 f x = x ( x + y + z ) 3 f x = 1 ( x + y + z ) + 3x 3 ( x + y + z ) 5 N , f y = 1 ( x + y + z ) + 3y 3 ( x + y + z ) 5 00 f z = 1 ( x + y + z ) + 3z 3 ( x + y + z ) 5 150, 100 f x + f y + f z = 3 ( x + y + z ) + 3x + 3y + 3z 3 ( x + y + z ) = O α=1.0, β=0.005, N 0 =10 α=1.0, β=0.005, N 0 =0 Twice of the first line t ( ) (1) f 1 (x, y) X, f 1 x + f 1 y = 0, (αf 1 ) x + (αf 1 ) ( f 1 y = α x + f ) 1 y = 0, 19.3,,,, αf 1 (x, y) X () f 1 (x, y) X f (x, y) X,, f 1 x + f 1 y = 0, f x + f y = 0, (f 1 + f ) x + (f 1 + f ) y ( f 1 = x + f ) ( 1 f y + x + f ) y = 0, f 1 (x, y) + f (x, y) X (3) (1) (), X X N 0 = 0, 19.3, N 0 =

55 47 0 :, :,, ( ), 300 1,, (),, 0.1, ( X ), ( Y ), X Y (mapping) 0.1 (,, ), ( ),, () (), X Y, X Y, ,, (), Y X, 0.,,,, (),, X Y, X Y,, Y X *1 * 300,, X Y Y (,, ),, 0.1,, , 400, 300,, 571? *1 Y X * Y X

56 48 0 : 57?, f, X Y, f : X Y (0.1) * , : Z :,,,, f g,,, f : X Y, X Y, f : x y (0.) x X, y Y, f, f :, g, g :, *4 f() = 400 g() =,,, ( ),,, ( ),,, ,, (, ) () 4?...,,,,, 0.7, ( ), R *5,, R R f, f : R R (0.3), f : (x, y) x + y (0.4), f(x, y) = x + y (0.5) () 0.8 3, (3 ) (3 ), R 3 R 3 *6, f, f : R 3 R 3 R (0.6) , *3 :, ; *4, *5 R R R! *6 R 6, 3 6, ( ),, R 3 R 3

57 0. 49, 0.9 x x f(x) = x f : ( (x 1, y 1, z 1 ), (x, y, z ) ) x 1 x + y 1 y + z 1 z (0.7), f ( (x 1, y 1, z 1 ), (x, y, z ) ) = x 1 x + y 1 y + z 1 z (0.8) *7 x, y α, β, f(αx + βy) = (αx + βy) = α(x) + β(y) = αf(x) + βf(y), f(x) (0.1) (),,,,,,,, 0., X W f : f : X W (0.9), f α K, x X, f(αx) = αf(x) (0.10) x, y X, f(x + y) = f(x) + f(y) (0.11), K 573, (0.10) (0.11), (0.10) (0.11), α, β K, x, y X, f(αx + βy) = αf(x) + βf(y) (0.1) 0.10 x = (x 1, x ) R, (,3), f, f : (x 1, x ) (, 3) (x 1, x ) = x 1 + 3x (0.13), f : (3, 1) (, 3) (3, 1) = 9 (0.14) *8 : f((3, 1)) = (, 3) (3, 1) = 9 (0.15), f, R R, f : x = (x 1, x ), y = (y 1, y ) α, β, f(αx + βy) = (, 3) (αx + βy) = (, 3) {(αx 1, αx ) + (βy 1, βy )} = (, 3) (αx 1 + βy 1, αx + βy ) = αx 1 + βy 1 + 3αx + 3βy (0.16), αf(x) + βf(y) = α(, 3) x + β(, 3) y = α(, 3) (x 1, x ) + β(, 3) (y 1, y ) = α(x 1 + 3x ) + β(y 1 + 3y ) = αx 1 + βy 1 + 3αx + 3βy (0.17) (0.16) (0.17), f(αx + βy) = f(αx) + f(βy) (0.18) 574, (0.1),,, f (0.1),, *7 *8 (0.15),,,,

58 50 0 : 0.11 R R g, 0.13 R R g : (x 1, x ) x 1 x (0.19), g((x 1, x )) = x 1 x (0.0), g((3, 4)) = 3 4 = 1 (0.1) g?, g((3, 4)) = g((6, 8)) = 6 8 = 48 (0.) F : (x, y) (x, y), :, x = (, 0) y = (3, 0), α = 1, β = 1, f(αx + βy) = f((5, 0)) = (5, 0), αf(x) + βf(y) = f((, 0)) + f((3, 0)) = (4, 0) + (9, 0) = (13, 0), g, (0.10), g((3, 4)) = g((3, 4)) = (3 4) = 4 (0.3), (0.1) (0.), (0.3) 575,, g ( )? * f(x), 3, F, F : f(x) 3f(x) (0.4), F ( f(x) ) = 3f(x) (0.5), F : f(x), g(x) α, β, 576 R R F : (1) F : (x 1, x ) (x 1, x ) () F : (x 1, x ) (0, x ) (3) F : (x 1, x ) (x, x 1 ) (4) F : (x 1, x ) ( x 1, x ) (5) F : (x 1, x ) (x 1 + 1, x ) F ( αf(x) + βg(x) ) = 3(αf(x) + βg(x)) = α 3f(x) + β 3g(x) = αf (f(x)) + βf (g(x)), F (0.1) 0.9, 0.1,,,,, 0.9, 0.1, f(x) = x 0.1 3x x 3x, 3, F 0.14 f(x) X f(x) X f (x) X,, f(x) d dx : f(x) f (x) (0.6), X X f(x), g(x) X, α, β R,, d ( ) αf(x) + βg(x) = α d dx dx f(x) + β d dx g(x), d/dx * 10 *9,, *10,, x < 0

59 f(x), ( ) (1) F : f(x) f(x) + 1 (0.7) () F : f(x) f(x) (0.8) (3) F : f(x) d f(x) (0.9) dx ( d ) (4) F : f(x) dx f(x) (0.30) x 1 f(x) F F : f(x) 1 0 f(x)dx (0.31),,,,,,, (0.10)(0.11), (0.10), αx X, (0.11), x + y X, X,,, X,, W,,,,,, (0.10) (0.11), f(x) = x, 0 x f(x) = x 3 f(x), x = 0 ), 1, 1,,,,,,,,, * 11 : 0.15 R, (x, y) R, f : (x, y) (x, y ) (0.3), R R,, (0.30),,,, ( ),, X, Y, : f : X Y (0.33),, Y X, f, 0.9, 0.1, 0.14, 0.10 ( ) 579, 3 ( 0.10, ) f(x), L : f(x) d dx f(x) d f(x) f(x) (0.34) dx L L (0.34) L, L = d dx d dx (0.35) *11,

60 5 0 :, P.37 (19.14), L, (19.14),,,,,,, (19.4), ( d dt + 1 ) x(t) = 0 (0.36), (19.11), ( d dt d dt ) x(t) = 0 (0.37), (19.51), ( ) x + y f(x, y) = 0 (0.38),,,,,,,, =0,, x, y, ( ) L, x, y, f(x, y, ), Lf(x, y, ) = 0 (0.39), ( ) 581 (0.39) : (0.39) L, (0.35) L 0.4,, x, y, f(x, y, ), L, 0 g(x, y, ), Lf(x, y, ) = g(x, y, ) (0.40) ( ) g(x, y, ) * 1, f f (explicit ) g 0, (0.40) (0.39),, 0.16, : d dx f(x) d dx f(x) + f(x) = x (0.41) L = d dx d dx + (0.4) g(x) = x (0.43) (0.40) (0.41) 0.17 : d dx f(x) d dx f(x) + f(x) = f(x) + x (0.44),? f(x), d dx f(x) d dx f(x) + f(x) f(x) = x (0.45) f 0.18 : d dx f(x) d f(x) + f(x) = f(x) + x (0.46) dx,,, *1

61 f(x), d dx f(x) d f(x) + f(x) = x (0.47) dx, (0.50) (0.40) t, x (f(x) ) 58 (1) () (3) R C,, t V (t) (, ),,, (0.41) f 1, f d dx f 1 d dx f 1 + f 1 = x (0.54) d dx f d dx f + f = x (0.55) 0.1 (0.54) (0.55), d dx (f 1 + f ) d dx (f 1 + f ) + (f 1 + f ) = x, f 1 + f (0.41), I(t) Q(t) 17.1, (17.40),,, V (t) = RI(t) + Q(t) C, V (t), (0.48) V (t) = V 0 sin ωt (0.49) (V 0, ω, ) (17.43), Q I, (0.48), R dq(t) dt + Q(t) C = V 0 sin ωt (0.50),, L = R d dt + 1 C (0.51) f = Q(t) (0.5) g = V 0 sin ωt (0.53) d dx (f 1 + f ) d dx (f 1 + f ) + (f 1 + f ) = x,,, 0 = x (0.56), f 1 + f (0.41) (0.41),,,,,,,, L, f 1 (x) L f 1 (x) = g 1 (x) (0.57), f (x) L f (x) = g (x) (0.58)

62 54 0 : g 1 (x), g (x), (L) a, b, a, b, al f 1 (x) + bl f (x) = ag 1 (x) + bg (x) (0.59), L ( ), L(af 1 (x)+bf (x)), L(af 1 (x) + bf (x)) = ag 1 (x) + bg (x) (0.60), ag 1 (x) + bg (x),, g 1 (x) g (x),,,, :,,,,,,, : R dq(t) dt R dq(t) dt + Q(t) C = V 0 i eiωt (0.65) + Q(t) C = V 0 i e iωt (0.66) (0.65), e iωt, e iωt, Q(t) = Q 0 e iωt, (0.65), RiωQ 0 e iωt + Q 0e iωt C e iωt, RiωQ 0 + Q 0 C = V 0 i Q 0, Q 0 = V 0 ( Rω + i/c), (0.65), Q 1 (t) = = V 0 i eiωt (0.67) (0.68) (0.69) V 0 ( Rω + i/c) eiωt (0.70) (0.66), 583 : Lf(x, y, ) = g(x, y, ) (0.61), f 0 (x, y, ), L Lϕ(x, y, ) = 0 (0.6) ϕ(x, y,... ), f 0 : f 0 (x, y, ) + ϕ(x, y, ) (0.63) Q (t) = V 0 ( Rω i/c) e iωt (0.71), (0.50), Q 1 (t) Q (t),, (0.50), : Q(t) = Q 1 (t) + Q (t) = V 0 V 0 ( Rω + i/c) eiωt + ( Rω i/c) e iωt (0.7), (0.61) , 0.19 (0.50), V 0 sin ωt : e iωt e iωt V 0 sin ωt = V 0 (0.64) i (1) (0.50), (0.7),, () (0.7), : : Q(t) = V 0{e iωt ( ωr i/c) + e iωt ( ωr + i/c)} (R ω + 1/C ) (0.73)

63 (3) (0.73), : : Q(t) = V 0 Rω cos ωt + (sin ωt)/c R ω + 1/C (0.74) (4) (0.74), : : Q(t) = CV 0 sin(ωt ϕ) (0.75) (ωrc) + 1 ϕ = arctan ωrc (0.76), α = β = 1 (0.11) , α, β (1) F (α(x 1, x ) + β(y 1, y ) ) = F ( (αx 1 + βy 1, αx + βy ) ) = (αx 1 + βy 1, αx + βy ) = α(x 1, x ) + β(y 1, y ) = αf ( (x 1, x ) ) + βf ( (y 1, y ) ) 5 ( ) X, E[X], E, : X, Y a, b, E[aX + by ] =? 6 ( ) X, V [X], V, :! 7 ( ) f(x) f(x + 1) F, () (3) F (α(x 1, x ) + β(y 1, y ) ) = F ( (αx 1 + βy 1, αx + βy ) ) = (0, αx + βy ) = α(0, x ) + β(0, y ) = αf ( (x 1, x ) ) + βf ( (y 1, y ) ) F (α(x 1, x ) + β(y 1, y ) ) = F ( (αx 1 + βy 1, αx + βy ) ) = (αx + βy, αx 1 + βy 1 ) = α(x, x 1 ) + β(y, y 1 ) = αf ( (x 1, x ) ) + βf ( (y 1, y ) ) F : f(x) f(x + 1) (0.77) F, : (0.10), (0.11) (0.11), f(αx + βy) = f(αx) + f(βy) (0.10), f(αx) + f(βy) = αf(x) + βf(y) (0.1), (0.1), α = α, β = 0 (0.10) (4) F (α(x 1, x ) + β(y 1, y ) ) = F ( (αx 1 + βy 1, αx + βy ) ) = ( αx 1 βy 1, αx + βy ) = α( x 1, x ) + β( y 1, y ) = αf ( (x 1, x ) ) + βf ( (y 1, y ) ) (5) F ( (0, 0) ) = F ( ( 0, 0) ) = F ( (0, 0) ) = (1, 0), F ( (0, 0) ) = (1, 0) = (, 0), 577 (1) f(x) = x, g(x) = x * 13, F (f(x) + g(x)) = F (x) = x + 1 *13 f(x) = x, f (x) = 1, f (x) = 0,,, f (x) = 0, f (x) = f (4) = = 0 0, 0

64 56 0 :, F (f(x)) + F (g(x)) = F (x) + F (x) = x +, () f(x) = 1, g(x) = x, F (f(x) + g(x)) = F (1 + x) = (1 + x) F (f(x)) + F (g(x)) = F (1) + F (x) = 1 + x, (3), d ( ) dx αf(x) + βg(x) = α d d f(x) + β dx dx g(x) (4) f(x) = x, g(x) = x 578, F (f(x) + g(x)) = F (x) = = 4 F (f(x)) + F (g(x)) = F (x) + F (x) = =, = α F (αf(x) + βg(x)) = 1 0 f(x)dx + β 1 = αf (f(x)) + βf (g(x)) g(x)dx ( αf(x) + βg(x) ) dx (1), () L, ( 0 ) g(x, y, ), Lf(x, y, ) = g(x, y, ), f(x, y, ) (3), L, Lf = L(f 0 + ϕ) = Lf 0 + Lϕ (0.78) f 0 (0.61), Lf 0 = g (0.79),, ϕ (0.6), Lϕ = 0 (0.80), (0.78), Lf 0 + Lϕ = g + 0 = g (0.81), (0.78), L(f 0 + ϕ) = g (0.8), f 0 + ϕ C C x = 1 i, α = 1 + i αx = (1 i)(1 + i) = x = 1 i, y = 1 + i x + y = (1 i) + (1 + i) = (R )... R C,,..., f = f 0 + ϕ (0.61)

65 57 1 3: 1.1 *1 ( ), : {x 1, x,, x n } (1.1), p 1, p,, p n K (1.) 1.1 (x y) (p 1 x + p y), (p 1 p ) 0, 0, p 1 x 1 + p x + + p n x n = 0 (1.3), p 1 = p = = p n = 0 (1.4), {x 1, x,, x n }, ( ),, 0, 0, 0 (1.3) 1.1, p 1 = p = 0, p 1 x + p y = 0, {x, y}, x, y, a, y = ax (1.6), ax y = 0 (1.7) 585 3,, x, y 0, y 0, 0,,,, {x, y} (), 1.1 x, y 0 x, y,, (1.3) p 1, p,, p n, (1.4), p 1 x + p y (p 1, p ) (1.5), p 1 x p y 586 (1) () (3) *1,, 587 A

66 58 1 3: 1. 3, (p 1, p, p 3) 0, , 1, {x 1, x,, x n }, p 1, p,, p n K p 1 x 1 + p x + + p n x n = 0, p 1 = p = = p n = 0, 1.1,,, 3, ax + by z = 0 (1.10), x, y, z 0, 0 ( z 1 ),, {x, y, z} (),,,,,,? : 1. 3 x, y, z 0, x, y, z, p 1 x + p y + p 3 z (1.8) (p 1, p, p 3 ), p 1 x p y p 3 z ( 1.) *, p 1 = p = p 3 = 0, p 1 x + p y + p 3 z = 0, {x, y, z}, x, y, z, x, y ( 1.3), a, b, z = ax + by (1.9) *, 6, O, O 3 p 1 x, p y, p 3 z, 1.3 : { x 1 = R , x = 3, x 3 = 4 } (1.11) 3 4 5,?,, (1.3), p 1, p, p 3, p 1 x 1 + p x + p 3 x 3 = 0 (1.1), p 1 = p = p 3 = 0 (1.1), (1.1) (1.11), 1p 1 + p + 3p 3 = 0 (1.13) p 1 + 3p + 4p 3 = 0 (1.14) 3p 1 + 4p + 5p 3 = 0 (1.15) (p 1, p, p 3 ) (, ),, *3, (p 1, p, p 3 ) = ( 1,, 1) (1.13) *3,,

67 (1.15), p 1 = p = p 3 = 0 (1.1) 18.7, P, P :, {f 1 (x) = x, f (x) = x, f 3 (x) = 1} (1.18) 588 R (1) e 1 = (1, 0), e = (0, 1), R {e 1, e } () R,, {e 1, e } (3) R,, R n (n ), : n,, 3,, n, n *4 0, *5 589 { x 1 = R 3 3 : 1 3, x = 3, x 3 = 4 } (1.16) 1 4 1, : (18.8), P, P = {f(x) = px + qx + r p, q, r R} (1.17), : p 1 f 1 (x) + p f (x) + p 3 f 3 (x), p 1 x + p x + p 3 0, p 1 x + p x + p 3 = 0 (1.19) (1.19) x = 0, 1, 1, p 3 = 0 p 1 + p + p 3 = 0 p 1 p + p 3 = 0, p 1 = p = p 3 = 0, {f 1 (x), f (x), f 3 (x)}, 0, x 0, p 1 = p = p 3 = 0, x 0, P P,, (1) {g 1 (x) = 1 + x, g (x) = 1 x, g 3 (x) = x + x } () {h 1 (x) = 1 + x, h (x) = 1 x, h 3 (x) = 1} * *5,?, (A λe)x = 0 x = 0 det(a λe) = 0,, A λe, 591 x f(x) = a cos x + b sin x X (a, b ), X = {f(x) = a cos x + b sin x a, b R}(1.0) (1) X R

68 60 1 3: () {sin x, cos x} X,? (3) {sin x, sin x} X,? (4) sin(x + π/3) X *6 (5) {sin x, sin(x + π/3)} X,? (6) {sin x, sin(x + π)} X,? (7) {sin x, sin(x + π/3), cos x} X,? 59 : {x 1, x,, x n } 1 0,, f(x) = f 1 (x) + 3f (x) + f 3 (x) (1.3) P ( ) 3 : B = {g 1 (x) = 1+x, g (x) = 1 x, g 3 (x) = x+x } P R, (1) e 1 = (1, 0), e = (0, 1), {e 1, e } R () g 1 = (1, 1), g = (1, 1), {g 1, g } R X B, X B 1.6 R n,, B X 593 5, e 1 = (1, 0, 0,, 0) (1.4) e = (0, 1, 0,, 0) (1.5) e n = (0, 0, 0,, 1) (1.6), P ( ) P B = {f 1 (x) = x, f (x) = x, f 3 (x) = 1} (1.1), P?, 1.4, B, P, f(x), a, b, c, f(x) = ax + bx + c f(x) = af 1 (x) + bf (x) + cf 3 (x), f(x) B, f(x) = x + 3x + 1 (1.) *6,,,,,,,,, 0 {e 1, e,, e n } (1.7), 595(1) R n 595(1) {e 1 = (1, 0), e = (0, 1)}, R 596 R 4? 597 x f(x) = A sin(x + B) Y (A, B ), Y = {f(x) = A sin(x + B) A, B R} (1.8) (1) X = {f(x) = a sin x + b cos x a, b R}

69 1.3 61, Y = X ( :, X, Y Y = X, Y X X Y, ) () {sin x, cos x} X 1.3,,,,,,,,,,,,, 3, 4, P ,,,, X,, X,,,,, 1.7 R 595, R,, n, R n n, R n n ( ) R n n ( ),,, K n *7!,, K n X, B = {v 1, v,, v n } (1.9), X x, x = x 1 v 1 + x v + + x n v n (1.30), n ( ): x 1, x,, x n (1.31), (1.9) (1.30), x x 1, x,, x n, x 1 x x =. x n (1.33) x (coordinate), x 1 x (component), 1.4 :,,,,! (1.33),?, (1.33) x X,, (1.33) (1.33), (1.30), 598 (1)? (),? *7 K n = K K K (1.3),, K, R C

70 6 1 3: 599,? 600, {ϕ 1 (x) = (1 + x), ϕ (x) = 1 + x, ϕ 3 (x) = 1} (1.37) 1.4 X, ( )K n, f(x) = x + 3x + 1?,, X, (1.30), K n,,,,,,,, P ( ) B,, B = {f 1 (x) = x, f (x) = x, f 3 (x) = 1} (1.34), f(x) = x + 3x + 1 (1.35), f(x) = f 1 (x) + 3f (x) + f 3 (x) (1.36), (, 3, 1) f(x),,, 3, R 3 1.4,,,, : 0.10, R R, y = (, 3) (x 1, x ) (1.38), x, y 1, y = ( 3) [ x1 x ] (1.39) ( 3) 1, 1.9 : P = {f(x) = px + qx + r p, q, r R} (1.40) 1 : P 1 = {f(x) = ax + b a, b R} (1.41), d/dx, P P 1, P, {v 1 (x) = x, v (x) = x, v 3 (x) = 1} (1.4) P 1 : {w 1 (x) = x, w (x) = 1} (1.43)

71 1.4 63, P px + qx + r, p q (1.44) r 601 X : X = {f(x) = a sin x + b cos x a, b R} (1.5) X R R 3,, P 1 f(x) = ax + b, [ ] a b R,, (1.45) d dx : px + qx + r ax + b (1.46),, a = p (1.47) b = q (1.48),, [ ] a = b [ ] p q (1.49) r,, x + 3x + 4, P ( (1.4)) 1 3 (1.50) 4, (1.49), L, g(x, y, ), [ ] [ ] Lf(x, y, ) = g(x, y, ) (1.55) = (1.51) , g 0, g P 1 ( (1.43)), 1 0, x + 3,, x + 3x + 4, f(x, y, ), (1.49),,,, : (1) f(x) X, d dx : f(x) f (x) (1.53), X X () {v 1 (x) = sin x, v (x) = cos x} X (3), : [ ] (1.54),,,,,,,,, (0.40), g(x, y, ) f, g, L A, (1.55), Af = g (1.56),,!,,!,,

72 64 1 3:, 8 X, {x, y} x 0, α, y = αx :! 588 (1) p 1, p R, p 1 e 1 + p e = 0, p 1 (1, 0) + p (0, 1) = (p 1, p ) = 0, p 1 = p = 0 {e 1, e } () {(1, 1), (1, 1)} (3) {(1, 1), ( 1, 1)} 9 R,, 0 : 10 X Y X Y Y, Y Y, Y :, 11 ( ) X Z X Z Z, Z Z, Z, (1) a 1, a, b 1, b, f 1 (x) = a 1 cos x + b 1 sin x (1.57) f (x) = a cos x + b sin x (1.58) f 1 X f X, α R, αf 1 (x) = αa 1 cos x + αb 1 sin x X (1.59), f 1 (x) + f (x) = (a 1 + a ) cos x + (b 1 + b ) sin x X(1.60) 1.5 X R () p, q R, (1) () (3) 587 A, p 1 x 1 +p x + +p n x n = 0 p 1 = p = = p n = 0 p 1 = p = = p n = 0, p 1 x 1 + p x + + p n x n = 0, A p sin x + q cos x = 0 (1.61) x = 0 q = 0, x = π/ p = 0, p = q = 0 {sin x, cos x} (3) {sin x, sin x} p sin x + q( sin x), p =, q = 1 0 {sin x, sin x} (4) ( sin x + π ) = sin x + cos x (1.6) (1.0) a = 3/, b =

73 / sin(x + π/3) X (5) p, q R ( p sin x + q sin x + π ) 3 ( = p + q ) 3q sin x + cos x (1.63) {sin x, cos x},, {sin x, cos x}, 0, p + q = 0 3q = 0, p = q = 0, {sin x, sin(x + π/3)} (6) sin x + sin(x + π) = sin x sin x = 0 {sin x, sin(x + π)}, 0 0 {sin x, sin(x + π)} (7) B, f(x) = px +, q 1 = q = q 3 = 0, qx + r, q 3 = p, q 1 q + q 3 = q, q 1 + q = r (q 1, q, q 3 ), f(x) = q 1 g 1 (x) + q g (x) + q 3 g 3 (x) (1.65), P B, B P 595 (1) {e 1, e } 588, x = (x 1, x ) R, x = (x 1, x ) = (x 1, 0) + (0, x ) = x 1 (1, 0) + x (0, 1) = x 1 e 1 + x e, R {e 1, e } {e 1, e } R () p 1, p R, p 1 g 1 + p g = 0, ( sin x + π ) = sin x + cos x, ( sin x + π ) sin x cos x = 0 {sin x, sin(x + π/3), cos x}, 0 0 {sin x, sin(x + π/3), cos x} 59 (!) q 1, q, q 3 R, q 1 g 1 (x) + q g (x) + q 3 g 3 (x) = 0 (1.64), q 1 (1 + x) + q (1 x) + q 3 (x + x ) = 0, q 3 x + (q 1 q + q 3 )x + (q 1 + q ) = 0 0, q 3 = 0, q 1 q + q 3 = 0, q 1 + q = 0 p 1 (1, 1) + p (1, 1) = (p 1 + p, p 1 p ) = 0, p 1 = p = 0 {g 1, g }, R x = (x 1, x ), p 1, p (x 1, x ) = p 1 (1, 1) + p (1, 1) (1.66)? (1.66), (p 1 + p, p 1 p ), x 1 = p 1 + p x = p 1 p, (1.66), p 1 = x 1 + x, p = x 1 x, x = x 1 + x g 1 + x 1 x g (1.67), R {g 1, g } {g 1, g } R

74 66 1 3: 596 {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} 597 (1) Y A sin(x + B), A sin(x + B) = A(sin x cos B + cos x sin B) = A cos B sin x + A sin B cos x A cos B = a, A sin B = b, A sin(x + B) = a sin x + b cos x X (1.68), Y X (1.69), X a sin x + b cos x,, a sin x + b cos x = a + b sin(x + δ) δ, (a, b) x (a, b) a + b = A, δ = B, a sin x + b cos x = A sin(x + B) Y (1.70), X Y (1.71), f(x) X, f (x) X ) () ( ) (3) X {v 1 (x) = sin x, v (x) = cos x}, X f(x) = a sin x + b cos x, [ a b] f(x), f (x) = a cos x b sin x = b sin x + a cos x,,, [ ] b a (1.7) (1.73) (1.7) (1.73), [ ] b = a [ ] [ 0 1 a 1 0 b] (1.74), (1.54) (1.69), (1.71), Y = X () {sin x, cos x} 591, 3, (1.68), Y...,, {sin x, cos x}, {sin x, cos x} Y Y = X,,, X 598 (1) () 1.5, {x, x, 1}, 3 3, ,, 600 f(x) = x + 3x + 1 = (x + 1) (x + 1), (, 1, 0) 601 (1) (d/dx,,

75 67 4:.1,,,,,, 60, 5,,,,, *1,, 18,, 603 f X α R u, u 1, u, v, v 1, v X (1) f(u, v), v, u : 4) 5) () : 3) 4) f(u, αv) = αf(u, v) (.) (3) : 3) 5) f(u, v 1 + v ) = f(u, v 1 ) + f(u, v ) (.3), * : (4) f(u, v), u, v (5) : 4) 0 = 00 R X, f(u, 0) = f(0, u) = 0 (.4) f : X X R (.1) 1) 5), f X, u, v, u 1, u X 1) f(u, u) 0 ) f(u, u) = 0 u = 0 3) f(u, v) = f(v, u) 4) α R, f(αu, v) = αf(u, v),,,,,, 5) f(u 1 + u, v) = f(u 1, v) + f(u, v).1 u, v R 1) 5), f u = (x 1, y 1 ), v = (x, y ) (.5) *1, 3 *,, (u, v) R R *3 *3,

76 68 4:, R R R f, f(u, v) = x 1 x + y 1 y (.6) f,,?,,? 1) f(u, u) = x 1 + y 1, x 1 y 1, 0 1) 604, (.6) f ) 5) 605 R R R u = (x 1, y 1 ), v = (x, y ) (1) f : f(u, v) = x 1 x + 3y 1 y (.7) () f : f(u, v) = x 1 x (.8) : ) (3) f : f(u, v) = x 1 + y 1 + x + y (.9) : 4) 5) x 1 *4 f(x) X f(x), g(x) X, 1 1 f(x)g(x)dx (.10) 1,.1 604, (.6),, (.7) R,, (.10), f,, f(u, v) u v (.11) (u, v) (.1) u, v (.13) u v (.14) (.11),, (.1), (.13),, (.14) 607 u = (1, ), v = (3, 4) (1) R (.6), u v = 5 () R (.7), u v = x 1 f(x) X X (.10) f(x) = x X (.15) g(x) = x + 1 X (.16), f(x), g(x) 1/3., (metric space), X u R, (.6) *4,, 1) u u, u u u

77 . 69 *5, u *6, u = u u (.17),, (.17), u, u = 0 (.17),,,,, , :, u = (1, ),, 5, (.7) 14!,, u = (1, ) 5,, :,,,,, 1,,, (1, ), 5,,,,,,,,, 1, (1, ) 5 611, 61 (1) (),,, 61 u = (1, ), v = (3, 4) (1) R (.6), u = 5, v = 5 () R (.7), u = 14, v = x 1 f(x) X X (.10), X f(x) = x X (.18) g(x) = x + 1 X (.19), f(x) = 1/5, g(x) = 4/3 *5, X R, u u *6 u 614 u, u u X, 0 (.0)

78 70 4:,, 0,, a, b, a b = a b cos θ (.1),,,, (.1), a, b, α a = αb (.5), a b,,,,,, (.1) (.1) 615, (.1),, (.1),, 616, : 3,,, 617 u = (1, ), v = (3, 4), a, b, (1) R (.6) u v θ : (.3) () R (.7) u v a b a b (.) θ : (.3), x 1 cos θ := a b a b (.3) (.1), a, b θ,, (.10) X f(x) = x X g(x) = x + 1 X, f(x) g(x) θ *7 : (.3),, (.), X, 0 a, b, a b = 0 (.4), a b *7 θ, f(x) g(x),,!,,

79 , 1 1, (.3), 1 1 (1) X a, b a, b t, f(t) f(t) = (ta + b) (ta + b) (.6) : p 1 v 1 = p v = 0 (.33) () p 1 = p = 0 (3) {v 1, v } 61, : t, f(t) 0 (.7) () : f(t) = a t + (a b)t + b (.8) (3) t f(t) = 0,, : (.7) (4) : (a b) a b 0 (.9) : (5) : a b a b (.30) : X, (6) : a b a b a b (.31) (7) : 1 a b a b 1 (.3).3 i, j, δ ij, δ ij = { 1 i = j 0 i j (.34) δ ij,, δ 1 = 0, δ 33 = ? 3 x, y, z 1 e 1, e, e 3 e i e j = δ ij (.35) (.3) 60, v 1, v, v 1 v = 0 p 1, p, p 1 v 1 + p v = 0 (1) v 1 v.4 X, ( ), 1,, {e 1, e,..., e n } X (.36) (n, e i e j = δ ij (.37)