2011de.dvi

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1 211 (

2 , 211.,,,,.,,,.,.,.,,.,.,,.., Web. kikuchi/prttime/wkym211/de/211de.html

3 1.,,,.,.,,,.,,,, (.,,.,,. 1.1.,,. 1.1 (.,.,,,.,., m,, x. (, x<, x.,, kx., k (,,. t, (. m d2 x = kx. (1.1 dt2 1.,.,. k,,., ω = m, x = cos ωt, t = x = cos =, = ω sin =, dt m d2 x dt = m d2 2 dt ( cos ωt =m 2 ω2 ( cos ωt = m k cos ωt = kx. m, x = x(t = cos ωt (1.1 x( =, ( = dt. 3

4 ,,, b., x = b sin ωt ω, dt = b ω cos ωt = b cos ωt,, t = = b cos = b. ω dt, m d2 x dt 2 ( = m d2 b sin ωt = m b dt 2 ω ω ω2 ( sin ωt = k b sin ωt = kx. ω, x = x(t = b sin ωt (1.1 x( =, ( = b ω dt.,, b., x = cos ωt + b sin ωt, ω ( m d2 x dt = m d2 d2 b ( cos ωt+m sin ωt =( k cos ωt+ ( k bω 2 dt2 dt 2 ω sin ωt = kx., x( = cos + b sin =, ω dt ( = ω ( sin + b ω cos = b. ω, x = x(t = cos ωt + b sin ωt (1.1 x( =, ω dt ( = b., x( =, ( =, x( =, dt dt ( = b., (1.1 x d2 x 1, dt2, (1.1 x( =, ( = b dt x(t = cos ωt + b sin ωt,. ( ω ( (1 y = y(t = x(t cos ωt + b 2 ω sin ωt + m k ( (t ( ω sin ωt + b cos ωt dt 2, x(t (1.1, dt =. (2 (1, x(t = cos ωt + b sin ωt. ω 1.3 (., m,, ( g., h>, 2 4

5 .,, t x = h 1 2 gt2, v = gt., m d2 x = mg x( = h, v( = ( = dt 2 dt.,, g v, 1 x.,.,,,,,., v,., v. m dv = mg kv. (1.2 dt, k,.,., v( =., (1.3., m dv dt + kv = mg. e k m t, m d dt ( e k m t v = me k m t dv dt + m k ( m e k k m v = e m t m dv dt + kv = mge k m t. m, v( =, t [ e k m t v = g e k m ] t m s ds = g k e k m s = mg k (e k m t 1., v = mg k k (1 e m t. ( km e km t, v( = mg dv (1 1 =, k dt = mg k = ge k m t,., v> mg mg, v k k., lim v = mg,, mg t + k k.,. 1.4,, v., v >.,, (1.2 v( = v, t.,,, v.,,. 5

6 1.5 (., 2., 2, (,.,.,,,.,. x 1, t x = x 2 R 3. x = x x x 2 3 x 3 x,,., M, m, G>, (. m d2 x dt = GMmx. (1.3 2 x 3 (x = (,,., x. G. r = x, x x,, 1,,,, 2. g = GM, (1.3,. d 2 x dt 2 = gx x 3. (1.4 v, v =., dt l = m(x v., x v x, v., t, x, v., x v,.., dv dt = d dt.,., d(x v dt = dt v + x dv dt = v v + x dl d(m(x v = dt dt., = ( = d2 x dt dt 2 ( gx =. (1.5 x 3 =. ( , b = b 1 b 2 b 3 R 3 b = 1 b b b 3. (1.5 l = x v t 6

7 , d dt (x l = d (x (x v = (x v =v (x v =. dt dt, t,,, t x v.,., (,, ( t x 1 x = R 2 v 1., v = =. dt x 2, 1 2 (x v =1 2 (x 1v 2 x 2 v 1., (. (., 3, R 3 1., (1.5 2m (1.6,, t.,.,,, t x v = x 1 v 2 x 2 v 1 >.,., { x1 = r cos θ,, r = x x2 2,, x 2 = r sin θ. v 2 (1.7., v 1 = 1 dt = dr dθ cos θ r sin θ dt dt, v 2 = 2 dt = dr dθ (1.8 sin θ + r cos θ dt dt. ( 2 dv 1 dt = d2 r dθ dθ cos θ 2dr sin θ dt2 dt dt r cos θ r sin θ d2 θ dt dt, ( 2 2 dv 2 dt = d2 r dθ dθ sin θ +2dr cos θ dt2 dt dt r sin θ + r cos θ d2 θ dt dt. 2 (1.9 <x 1 v 2 x 2 v 1 ( dr dθ = r cos θ sin θ + r cos θ dt dt ( dr dθ r sin θ cos θ r sin θ = r 2 dθ dt dt dt. 7

8 , dθ dt, (1.4, ( dθ cos θ >.., dt sin θ (1.11 r,, d dt ( sin θ cos θ ( d 2 2 r dθ dt r = g 2 dt r, 2 (1.1 2 dr dθ dt dt + r d2 θ =. dt2 (1.11 ( r 2 dθ =2r dr dθ dt dt dt + d2 θ r2 dt =. 2 r 2 dθ dt = c (. (1.12 t dθ >, c>. (1.1, dt d 2 r dt c2 2 r = g 3 r. ( dθ >, t θ = θ(t t = ψ(θ, t = ψ(θ dt r = r(t, r ( θ., dr dθ = dr dt dt dθ = r2 dr c dt. (1.14, u = 1., r du ( dθ = 1r dr 2 dθ = 1 dr c dt. (1.15, d 2 u dθ 2 = 1 c, (1.13,, d dθ ( dr = r2 d 2 r dt c 2 dt. ( d 2 u dθ 2 + u = g c 2. (1.17, u = g c (, ũ = u u 2, d2 u dθ =, 2 ( ( d 2 ũ d 2 dθ + ũ = u d 2 2 dθ + u u 2 dθ + u 2 = g c g =. ( c2 8

9 , 1.1, α, β R, ũ = α cos θ + β sin θ. (1.19, δ = α 2 + β 2, cos θ 1 = α δ, sin θ 1 = β δ θ 1,., h = c2 g, ε = c2 δ g θ 1 > θ, r = 1 u = 1 r = u = g c 2 + δ cos(θ θ 1 (1.2., u = 1 h (1 + ε cos(θ θ 1, 1+ε(θ h., ε = 1+εcos(θ θ 1, r = h, dθ dt = c h 2,., θ = θ θ 1, x 1 = r cos θ, x 2 = r sin θ., R 2, x 1 x 2 θ 1, x 1 x 2, x 1 x 2., h = r(1 + ε cos θ, h 2 = r 2 (1 + ε cos θ 2 = r 2 (1 + 2ε cos θ + ε 2 cos 2 θ = r 2 (1 + 2ε cos θ(1 + ε cos θ ε 2 cos 2 θ = r 2 +2hε x 1 ε 2 x 2 1 =(1 ε2 x hε x 1 + x 2 2., <ε<1, ( (1 ε 2 x 1 + hε 2 + x 2 1 ε 2 2 = h 2 + h2 ε 2 1 ε = h2 2 1 ε. 2, = h 1 ε, b = h, 2 = 1 ε 2 hε 1 ε 2, ( x x2 2 =1. ( b2 <ε<1 1 ε 2 > 1 ε 2., ( = h 1 ε, 2 h ( b =., ε =, 1 ε 2.,., ε =1, ( x 2 2 = 2h x 1 h. (

10 , x 1., ε>1, <ε<1,, = h ε 2 1, b = ( (ε 2 1 x 1 hε 2 x 2 ε = h2 ε 2 1. h ε2 1, = hε ε 2 1, ( x x2 2 =1. ( b2, 2 x 1. 2,., s hs 1+εs = h h ε ε s ε,, s> 1 ε, 1+εcos θ > cos θ > 1 ε, x 1 = h cos θ 1+εcos θ h 1+ε < hε ε 2 1 =., x 1 <. ε., <ε<1, ε = = 2 b 2, ε>1, ε = = 2 + b 2., ε 1, 2,.,,,, θ 1 = θ( =,,., ε<1.,, πb, T., ( (x 1v 2 x 2 v 1 = 1 dθ r2 2 dt = c., 2, T = 2πb c = 2π c h 1 ε h = 2 1 ε 2 2πh2. (1.24 c(1 ε ( T 2 = 4π2 h 4 3 c 2 (1 ε 2 = 4π2 h h = 4π2 3 c 2 1 ε 2 g 3. (1.25,.,. 1

11 , E., E m = 1 2 m v 2, E p = GMm = gm r r E = E m + E p., E m,., (1.8, (1.12 (1.14, ( 2 ( 2 dr v 2 dθ dr = cos θ r dt dt sin θ dθ + sin θ + r dt dt cos θ ( 2 ( 2 ( ( 2 (dr ( 2 (dr 2 dr dθ dθ = + r 2 = + r 2 = c2 + r 2. dt dt dt dθ r 4 dθ, r = h 1+ε cos θ, dr dθ = hε sin θ (1 + ε cos θ 2., v 2 = c2 (1 + ε cos θ 4 h 4 = g h (1 + 2ε cos θ + ε2. ( h 2 ε 2 sin 2 θ (1 + ε cos θ + h 2 4 (1 + ε cos θ 2, E = E m + E p = 1 2 m v 2 gm r = gm 2h (1 + 2ε cos θ + ε2 gm h (1 + ε cos θ =gm 2h (ε2 1.,, ε<1,.,,.,,,,,. 11

12 ( 12

13 ,,., x 1,...,x n f (. f. 1, 2,,.,.,, m, R m+2 D R m+2 Φ(x, y, u 1,...,u m,. ( Φ x, y, y,...,dm =. (1.26 m (1.26 x, y.,.,.,.., 1.5 (1.3, f = f 1. f r, 1.,,, (1.26., 1.1 (1.1 2, 1.3 (1.2 1.,,. 2.,, 1., 1.1 (1.1, x 1 = x, x 2 =,. dt 1 = x 2, dt 2 = k (1.27 dt m x 1., 1., R 3 D R 3 Φ(x, y, u,. ( Φ x, y, =. (

14 Φ(x, y, u = u, R 2 D R 2 F (x, y u = F (x, y,. = F (x, y. (1.29, (1.28., f (x,y D (y = f(x (x 1.,. y 1 4 ( 2 =. y = f(x, f( = 1., ( = ±2 (,, f(x =(x+1 2, f(x =(x 1 2 ( = 2, ( = 2 1 y = f(x (.,, x,, y = f(x D y.,,.,,.,, 1., 1.,. 1.6 I x,i y R, ϕ : I x R, ψ : I y R, I y ψ(y.,. ( x y = ϕ(xψ(y. (1.3 I x I y, (1.3 y = f(x, y = f(x., F : I x I y R. F (x, y =, F (x,y =., F x, I x I y F y x ϕ(sds y 1 dt. (1.31 y ψ(t = ϕ(x, F y = 1 ψ(y, I y ψ(y., ( 1.1,

15 , x I x, I x, y I y, I y I x,,i y, R, I x, C 1 f : I x, R, 1. (i f(x ( =y. ( x x (ii I x, I y,, y = f(x F (x, y =. y y df (iii = F x(x, f(x F y (x, f(x = ϕ(xψ(f(x. (i, (iii, y = f(x (1.3 y = f(x ( I x = I y =(, + (,. = y2 x. (1.32 f(1 = 1 y = f(x. (1.31 F : (, + (, + R. ds y [ ] y F (x, y = 1 s dt 1 1 t = [log 2 s]x 1 + = log x (1.33 t 1 y y = f(x F (x, y =, 1 (, + 1 I x,i y (, + 1, y = f(x = 1 log x I x f(1 = 1 (1.32., I x, =(,e, I y, =(, (,e y = f(x = f(1 = 1 ( log x,., ψ(y = y I y, f(x =y ( (1.3 y = f(x., I x = I y = R,. =3y 2 3. (1.34, t R, t 2 3 =(t , ϕ(x =1, ψ(y =3y 2 3, = ϕ(xψ(y. (1.34 f( =.,, y = f(x =x 3., ψ( = =, y = f(x =( (1.34.,. 15

16 1.1 (1 y = f(x =x 3, y = f(x =( (1.34 f( =. (2 f : R R. { x 3, x, f(x =, x <., y = f(x (1.34 f( =., f( = (1.34 y = f(x. 1.,, J R, F : J R F (t t (t J.,. ( y = F. (1.35 x (, x y x J R 2, y = f(x (1.35 x y., u = y, y x, u x x x., y = xu,,. = u + xdu = F (u. du = F (u u. (1.36 x, ϕ(x = 1 x, ψ(u =F (u u, D = {(x, u R2 ; x,u J}, (1.36., u = y, ( x x D, (1.36 u = g(x u = g(x ( u., f(x =xu = xg(x, y = f(x (1.35 y = f(x J =(, +,. = x + y x. (

17 , x x, x + y =1+ y, F : J R F (t = x x 1+t (t J, J F (t t, (1.37., u = y, (1.37. x du = F (u u x = 1 x. (1.38, (1.37 f(1 = 1 y = f(x., (1.38 g(1 = 1 =1 u = g(x. ( u, g(1 = 1, g(u = [log s] x = log x +1., (1.37 f(1 = 1 y = f(x =xg(x =x(log x y = f(x =x(log x +1 (1.37 f(1 = 1,. J R, F (t =t t J, (1.35., u J F (u =u., f(x =u x ( 1., x R \{}, x x, y x = u, df ( y = u = F (u =F., y = f(x =u x (1.35 x f(x =u x. 17

18 ( 18

19 , m, D R m+2, Φ(x, y, u 1,...,u m D, y, u 1,...,u m 1.,. ( Φ x, y, y,...,dm =. (1.39 m, Φ y, u 1,...,u m,, 1,,,., I R, ϕ, ψ : I R., I., b R, f( =b y = f(x I 1.. ( f(x =b exp ϕ(tdt + = ϕ(xy + ψ(x. (1.4 ( exp ϕ(sds ψ(tdt, x I. (1.41 t.. ( exp ( x exp ϕ(tdt ϕ(xy = ψ(x. ϕ(tdt,.. ( exp ( yϕ(x exp = d ( ϕ(tdt = ψ(x exp ϕ(tdt. ( ϕ(tdt yϕ(x exp ( ( y exp 19 ϕ(tdt. ϕ(tdt

20 ,. ( ( d x ( y exp ϕ(tdt = ψ(x exp ϕ(tdt. (1.42, y = f(x f( =b, x,. ( ( t f(x exp ϕ(tdt b = exp ϕ(sds ψ(tdt., y = f(x. ( ( ( f(x =b exp ϕ(tdt + exp ϕ(sds exp ( ( = b exp ϕ(tdt + exp ϕ(sds ψ(tdt. t t ϕ(sds ψ(tdt 1.16 I = R,. = y + x. (1.43, b R. ϕ, ψ : I R, ϕ(x =1, ψ(x =x (x I = R, f( = b y = f(x. ( ( f(x =b exp dt + exp ds tdt = be x + t exp(x tdt = be x + (x te t dt = be x + x = be x + x(e x 1 [te t ] x + =(b +1e x x 1. t e t dt te t dt e t dt = be x + x(e x 1 xe x +(e x y = f(x =(b +1e x x 1 (1.43 f( = b,. (1.4,,, I ψ(x =. = ϕ(xy. (1.44, f( =b. ( f(x =b exp ϕ(tdt, x I. (1.45 2

21 , (1.44., b,., I y = {y R ; y b < b }., F : I I y R. y dt x F (x, y = ϕ(sds t = ϕ(sds log y + log b, (x, y I I y. b, U I U b V I y V, x (1.44 y = f(x U V F (x, y =, log y = log b + ϕ(sds (., y = b exp ϕ(sds, y V I y, y b,. ( y = f(x =b exp ϕ(sds.,, 1,. (1 1, y = f(x ϕ I,,, (, b, y = f(x U I, U. (2 (, b, b =,.,,., I R, α, 1, ϕ, ψ : I R., I, b>., U I U, f( =b y = f(x U 1 (. = ϕ(xy + ψ(xyα. (1.46, α, b, f( =b, U I, U 1.. b>, c = b 1 α >., u = y 1 α., du =(1 αy α,. du =(1 αy α (ϕ(xy + ψ(xy α =(1 α(ϕ(xy 1 α + ψ(x. 21

22 , u. du =(1 α(ϕ(xu + ψ(x. (1.47, 1, 1.15, I g( =c (1.47 I 1. g, c>, U I U, u = g(x > (x U., f(x =g(x 1 1 α, y = f(x f( =b (1.46., α, 1 α, b, c = b 1 α., b c. b>., b<., c<., u = y 1 α, g( =c (1.47 I 1. g, c<, U I U, u = g(x < (x U., f(x 1 α = g(x f(x, U f(x =g(x 1 1 α, y = f(x f( =b ( I = R,. = y xy2. (1.48, b, f( = b. c = b 1 2 = b 1, u = y 1 2 = y 1, u. du =(1 2(u x = u + x. (1.49, 1, g( = c = b 1 (1.49 u = g(x. ( ( g(x =c exp ( 1dt + exp ( 1ds tdt t = 1 b e x + te ( x+t dt = 1 b e x + e x ( [te t ] x = 1 b e x + e x (xe x (e x 1 = b +1 e x + x 1. b e t dt, h(x =(b+1e x +b(x 1 (x I = R., h( = (b+1e +b ( 1 = (b +1 b =1>., h (x = (b +1e x + b, b = 1, h (x =b = 1, b 1, h (x =, e x = b., b +1 1 <b<, h (x = x, h ( = (b+1e +b = 1 <, h, h (x <. b< 1, b> 22

23 b, b +1 >, x = log b +1 h (x = (b +1e x + b = b., (b +1e x = b, h(x =b + b(x 1 = bx., h (x =(b +1e x, b>, h, x<x h (x <, x>x h (x >, h ( = 1 < x >., x = x h, bx >., x h(x >. b< 1, h, x<x h (x >, x>x h (x <, h ( = 1 < x <., x = x h, bx >., lim h(x, lim h(x x + x, (. lim x + h(x = lim x h(x = { +, b >,, b <, { +, b > 1 b<,, b < 1. h( = 1 >, 1 b<, h(β = β> 1., b< 1, x <, h(α =h(β = α<x < α β> 1., U I = R. I = R, b >, U = (,β, 1 b<, (α, β, b < 1., x U h(x >, g(x., f(x =g(x 1, y = f(x f( = b (1.48 U. 1.2 b., R h(x =(b +1e x + b(x 1, lim h(x lim h(x. x + x b 1.21 y = f(x = (1.48 f( = b (b +1e x + b(x 1,.,, α =, 1, α =1, 1.,, 1,. 23

24 1.22 I R, ϕ, ψ, ρ : I R, ρ., I, (. = ϕ(xy2 + ψ(xy + ρ(x. (1.5, I 1 I I 1, I 1 y = f 1 (x., b f 1 ( b, U I 1, U, f( =b U 1.. y = f 1 (x (1.5,. df 1 (x =ϕ(xf 1(x 2 + ψ(xf 1 (x+ρ(x, x I 1., u = y f 1., u. du = d (y f 1(x =(ϕ(xy 2 + ψ(xy + ρ(x (ϕ(xf 1 (x 2 + ψ(xf 1 (x+ρ(x = ϕ(x(y 2 f 1 (x 2 +ψ(x(y f 1 (x = ϕ(x(y f 1 (x(y + f 1 (x + ψ(x(y f 1 (x = ϕ(xu(u +2f 1 (x + ψ(xu., u,. du = ϕ(xu2 +(2ϕ(xf 1 (x+ψ(xu. (1.51. α =2, b f 1 (, U I 1, U, g( =b f 1 ( (1.51 u 1., f(x =g(x+f 1 (x, y = f(x f( =b (1.5 U I = R,. = y2 +2y 3. (1.52,., f 1 (x = 1(.,. df 1 (x == =f 1 (x 2 +2f 1 (x 3. 24

25 , y = f 1 (x =1 (1.52., u = y f 1., u. du = u2 +( u = u 2 +4u. (1.53 (, α =2., b b 1=f 1 (., b 1 = b 1=b f 1 (., v = u 1 2 = u 1., v. dv =(1 2(4v +1= 4v 1. (1.54, c = 1 = 1 1., h( = c = (1.54 b 1 b 1 b 1 v = h(x. ( ( h(x =c exp ( 4dt + exp ( 4ds ( 1dt t = 1 b 1 e 4x e 4(x t dt = 1 b 1 e 4x e 4x e 4t dt = 1 b 1 e 4x e 4x [ 1 4 e4t ] x = 1 b 1 e 4x 1 4 e 4x (e 4x 1 = b +3 4(b 1 e 4x 1 4 = (b +3e 4x (b 1. 4(b 1, k(x =(b+3e 4x (b 1 (x I = R., k( = (b+3e (b 1 = 4 >., b = 3, k(x =(( 3 + 3e 4x (( 3 1 = 4 >. b 1, 3, k(x =, e 4x = b 1., 3 <b<1, b +3 k(x = x., k( = 4 >, k, x k(x >. b< 3, b>1, x = 1 b +3 log 4 b 1, k(x =. b>1, b +3>b 1 > b +3 b 1 > 1, x >. b< 3, b 1 <b+3<, b +3 b 1 < 1, x <., lim k(x = (b 1, lim x + k(x = x { +, b > 1 3 <b<1,, b < 3. b 1, 3, k I, k(x = x 1., U I = R. (,x, b > 1, U = I = R, 3 b<1, (x, +, b < 3. 25

26 , x U I x k(x >., 1 h(x = 4(b 1 k(x =(b +3e 4x (b 1., U 4(b 1 u = g(x =h(x 1 4(b 1 = (b +3e 4x (b 1 g( = b 1=b 1 (1.53 U., f( = b (1.52 U. y = f(x =g(x+f 1 (x =, b = 3, f(x = = 4(b 1 + ((b +3e 4x (b 1 (b +3e 4x (b 1 3(( 3 1 (( 3 1 4(b 1 (b +3e 4x (b 1 +1 = (b +3e 4x +3(b 1 (b +3e 4x (b 1. = 3 =b ( = y2 +2y 3. (1 y = 3(,, b 3 b, f( = b y = f(x. (2, b 1, 3 b, f( = b y = f(x., J R, ϕ : J R C 1., (. y = x ( + ϕ. (1.55, c J,. y = cx + ϕ(c, x R. (1.56., ϕ C 2, ϕ (c (c J. c J, y = cx + ϕ(c, c =x + ϕ (c, x, y (x(c,y(c. x(c,y(c c C 1, c y = f(x,.. 26

27 .,. c J, y = f(x =cx + ϕ(c (x R, = c,. y = cx + ϕ(c =x ( + ϕ., y = f(x =cx + ϕ(c (1.55., ϕ C 2, ϕ (c (c J, R (1.55., c J, x, y. { y = cx + ϕ(c, = x + ϕ (c. (x(c,y(c., (x(c,y(c y = cx + ϕ(c, c., =x(c+ϕ (c, dc = ϕ (c., y(c =cx(c+ϕ(c,. dc = x(c+c dc + ϕ (c, c J., =x(c+ϕ (c, dc = c dc., x(c,y(c C1., J, x (c, I = {x(c R ; c J}, I R, c I C 1., y x I,.,. = dc dc = dc dc = c. y(c =cx(c+ϕ(c =x(c + ϕ (., c J {(x(c,y(c ; c J} (1.55., y = f(x (x(c,y(c, c,, y = cx + ϕ(c.,,.,,,. 27

28 1.26 J = R,. y = x ( 2. (1.57 ϕ(t = t 2 C 2, ϕ (t = 2.,. c J = R, y = cx c 2 (1.57.,. y = cx c 2 c, =x 2c., x, y. { y = cx c 2, = x 2c. 2 c = x. 1,. 2 y = x 2 x ( x 2 2 = x 2 2 x2 4 = x2 4., y = f(x = x2 4 ( y = x ( 2. (1 y = f(x = x2 4. (2 y = x2 4 (2c, c2 (c R y = f (c (x,. 28

29 ,. D R 2, ϕ, ψ : D R.,. ϕ(x, y + ψ(x, y =. (1.58, 2., D (x,y D., ψ(x,y., 1, ψ(x, y,. = ϕ(x, y ψ(x, y. (1.59 ( 1, δ> ( δ, δ C 1 c (x c = c (y ( ( :( δ, δ D, c( = x, dc y dt ( (t ( δ, δ. ϕ(c (x (t,c (y (t dc(x dt (t+ψ(c(x (t,c (y (t dc(y (t =, t ( δ, δ. (1.6 dt, dc(x dc dt ( dt ( =, ψ(x,y dc(y ( dt dc(y ( =, ( =, dt., dc(x dt (., t x τ t = τ(x, c (y, x y x y = f(c =c (y (τ(x. f, (1.58.,,. dt (x = (t dt (x = dt (t = dt (t ϕ(x, y ψ(x, y., (1.59 (1.6., ( ,., ϕ(x,y, x y. 29

30 (1.58, ϕ, ψ,.., D = R ϕ, ψ : R 2 R., R 2 C 1 U : R 2 R, U U (x, y =ϕ(x, y, (x, y =ψ(x, y., x y,. ϕ(x, y + ψ(x, y =. (1.61, (x,y( R 2 ψ(x,y (. δ> ( δ, δ c (x x c =, c( =,. c (y y ϕ(c (x (t,c (y (t dc(x dt (t+ψ(c(y (t,c (y (t dc(y (t =, t ( δ, δ. (1.62 dt, ϕ(x, y = U x. (x, y, ψ(x, y = U (x, y, (1.62 y U x (c(x (t,c (y (t dc(x dt (t+ U y (c(x (t,c (y (t dc(y (t =. dt,. d dt U(c(x (t,c (y (t =., c {(c (x (t,c (y (t R 2 ; t ( δ, δ} R 2 C. C = {(x, y R 2 ; U(x, y =U(x,y }. (1.63, R 2 C (x,y (1.62. R 2 F F (x, y =U(x, y U(x,y ((x, y R 2, C R 2. C = {(x, y R 2 ; F (x, y =}. F (x,y =U(x,y U(x,y =, F y (x,y = U y (x,y =ψ(x,y, ( 1.1, 1.2 x V x, y V y V x,v y R V x C 1 f : V x R, f(x V y (x V x, f(x =y, 3

31 (x, y V x V y, y = f(x F (x, y =,,. = U x(x, f(x f(x = ϕ(x, U y (x, f(x ψ(x, f(x. (1.64, y = f(x (1.61 f(x =y., f(x =y (1.61, F (x, y = (x,y., ϕ(x,y, (x,y F (x, y = g(y =x (1.61 x = g(y., ϕ, ψ, ϕ, ψ : R 2 R C 1., 2. (i. (ii. ϕ(x, y + ψ(x, y =. (1.65 ϕ y (x, y = ψ(x, y. (1.66 x. ((i (ii. U : R 2 R U U (x, y =ϕ(x, y, (x, y =ψ(x, y x y., ϕ y (x, y = 2 U y x, (x, y, ψ x (x, y = 2 U x y (x, y, ϕ, ψ C1 2 U y x (x, y, 2 U x y (x, y., 2 U y x (x, y = 2 U (x, y x y., ϕ (x, y = ψ(x, y. y x ((ii (i. (x,y R 2., U : R 2 R. y U(x, y = ϕ(s, y ds + x ψ(x, tdt, y (x, y R 2. (1.67,. U d x (x, y = ϕ(s, y ds + y y ψ ψ(x, tdt = ϕ(x, y + (x, tdt x x x y y x y ϕ = ϕ(x, y + (x, tdt y y = ϕ(x, y +(ϕ(x, y ϕ(x, y = ϕ(x, y, U y (x, y = y ψ(x, tdt = ψ(x, y. y y 31

32 1.3 (ii (i, (x, y R 2 U,, x ϕ,, y ψ., (x,y R 2, y ψ,, x ϕ.,. ( 1.29 (i, (ii. ϕ (i grd U = C 2 U : R 2 R. ψ ( ϕ (ii rot =. ψ (i (ii,.,. (iii γ :[, 1] R 2 γ( = (x,y, γ(1 = (x, y C 1.,. 1 ( U(x, y = ϕ(x, y + ψ(x, y = ϕ(γ(t dt + ψ(γ(t dt. (1.68 dt γ U(x, y γ. (ii (i, (x,y (x, y C 1 1, U., C 1,., R 2, D., D (, ( (2x + y +(x +2y =. (1.69, (x,y R 2 (x,y (,., ϕ(x, y =2x + y, ψ(x, y = x +2y, ϕ(x,y, ψ(x,y 1., ϕ ψ (x, y =1, y (x, y =1. x, (1.69., U : R 2 R. U(x, y = ϕ(s, ds + y ψ(x, tdt = 2sds + y =[s 2 ] x +[xt + t2 ] y = x2 + xy + y 2, (x, y R 2. (x +2tdt 32

33 , (1.69 C, (x,y,. C = {(x, y R 2 ; x 2 + xy + y 2 = x 2 + x y + y 2 }. (1.7,. ( ( 1.32 (1 x 2 + xy + y 2 = 1 2 (x y 2 1 x, C 1 2 y, C,,. (2 C = {(x, y R 2 ; x 2 + xy + y 2 =} R 2., R 2. ϕ, ψ : R R, ψ(y (y R.,. = ϕ(xψ(y. (1.71,. ϕ(x 1 =. (1.72 ψ(y (, ϕ(x =, 1 =., ϕ, y x ψ(y ψ : R 2 R ϕ(x, y =ϕ(x, ψ(x, y = 1 ψ(y ((x, y R2, (x,y R 2, F : R 2 R. F (x, y = x ϕ(s, y ds + y y ψ(x, tdt = x ϕ(sds x dt ψ(t. (1.73, F F 1 (x, y =ϕ(x, (x, y =, (1.72 x y ψ(y., F (x, y = (x,y y = f(x f(x =y (1.71.,, D R 2, ϕ, ψ : D R.,. ϕ(x, y + ψ(x, y =. (1.74, D μ : D R,, μ. μ(x, yϕ(x, y + μ(x, yψ(x, y =. (

34 1.29, μ,. (μϕ y (x, y = (μψ (x, y. x, μ. ψ(x, y μ ( ψ ϕ (x, y ϕ(x, y μ(x, y+ (x, y (x, y μ(x, y =. (1.76 x y x y,, (1.74., μ, x., μ x μ : R R, (1.76. ψ(x, y dμ ( ψ (x+ ϕ (x, y (x, y μ(x =. x y, D = {(x, y R 2 ; ψ(x, y }, D x = {x R ; y R, (x, y D}, D μ = {x R ; μ(x }., x 1 D x D μ, (x 1,y 1 D y 1 x 1 V x D x D μ, y 1 V y, V x V y D V x,v y R, (x, y V x V y ψ(x, y,μ(x,. ( 1 ϕ ψ (x, y (x, y = 1 dμ (x. (1.77 ψ(x, y y x μ(x x, y., V x V y D V x,v y R, V x V y ψ(x, y, (1.77 y V y., (x 1,y 1 V x V y, μ(x 1 =1 (1.77 V x,. ( ( 1 ϕ ψ μ(x = exp (t, y (t, y dt, x V x,y V y. (1.78 x 1 ψ(t, y y x, V x μ., (x 1,y 1, (1.75 y = f(x, f(x 1 = y 1. y = f(x, ( (1.74., V x V y ϕ(x, y, 1 ϕ ψ (x, y (x, y y, g(y 1 =x 1 ϕ(x, y y x (1.74 x = g(y (x 2 +2xy +(2x 2 + xy =. (1.79 ϕ(x, y =x 2 +2xy, ψ(x, y =2x 2 + xy.,. ϕ y ψ (x, y (x, y =2x (4x + y = 2x y. x 34

35 , D = {(x, y R 2 ; ψ(x, y =2x 2 + xy }, D. ( 1 ϕ ψ (x, y (x, y = 2x y ψ(x, y y x 2x 2 + xy = 1 x., x., (x 1,y 1 =(1, 1., ψ(x 1,y 1 =ψ(1, 1 = =3., μ(x. ( 1 μ(x = exp t dt = exp( [log t] x 1 = exp( log x = 1 x. 1 μ(x (1.79,., U : R 2 R. U(x, y = (s +2 1ds + 1 ( x 2 = 2 +2x (x +2y +(2x + y =. (1.8 y ] x [ s 2 (2x + tdt = s + 1 (2xy + y2 2 2x 1 = x2 2 2 [ ] y 2xt + t xy + y U(x 1,y 1 =U(1, 1 =, (x 1,y 1 =(1, 1 U(x, y = y = f(x f(x 1 =f(1 = 1 = y 1 (1.79. ( ( 1.35 x2 y2 +2xy = 1 2 (xy 1 2 x, R y C R 2,. } C = {(x, y R 2 ; x2 y2 +2xy =., 1,. ϕ, ψ : R R,.,. = ϕ(xy + ψ(x. (1.81 (ϕ(xy + ψ(x =., ϕ(x, y =ϕ(xy + ψ(x, ψ(x, y = 1, (x, y R 2, ψ(x, y = 1, ϕ ψ (x, y =ϕ(x, (x, y =,. y x ( 1 ϕ ψ (x, y (x, y = 1 (ϕ(x = ϕ(x. ψ(x, y y x 1 35

36 , x., x R 2, μ : R R. ( μ(x = exp ϕ(tdt, x R. x, x R μ(x, (1.81. ( ( exp ϕ(tdt (ϕ(xy + ψ(x exp ϕ(tdt =. x x, y R, U : R 2 R. ( t y ( U(x, y = exp ϕ(sds (ϕ(ty + ψ(tdt exp ϕ(sds dt x x y x ( t ] x ( t ( = y [exp ϕ(sds + exp ϕ(sds ψ(tdt (y y exp ϕ(sds x x x x x ( ( ( t = y exp ϕ(sds 1 + exp ϕ(sds ψ(tdt x x x ( x (y y exp ϕ(sds x ( x ( t = y exp ϕ(sds + y + exp ϕ(sds ψ(tdt. x x x U(x,y =, (1.81 y = f(x U(x, y =.,. ( ( t y exp ϕ(sds = y + exp ϕ(sds ψ(tdt. x x x, (1.81 f(x =y y = f(x. ( ( ( t f(x =y exp ϕ(tdt + exp ϕ(sds exp ϕ(sds ψ(tdt x x x x ( x ( = y exp ϕ(tdt + exp ϕ(sds ψ(tdt. x x t 36

37 2.,.,,,, 1.,,, 2,,, prmeter,,. 2.1.,,.,. 2.1 m. x = x 1. R m, x R x. x m m x = x 2 j x = x2 m. (2.1 j=1,. (1 x R m, x, x =, x = =. R m. (2 x R m, R, x = x. (3 x, y R m, x + y x + y (., V : V x x R (1 (3, V. 2.2 m. (1 x = x 1., y = y 1. R m, x y x, y R. x m y m x, y = m x j y j = x 1 y x m y m. (2.2 j=1 37

38 , R m, x 2 = x, x (x R m.,. (i x R m, x, x, x, x =, x =. (ii x, y, z R m, R,. x + y, z = x, z + y, z, x, y + z = x, y + x, z, x, y = x,y = x, y. (iii x, y R m, x, y = y, x. (iv x, y R m, x, y x y (. (2 (1, R m,. 2.3 l, m, D R m. D f : D R l, L>, x, y D,. f(x f(y L x y. ( m, R >, D = {x R m ; x R}., f : D R C 1., f., x = x 1., y = y 1. D, <θ<1 θ,. x m y m f(x f(y = m j=1 f x j (y + θ(x y(x j y j. (2.4,. m f f(x f(y = (y + θ(x y(x j y j x j=1 j m ( 2 f m (y + θ(x y (x j y j x 2 j j=1 j=1 j=1 m ( 2 = f (y + θ(x y x y. x j 38

39 { } f, M mx (z x j ; z D, 1 j m M,. m f(x f(y M 2 x y = mm x y. (2.5 j=1, L = mm, f(x f(y L x y, f D.,. 2.5 l, m, D R m+1, f : D R l. f, L>, (x, y, (x, y D x R, y, y R m,. f(x, y f(x, y L y y. (2.6, D R m+1, D f, (x, y D, (x, y U D U, f U.,,. 2.6 m, R, b R m, α, δ >, D R m+1 R m+1. D = {(x, y R R m ; x δ, y b α}. (2.7, F : D R m D,, L>, x δ, y b, y b α x R, y, y R m,.,. F (x, y F (x, y L y y. (2.8 = F (x, y. (2.9 {, M mx{ F (x, y ;(x, y D} M, δ = min δ, α } M., I =[ δ,+ δ ] f( =b (2.9 y = f(x 1. 39

40 . y = f(x (2.9 f( =b I,. f(x =b + F (t, f(tdt, x I. (2.1, I (2.1 f., I + =[, + δ ] f. f : I + R m f (x =b (x I +., f 1 : I + R m. f 1 (x =b + F (t, f (tdt = b +, x I +,. x f 1 (x b = F (t, f (tdt = F (t, bdt F (t, bdt, x I +. (2.11 F (t, b dt M(x Mδ α., (x, f 1 (x D (x I +,., f 2 : I + R m. f 1 (x b M(x, x I +. (2.12 f 2 (x =b + F (t, f 1 (tdt, x I +. (2.13, x I +,. f 2 (x b = F (t, f 1 (tdt F (t, f 1 (t dt M(x Mδ <α., (x, f 2 (x D.,. f 2 (x f 1 (x = F (t, f 1 (tdt,. F (t, f (tdt F (t, f 1 (t F (t, f (t dt L f 1 (t f (t dt [ ] LM(t 2 x = = 2 LM(t dt LM(x 2, x I +. 2 f 2 (x f 1 (x LM(x 2, x I +. (

41 , I + {f n } n= n. f (x =b, f n (x =b +,. F (t, f n 1 (tdt, x I +,n 1. (2.15 f n (x b α, x I +,n, (2.16 f n (x f n 1 (x Ln 1 M(x n, x I +,n 1. (2.17 n! n =, f, (2.16., n =1, f 1, (2.16, (2.17., n>1, n 1., (2.16 t I + (t, f n 1 (t D., (2.15., x I +, f n (x b = F (t, f n 1 (tdt F (t, f n 1 (t dt M(x Mδ <α., (2.16., n 2 n 1 1,,. f n (x f n 1 = F (t, f n 1 (tdt F (t, f n 2 (tdt, (2.17. F (t, f n 1 (t F (t, f n 2 (t dt L f n 1 (t f n 2 (t dt [ ] L n 1 M(t n x = n! = Ln 1 M(x n. n! L Ln 2 M(t n 1 dt (n 1!, n, g n (x =f n (x f n 1 (x(x I +.,. g n (x Ln 1 M(x n n! Ln 1 Mδ n n!, x I +. (2.18, n, M n = Ln 1 Mδ n, M n g n n! n=1 n=1., M n = M (Lδ n = M L n! L (exp(lδ 1 < +, n=1 n=1 g n I + ( 1.2, 1.34., n=1 41

42 n, f n = f + n (f j f j 1 =f + j=1 n g j, {f n } n= I + f. f I + (2.9., f (2.1. ε>., F D,., δ 1 >, (x, y, (x, y D x x <δ 1, y y <δ 1,. F (x, y F (x, y < j=1 ε 2δ. (2.19 {f n } n= I + f, n 1, n n 1 1 n x I +,. { ε } f n (x f(x < min 2,δ 1. (2.2 (2.15, n n 1 n, x I +,. ( (b f(x x + F (t, f(tdt = (f(x f n(x + F (t, f n 1 (tdt F (t, f(tdt f(t f n (t + F (t, f n 1 (tdt F (t, f(tdt f(t f n (t + < ε 2 + F (t, f n 1 (t F (t, f(t dt ε dt ε 2δ 2 + δ ε = ε. 2δ ε>, I + x I+. f(x =b + F (t, f(tdt. I =[ δ,] (2.9 f( =b., I f( =b (2.9 1.,. 2.7 (. J R, J, ϕ, ψ, ρ : J R ϕ(x,ψ(x,ρ(x (x J,. ϕ(x ψ(x+ ϕ(tρ(tdt, x J, x. (2.21, x J, x x,. ( ϕ(x ψ(x+ ψ(tρ(t exp ρ(sds dt. ( t

43 . (2.21 ρ,. ϕ(xρ(x ψ(xρ(x +ρ(x ( 2, exp ( ( d x exp ρ(tdt ( = ϕ(xρ(x exp ( ψ(xρ(x exp ϕ(tρ(tdt ( ρ(tdt ρ(x exp ρ(tdt, x J, x. ϕ(tρ(tdt, x J, x. ρ(tdt,., x (x J, x,. ( ( exp ρ(tdt ϕ(tρ(tdt ψ(tρ(t exp,. ( ϕ(tρ(tdt exp = ρ(sds ( ψ(tρ(t exp (2.21,. ( ϕ(x ψ(x + ψ(tρ(t exp t t ( ψ(tρ(t exp ρ(sds ρ(tdt ϕ(tρ(tdt t ρ(s dt. t ρ(sds dt dt, x J, x. ρ(sds dt, x J, x y = f(x, y = g(x f(x b, g(x b α (x I f( =b, g( =b (2.9 I., f(x =g(x(x I +., x I + x,. f(x =b + g(x =b + F (t, f(tdt, F (t, f(tdt. f(x b, g(x b α (x I +,. f(x g(x = F (t, f(tdt F (t, g(tdt F (t, f(t F (t, g(t dt L f(t g(t dt, x I +. 43

44 ,, J = I +, ϕ(x = f(x g(x, ψ(x =,ρ(x =L (x I +,. ( f(x g(x + L exp Lds dt =, x I +., x I +, f(x =g(x. I f(x =g(x(x I. 2.8 I =[ δ,], {f n } n=. f (x =b, f n (x =b (1 n,. x t F (t, f n 1 (tdt, x I,n 1. f n (x b α, x I,n, f n (x f n 1 (x Ln 1 M( x n, x I,n 1. n! (2 {f n } n=1 I., f, y = f(x I f( =b ( J R, J, ϕ, ψ, ρ : J R ϕ(x,ψ(x,ρ(x (x J,. ϕ(x ψ(x+ x ϕ(tρ(tdt, x J, x., x J, x x,. ( t ϕ(x ψ(x+ ψ(tρ(t exp ρ(sds dt. x, I f( =b (2.9 y = f(x ,,.,,. 2.1 m, D R m+1, R, b R m (, b D., F : D R m R m C 1.,. = F (x, y. ( x

45 , δ >, I =[ δ,+ δ ], f( =b ( δ 1,α>, D 1 = {(x, y R R m ; x δ 1, y b α} D., F = F 1. F m., 1 j m j, F j : D 1 R C 1, L >, 1 k m k, F j (x, y y k L ((x, y D 1., 2.4, x δ 1, y b, y b α x R, y, y R m,.,. F j (x, y F j (x, y ml y y, 1 j m. F (x, y F (x, y m ml y y = ml y y., F., F C 1,., M>, (x, y D 1, F (x, y M {., δ = min δ 1, α }, 2.6, I =[ δ,+ δ ] M f( =b ( D = R 2, F : D R F (x, y =y ((x, y R 2., F C 1., D. = F (x, y =y. (2.24, =, b R, δ 1 =1, α = mx{ b, 1} >., D R 2. D 1 = {(x, y R 2 ; x 1, y b α}., M = mx{ F (x, y ;(x, y D 1 }, M = mx{2 b, b +1} > {., δ = min δ 1, α } >, I =[ δ,δ ] {f n } n= M. f (x =b, f n+1 (x =b + f n (tdt, x I,n. (

46 , x I, f 1 (x,f 2 (x,f 3 (x. f 1 (x =b + f 2 (x =b + f 3 (x =b + bdt = b + bx, (b + btdt = b + [bt + bt2 2 (b + bt + bt2 dt = b + 2 ] x = b + bx + bx2 2, ] x [bt + bt2 2 + bt3 6 = b + bx + bx2 2 + bx3 6., n,, n. n x j f n (x =b j!, x I. (2.26 j= n =,., n>, n 1., x I,. [ ] x n 1 t j n 1 x f n (x =b + f n 1 (tdt = b + b j! dt = b + t j+1 b (j + 1! j= j= n 1 = b + b j= x j+1 n (j + 1! = b + b j=1 x j n j! = b bx n, n, (2.26., n! n=. x I, n, bx n n! = b x n b δn. n! n! b δ n bx n, I., n! n! n= n= b δ n = b e δ bx n, ( 1.2, n! n! n= n= 1.2., {f n } n= I bx n f(x = = be x n! n=., y = f(x =be x I f( = b ( b. j= x j j!.,, (2.24 f( = b y = f(x,,, 1,, f( = b 1 y = f(x =be x.,,,.,,,. 46

47 =3y 2 3. (2.27 F (x, y = 3y 2 3, R, (, R 2.,, Φ(y =3y 2 3 (y R., y, dφ 2 1 (y =3 3 y 3 =2y 1 3., y, <θ<1 θ,. Φ(y Φ( y =2(θy 1 3.,. Φ(y Φ( lim y y = lim y 2(θy 1 3 =+., F (x, y =Φ(y =3y 2 3 (,,., =, y = f 1 (x =x 3, y = f 2 (x =( (2.27 f 1 ( = f 2 ( =, (,.,,.,,, m, R, b R m, α, δ >, D R m+1 R m+1. D = {(x, y R R m ; x δ, y b α}. (2.28, F : D R m D.,. = F (x, y. (2.29 {, M mx{ F (x, y ;(x, y D} M, δ = min δ, α } M., I =[ δ,+ δ ] f( =b (2.9 y = f(x ( 1.,,.,. 47

48 2.15 S R, {f n } n=1 S., S., {f n } n=1, ε>, δ>, n x <δ S x,. f n (x f n ( <ε., {f n } n=1 S, {f n} n=1 S.,,, I R, {f n } n=1 I,. (i {f n } n=1 I. (ii x I, {f n (x} n1., {f n } n=1 {f n k } k=1, I f : I R ( -.,. 48

49 2.2.,, n, R, b R n, α, δ >., D = {(x, y R R n ; x δ, y b α} R n+1, D = {(x, z R 2 ; x δ, z b α} R 2, F : D R n, G : D R R n, R,. G(x, z >, (x, z D, (2.3 F (x, y <G(x, y, (x, y D. (2.31 M >, M mx{ G(x, z ;(x, z D },, { δ = min δ, α }, I = [, + δ ] M b = f(, b = g( y = f(x, z = g(x 1., x I, f(x g(x. = F (x, y, (2.32 dz = G(x, z. (2.33, n, R, δ>, I =[, + δ], I =[, + δ, f : I R n R n C 1., I x f(x R x I,.. f(x = f (1 (x. f (n (x R n (x I., 1 j n n j f (j I C 1, f(x = (f (j (x 2 (x I., ϕ : R n R ϕ(y = y (y R n, ϕ R n \{}, 1 j n j y R n \{},. j=1 ϕ (y = 1 y j 2 (y y2 n 2 2yj = y j ϕ(y. (

50 , I ψ : I R ψ(x = f(x (x I., c I, f(c, ψ x = c,., ψ x = c. dψ (c = n j=1 f (j (c f(c (j df (c. (2.35, c I f(c =., f (j (c = (1 j n, ε>, <δ 1 + δ c δ, <h<δ 1 h 1 j n j,. f (j (c + h = df (j (ch + r(j (h, r (j (h h < ε n.,. f(c + h f(c df h (c = f(c + h df h (c f(c + h df h (c = n ( r (j 2 (h < n ε2 h n = ε. j=1, f(c =, ψ x = c, ψ x = c D + ψ(c, D + ψ(c = df (c., f(c =, D + ψ(c = lim D df +ψ(x. x c+ (c, df (c >, ε = 1 df 2 (c, <h<δ 1 h,. f(c + h > df h (c 1 df 2 (c = 1 df 2 (c >., f(c + h >, (2.35. lim h + dψ (c + h = lim h + n j=1 f (j (c + h f(c + h h = lim h + f(c + h ( df = (c 1 n j=1 = df (c n j=1 = D +ψ(c. (j df (c + h f (j (c + h h df (j (j df (c (c (j df (c + h 5

51 df (c =, <h<+ δ c h, f(c + h, (2.35. dψ (c + h = n f (j (j (c + h df (c + h f(c + h j=1 1 n n ( (f f(c + h (j (c + h 2 df (j 2 (c + h = df (c + h. j=1, f(c + h = D + ψ(c + h = df (c + h, <h<+ δ c h. D + ψ(c + h df (c + h. f C 1, j=1 df df lim (x = (c =.,. x c+ lim D +ψ(c + h ==D + ψ(c. h +, ψ D + ψ I., m. c I f(c >g(c c., m = inf{x I ; f(x >g(x}. (2.36 m =, f( = b = g( f(m = g(m. m >, x<m f(x g(x f(m g(m., m, l m x l <m+ 1 l f(x l >g(x l x l., f(m = lim f(x l lim g(x l =g(m., f(m = g(m l l., m = + δ x I f(x g(x,., m<+ δ., I ψ : I R ψ(x = f(x (x I, 2.18 ψ I =[, + δ, x = m. ψ D + ψ., D + ψ(m < dg (m. f(m, (2.35. n f (j (m df (j D + ψ(m = f(m (m df (c. j=1 51

52 f(m =, D + ψ(m = df (m.,. D + ψ(m df (m = F (m, f(m., dg (m =G(m, g(m., f(m = g(m, (2.31 F (m, f(m <G(m, f(m =G(m, g(m., D + ψ(m F (m, f(m <G(m, g(m = dg (m. m<+ δ,, <δ 1 <+ δ m δ 1, m<x<m+ δ 1,. f(x f(m D + ψ(m x m < 1 ( dg 2 (m D +ψ(m, g(x g(m x m dg (m < 1 ( dg 2 (m D +ψ(m.,. f(x f(m x m < 1 2 ( dg (m+d g(x g(m +ψ(m < x m., f(x f(m <g(x g(m, f(m = g(m, m<x< m + δ 1, f(x <g(x., m, l> 1 l δ 1, m x l <m+ 1 l f(x l >g(x l x l.., x I f(x g(x. 2.17, n, R, b R n, α, δ >., D = {(x, y R R n ; δ x, y b α} R n+1, D = {(x, z R 2 ; δ x, z b α} R 2, F : D R n, G : D R R n, R,. G(x, z >, (x, z D, (2.37 F (x, y <G(x, y, (x, y D. (2.38 M >, M mx{ G(x, z ;(x, z D },, { δ = min δ, α }, I = [ δ,] M 52

53 b = f(, b = g( y = f(x, z = g(x 1., x I, f(x g(x. = F (x, y, (2.39 dz = G(x, z. ( (1 n, R, δ>, I =[ δ, ], I =( δ, ], f : I R n R n C 1, ψ(x = f(x (x I., x I, ψ., x = c I ψ D ψ(c.,, f(c ψ x = c, f(c =, D ψ(c = df (c., ψ D ψ. (2 (1,

54 ( 54

55 2.3.,., n, D R n+1, F : D R n R n, R, b R n, α, δ >, D = {(x, y R R n ; x<+ δ, y b < α} D., I =[, + δ f( =b y = f(x. = F (x, y. (2.41, lim f(x x +δ =b, (+δ, b D, f(+δ =b, y = f(x (2.41 [, + δ].. = + δ. [,, (2.41 y = f(x. f(x =b + F (t, f(tdt, x [,. (, b D, α 1,δ 1 >, D 1 = {(x, y R R n ; x < δ 1, y b <α 1 } D., F D D 1., b = lim f(x, F (, b = lim F (x, f(x., x x [, ] x F (x, f(x, [, ] x F (t, f(tdt,. df (x =F (x, f(x, x [, ]., y = f(x (2.41 [, ]., (, b (2.41 g( =b y = g(x, n, D R n+1, F : D R n R n.,, R <, b, b R n, (, b, (, b D.,. = F (x, y. (

56 , δ>, (2.42 [,, [, + δ f( =b, g( =b y = f(x, y = g(x, lim f(x =b., x [, + δ h. h(x = { f(x, x [,, g(x, x [, + δ., y = h(x (2.42 h( =b [, + δ , y = h(x (2.42 [, ], h(x =b + F (t, h(tdt, x [, ]., h x =., g,. g(x =b + F (t, g(tdt = b + F (t, h(tdt, x [, + δ., x [, + δ,. ( h(x =g(x = b + F (t, h(tdt + = b + F (t, h(tdt. F (t, h(tdt, y = h(x (2.42 [, +δ h( =b,, x =. 2.21, 2.22,., n, D R n+1, F : D R n R n, R, b R n, α, δ >, D = {(x, y R R n ; δ <x, y b < α} D., I =( δ, ] f( =b y = f(x. = F (x, y. (2.43, lim f(x x δ =b, ( δ, b D, f( δ =b, y = f(x (2.43 [ δ, ]. 56

57 2.24 n, D R n+1, F : D R n R n.,, R >, b, b R n, (, b, (, b D.,. = F (x, y. (2.44, δ>, (2.44 (,], ( δ, ] f( =b, g( =b y = f(x, y = g(x, lim f(x =b., x ( δ, ] h. { f(x, x (,], h(x = g(x, x ( δ, ]., y = h(x (2.44 h( =b ( δ, ]. 2.21, , 2.22, 2.23, n, D R n+1, F : D R n R n., R, b R n, (, b D.,. = F (x, y. (2.45, I 1,I 2 R I 1 I 2, y = f(x, y = g(x (2.45 f( =b, g( =b I 1, I 2, x I 1, g(x =f(x., y = g(x y = f(x I 2., 1, 2 R 1 < 2, y = f(x (2.45 I 1 =( 1, 2 ]., b 2 = f( 2, ( 2, b 2 D, 2.14, δ>, [ 2, 2 + δ g( =b 2 (2.45 [ 2, 2 + δ., ( 1, 2 + δ h. { f(x, x (1, 2, h(x = g(x, x [ 2, 2 + δ., y = h(x (2.45 y = f(x I 2 =( 1, 2 + δ., (2.45 I 1 =( 1, 2 ] y = f(x I 1 I 2 I 1., I 1 [ 1, 2, [ 1, 2 ], I 1 I 2 I 1.,,. 57

58 2.27 n, D R n+1, F : D R n R n.,. = F (x, y. (2.46, (, b D, f( =b y = f(x., (, b D, I I (2.46 f( =b I (I,f. F = {(I,f; I, y = f(x (2.46 f( =b I }. (2.47, Ĩ R. Ĩ =, f : Ĩ Rn. f : I R n (I,f F I. (2.48 f(x =f(x, x I, (I,f F. (2.49, y = f(x f( =b (2.46, Ĩ (2.46 f( =b. y = Ĩ(x (2.46 Ĩ, (1 x Ĩ, f(x x I, (I,f F (I,f., y = f(x f( =b (2.46 Ĩ. (2 J R Ĩ J Ĩ J., y = f(x J.. (1 x I 1,I 2 (I 1,f 1, (I 2,f 2 F., f 1 (x =f 2 (x., δ >, (2.46 f ( =b I =[ δ,+ δ ] y = f (x 1. x I,, f 1 (x =f 2 (x =f (x., x>+ δ., [, x] I 1,I 2.,. M = sup{c ; f 1 (t =f 2 (t t [, c]}. (2.5 58

59 , M =+., f M + δ., M x, c<m c t [, c] f 1 (t =f 2 (t., f 1 (c =f 2 (c, f 1 (M =f 2 (M., b = f 1 (M = f 2 (M, M x (M,b D., δ 1, [M,M + δ 1 ] I 1,I 2 g(m =b (2.46 [M,M + δ 1 ] 1., t [M,M + δ 1 ] t, f 1 (t =f 2 (t. M., M>x, f 1 (x =f 2 (x. x< δ., x Ĩ, x I, (I,f F (I,f, I, δ>, (x δ, x + δ I., t (x δ, x + δ t f(t =f(t. y = f(x I (2.46, y = f(x (x δ, x + δ, (2.46. x Ĩ, y = f(x (2.46 Ĩ f( =b. (2, g : J R n, y = g(x J g( =b (2.46, (J, g F., J Ĩ, Ĩ J J = Ĩ., J., g D = R 2,. = y2. (2.51 D y 2 C 1, 2.1,, b R, f( =b b (2.51. y = f(x = b( x+1,, (, b (2.51 f( =b., Ĩ R. (,+ 1, b >, b Ĩ = (, +, b =, ( + 1b, +, b <. b 2.3 (1 y = f(x = (2.51 f( =b b( x+1. (2 b> lim f(x, b< lim f(x. x (+ 1 b x (+ 1 b + 59

60 ,,,,, n, I R, 1 j, k n, ϕ j,k : ϕ 1,1 (x ϕ 1,n (x ψ 1 (x I R, ψ j : I R, ϕ(x =.., ψ(x =. ϕ 1,n (x ϕ n,n (x ψ n (x.,. = ϕ(xy + ψ(x. (2.52, I b R n, (2.52 f( =b y = f(x I 1.. F (x, y =ϕ(xy + ψ(x (x I, y R n., I, b R n, b, [ δ, + δ] I δ > ε>, M mx{ ϕ j,k (x ; x δ, 1 j, k, l n} M >, y = y 1., y = y 1. R n,. y n y n F (x, y F (x, y = (ϕ(xy + ψ(x (ϕ(xy + ψ(x 2 n = ϕ(x(y y 2 n 2 = ϕ j,k (x(y k y k j=1 k=1 ( n n ( n ϕ j,k (x 2 y k y k 2 j=1 k=1 k=1 n n 2 M 2 y k y k 2 = n 2 M 2 y y 2. k=1, L = nm >, F (x, y F (x, y L y y, (2.52., y = f(x I, 1., f( = b (2.52, I. I =(, + (, =, + =+, f( =b y = f(x J =(, +., + +., + < +., + =+, + R. 6

61 , M 1,M 2 >.,. M 1 mx{ ϕ j,k (x ; x [, +], 1 j, k n}, M 2 mx{ ψ j (x ; x [, +], 1 j n}. dz = nm 1z + nm (2.53 g( = b z = g(x. g(x = b e nm1(x + e nm1(x t ( nm 2 +1dt = b e nm1(x nm (e nm1(x 1, x I =(, +. (2.54 nm 1, G(x, z =nm 1 z + nm 2 +1 (x I, z R, x [, + ] x y = y 1. y n R n,. F (x, y = ϕ(xy + ψ(x ϕ(xy + ψ(x = n n 2 n ϕ j,k (xy k + ψ j (x 2 j=1 k=1 j=1 n n n n ϕ j,k (x 2 y k 2 + ψ j (x 2 j=1 k=1 j=1 j=1 nm 1 y + nm 2 <G(x, y., 2.17, x [, +. f(x g(x g( + = b enm 1( + nm (e nm1( + 1 < +. nm 1, f [, +., g [, +,, ε>, <δ 1 < + δ 1, + δ 1 <x 1 <x 2 < +,. <g(x 2 g(x 1 = 2 x 1 G(t, g(tdt < ε. 61

62 ,. f(x 2 f(x 1 = 2 x 1 G(t, f(t dt = 2 2 x 1 x2 x 1 x 1 (nm 1 g(t+ nm 2 +1dt = 2 F (t, f(tdt F (t, f(t dt x 1 (nm 1 f(t + nm 2 +1dt 2 x 1 G(t, g(tdt < ε., b = lim f(x R n. ( x + +, b I R n, <δ < + + δ, x [ +, + + δ f 1 ( + =b (2.52 y = f 1 (x., (2.52 f( =b y = f(x (, + + δ., J =(, + f( =b (2.52., + = +., = (1, >., N 1,N 2 >. N 1 mx{ ϕ j,k (x ; x [,], 1 j, k n}, N 2 mx{ ψ j (x ; x [,], 1 j n}., H(x, z =nn 1 z + nn 2 +1 (x [,], z R,. dz = H(x, z. (2.55, h( = b y = h(x. (2 (1 h, >. 62

63 2.4.,.,, l, m, R, b R m, λ R l, α, β, δ >, D R l+m+1 R l+m+1. D = {(x, y, μ R R m R l ; x δ, y b α, μ λ β}. (2.56, F : D R m, D μ,, L>, x δ, y b, y b α, μ λ β x R, y, y R m, μ R l,.,. F (x, y, μ F (x, y, μ L y y. (2.57 = F (x, y, μ. (2.58, M mx{ F (x, y, μ ;(x, y, μ D} M>, δ > { δ = min δ, α }., μ λ β μ R l f(, μ =b M (2.58 I =[ δ,+ δ ] y = f(x, μ, f I {μ R l ; μ λ β}.. μ λ β μ R l., 2.6, I f(, μ =b (2.58 1,. f(x, μ =b + F (t, f(t, μ, μdt, x I. (2.59, x + δ, μ λ, μ λ β x R, μ, μ R l,. f(x, μ f(x, μ = F (t, f(t, μ, μdt F (t, f(t, μ, μ dt F (t, f(t, μ, μdt F (t, f(t, μ, μ dt + F (t, f(t, μ, μ dt F (t, f(t, μ, μ dt + F (t, f(t, μ, μ F (t, f(t, μ, μ dt F (t, f(t, μ, μ F (t, f(t, μ, μ dt. 63

64 , (2.57,. F (t, f(t, μ, μ F (t, f(t, μ, μ L f(t, μ f(t, μ, t + δ., F D, D., ε>, δ 1 >, (x, y, μ, (x, y, μ D, x x, y y, μ μ <δ 1,. F (x, y, μ F (x, y, μ <ε.,, μ μ <δ 1, t + δ t R,. F (t, f(t, μ, μ F (t, f(t, μ, μ <ε., x + δ x R, μ λ, μ λ β, μ μ <ε μ, μ R l,. f(x, μ f(x, μ εdt + = ε(x + L f(t, μ f(t, μ dt L f(t, μ f(t, μ dt.,,. ( f(x, μ f(x, μ ε(x + Lε(t exp = ε(x +εe Lx L(t e Lt dt t Lds dt ( [ (t = ε(x +εe ] Lx e Lt x + = ε(x +εe Lx ( (x e Lx + e Lt dt ] x [ e Lt = εelx L (e L e Lx = ε L (el(x 1 ε L (elδ 1., δ x x R,., x, x I, x x < ε M, μ λ β μ Rl,. f(x, μ f(x, μ = F (t, f(t, μ, μdt x x x x x F (t, f(t, μ, μ dt Mdt = M(x x <M 64 L ε M = ε.

65 ε M <x x f(x, μ f(x, μ < ε., x, x I, x x < ε M, μ λ, μ λ β, μ μ <δ 1,. f(x, μ f(x, μ f(x, μ f(x, μ + f(x, μ f(x, μ <ε+ ε L (elδ 1 = ε elδ + L 1. L, f I {μ R l ; μ λ β}., m, R, b ( R m, α, δ >, D R m+1 R m+1. D = {(x, y R R m ; x δ, y b ( α}. (2.6, F : D R m D,, L>, x δ, y b (, y b ( α x R, y, y R m,.,. F (x, y F (x, y L y y. (2.61 = F (x, y. (2.62 { α }, M mx{ F (x, y ;(x, y D} M>, δ = min δ, { 2M, I =[ δ,+ δ ], J = b R m ; b b ( α }., b J 2, f(, b =b (2.62 I y = f(x, b 1, f I J, f I J.. J = { z R m ; z α }, D 1 R 2m+1. 2 D 1 = {(x, z, b R R m R m ; x δ, z J, b J}. (2.63, (x, z, b D 1, (z + b b ( z + b b ( α 2 + α 2 = α, (x, z + b D., G : D 1 R m. G(x, z, b =F (x, z + b, (x, z, b D 1. (

66 , mx{ G(x, z, b = F (x, z + b ;(x, z, b D 1 } M., x δ x R, z, z J, b J,. G(x, z, b G(x, z, b = F (x, z + b F (x, z + b L (z + b (z + b = L z z., G D 1 b.,. dz = G(x, z, b. (2.65, 2.33, b J, g(, b = (2.65 I z = g(x, b 1, g I J, g I J., x I, b J, f(x, b =g(x, b+b., f I J, f(, b =g(, b+b = + b = b., y = f(x, b. = d d d f(x, b = (g(x, b+b = g(x, b = G(x, g(x, b, b =F (x, g(x, b+b =F (x, f(x, b., y = f(x, b (2.62 f(, b =b I, I J. 66

67 3.,.,,. 3.1.,.,.,, 1.,. n, K C R., K n M(n, K. 1,1 1,n M(n, K = A =.. ; j,k K, 1 j, k n. (3.1 n,1 n,n 3.1 n, I R, A : I M(n, C M(n, C.,. = A(xy. (3.2, I., b C n, f( =b (3.2 I y = f(x 1.. y = y R + iy I (y R, y I R n, A(x =A R (x+ia I (x (A R (x,a I (x M(n, R., z R 2n, B(x M(2n, R. ( y R z =, (3.3 y I ( A(x B(x = B(x B(x A(x. (3.4, B(x, I.,. = R + ii, A(xy =(A R (x+ia I (x(y R + iy I =(A R (xy R A I (xy I +i(a I (xy R + A R (xy I. 67

68 , dz = B(xz. (3.5 (, b = b R + ib I (b R, b I R n, c = b R b I, c R 2n., 2.31, ( c R 2n, g( =c ( I z = g(x 1., z = z (1 (z (1, z (2 R n g (1 (x, g(x = z (2 g (2 (x (g (1 (x,g (2 (x R,. dz (1 = AR (xz (1 A I (xz (2, dz (2 = AI (xz (1 + A R (xz (2., y = z (1 + iz (2, f(x =g (1 (x+ig (2 (x,. = dz(1 idz(2 + =(A R (xz (1 A I (xz (2 +i(a I z (1 + A R z (2 =(A R (x+ia I (x(z (1 + iz (2 =A(xy., g (1 ( =b R, g (2 ( =b I, f( =g (1 (+ig (2 ( =b R +ib I = b C n.,.,,. 3.2 n, I R, I., A : I M(n, R I M(n, R,. = A(xy. (3.6 (1 f : I R n, g : I R n, y = f(x, y = g(x (3.6., y = f(x+g(x, y = cf(x (c R (3.6., (3.6 I F R. F R = {f : I R ; y = f(x (3.6 }. (3.7, F R R. (2 I, Φ : R b f b (x F R, b R n, y = f b (x f( =b I., Φ R., dim F R = n. 68

69 . (1 x I,. d df (f + g(x = (x+dg (x =A(xf(x+A(xg(x =A(x(f + g(x, ( d df (cf(x =c (x = c(a(xf(x = A(x(cf(x., y =(f + g(x, y =(cf(x (3.6., F R R. (2 b, b R n, c R, (1, y = f b (x+f b (x, y = c(f b (x (3.6., f b (+f b ( =b + b, c(f b ( = cb., 2.31, f b (x+f b (x =f b+b (x, c(f b (x = f cb (x., Φ., =f b (x, =f b ( =b, Φ., g(x F R, b = g(, 2.31 g(x =f b (x., Φ., Φ, dim F R = dim R n = n., I R,. 3.3 n, I R, I., A : I M(n, C I M(n, C,. = A(xy. (3.8 (1 f : I C n, g : I C n, y = f(x, y = g(x (3.6., y = f(x+g(x, y = cf(x (c C (3.8., (3.8 I F. F = {f : I C ; y = f(x (3.6 }. (3.9, F C. (2 I, Φ : C b f b (x F, b C n, y = f b (x f( =b I., Φ C., dim F = n , 3.1, 3.2.,., 1,, 1. 69

70 3.4 n, I R, 1 j n j, : I C.,. j d n y + 1(x dn 1 y n + + n 1 n 1 + n(xy =. (3.1 (3.1 I y = f(x, g : I C n. g (x =f(x,g 1 (x = df (x,...,g n 1(x = dn 1 f (x. (3.11 n 1, A : I M(n, C A(x = , x I. ( n (x n 1 (x 1 (x,. dz = A(xz. (3.13, (3.1 y = f(x, (3.11 C n g (x g 1 (x g(x =, z = g(x (3.13.,. g n 1 (x g (x g 1 (x z = g(x =. (3.13, y = g (x (3.1 g n 1 (x. g (x (3.11. (3.1 y = f(x, g(x =. g n 1 (x C n, j n 2 j,. dg j (x = d ( d j f (x = dj+1 f j (x =g j+1(x. j+1 7

71 ,. dg n 1 (x = d ( d n 1 f (x = dn f n 1 (x n = 1 (x dn 1 f (x n 1(x df n 1 (x n(xf(x = n (xg (x n 1 (xg 1 (x 1 (xg n 1 (x. g (x, z = g(x =. (3.13., (3.13 g n 1 (x g (x z = g(x =. g n 1 (x j,., f(x =g (x., 1 j n 1 d j f (x =g j(x. j df, j = 1, (x =dg (x =g 1(x,., 1 <j n 1, j 1.,. d j f (x = d ( d j 1 f (x = dg j 1 j j 1 (x =g j(x., j.,. d n f (x = d ( d n 1 f n (x = dg n 1 n = n (xg (x n 1 (xg 1 (x 1 (xg n 1 (x = 1 (x dn 1 f (x n 1(x df n 1 (x n(xf(x., y = f(x (3.1.,, f(x g(x.,, n, I R, j, b = b. b n 1 : I C (1 j n 3.4., I C., (3.1 I y = f(x, j n 1 j, 1. d j f (x =b j. (3.14 j 71

72 . A(x 3.12, b, g( =b (3.13 I 1. g(x, f(x =g (x, j n 1 j, g j ( =b j, dj f ( =g j(, j 3.4, y = f(x (3.14 (3.1 1., C n, R n {b (1,..., b (k b (n }, b (k =. K n (1 k n, K = C R, b (k n 1 d j f k ( j =bk j (1 k n, j n 1 y = f k (x,.., b (k, f k (x,,, f k (x,.,. 3.6 n, I R, 1 k n k, f k : I C C n 1., f 1,...,f n,. f 1 (x f 2 (x f n (x df 1 W (x = det (x df 2 (x df n (x.... (3.15 d n 1 f 1 d (x n 1 f 2 d (x n 1 f (x n 1 n 1 n n, I R, 1 k n k, : I R.,. k n n + 1(x dn 1 y n n 1(x + n(xy =. (3.16, y = f j (x(1 j n (3.16 n., W (x f 1 (x,...,f n (x. (1 I W ( n, I, W (. (2 {f 1,...,f n } (3.16 F R, I, W (. 72

73 . (1, W (x f k.,,. f 1 (x f n (x n 1 dw (x =.. ( ( d d det j f 1 d d (x j f 1 (x j j j=.. d n 1 f 1 d (x n 1 f n (x n 1 n 1 ( d d j f k, j n 2, (x = dj+1 f k (x (1 k n. j j+1 j +1 j +2.,,. f 1 (x f n (x dw (x = det.. d n 2 f 1 d (x n 2 f n (x j 2 j 2 d n f 1 d (x n f n (x n n f 1 (x f n (x n = j (x det.. d j=1 n 2 f 1 d (x n 2 f n (x n 2 = 1(xW (x. n 2 d n j f 1 d (x n j f n (x n j n j, y = W (x I 1, W (x. ( W (x =W ( exp ( 1 (tdt, x I. (3.17, W (, I, W (. f k (x (2 1 k n k, g k (x =.., f 1,...,f n d n 1 f k (x n 1, (c 1,...,c n (,..., c 1...,c n,. c 1 f 1 (x+ + c n f n (x =., j n 1 j,. d j f 1 c 1 (x+ + c d j f n j n (x =. j 73

74 ,. c 1 g 1 (x+ + c n g n (x =., I c 1 g 1 (+ + c n g n ( ={}., g 1 (,..., g n ( R n, W ( = det(g 1 (,...,g n ( =., I W ( =., g 1 (,...,g n (., (c 1,...,c n (,..., c 1,...,c n, c 1 g 1 ( + + c n g n ( =., j n 1 j, j +1,. c 1 d j f 1 j ( + + c n d j f n j ( =., h(x =c 1 f 1 (x+ + c n f n (x., y = h(x (3.16 I,. d j h j ( =c 1 d j f 1 j ( + + c n d j f n j ( =, j n 1., 3.5, I h(x = (., j n 1 j I,. d j h ( =c d j f 1 j 1 (+ + c d j f n j n ( =. j, c 1 g 1 (+ + c n g n ( =, g 1 (,...,g n ( R n., W ( = det(g 1 (,...,g n ( =. 74