IA September 25, 2017 ( ) I = [a, b], f (x) I = (a 0 = a < a 1 < < a m = b) I ( ) (partition) S (, f (x)) = w (I k ) I k a k a k 1 S (, f (x)) = I k 2

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1 IA September 5, 7 I [, b], f x I < < < m b I prtition S, f x w I k I k k k S, f x I k I k [ k, k ] I I I m I k I j m inf f x w I k x I k k m k sup f x w I k x I k inf f x w I S, f x S, f x sup f x w I x I x I I k I k I k inf x inf f x x I k x I k inf x inf x I k x I k f x w I k + w I k w I k inf f x w I k inf f x x I k x I k w I k + inf x I k f x w I k S, f S, f S, f S, f,,, S, f S, f S n, f S n, f S, f S, f

2 lim S, f f x b I f x I lower integrl lim S, f f x f x I upper integrl b I f x f x f x f x I I f x I integrble Riemnn, integrble in the sence of Riemnn b f x f x I f x I integrl of fx on I S, f f x { x : x : w I k > { sup x Ik f x inf x Ik f x m w I k w I k b k S, f 3 I [, ] f x { x {,, 3, } f x, g x I f x + g x I f x + g x I f x + g x I f x c cf x cf x c f x I I inf f x + g x inf f x + inf g x x I k x I k x I k S, f + S, g S, f + g S, f + g S, f + S, g

3 I [, b] f x uniformly continuous ε > δ > I, f x, y I, x y < δ f x f y < ε x I, ε > δ > D R n f x f x, x,, x n D ε > δ > x, y D d x, y < δ f x f y < ε ε > δ > x, y D { f x f y ε d x, y < δ x i, y i D δ > δ > d x i, y i < δ i f x i f y i ε x i D x i x D d y i, x i y i x y f x x x x i x, y i x f x x i f x i f x < ε f y i f x < ε f x i f y i < ε I [, b] ε > δ > x y < δ x y δ f x f y ε f x f y < ε I k δ sup f x inf f x x I k x I k < ε S, f S, f sup f inf f w I k x I k x I k εw I ε f x f x < εw I k 3

4 w mx w I k k S, f S, f f Drbowx f x: I I [, b] : { < < < r b} I I k [ k, k ]: S, f x I r k I k m inf f x w I k x I k k m S, f x sup f x w I k x I k k S S f x, f x f x I I f x I f x I f x I [, b] < c < b, I I I, I [, c], I [c, b] f x I I, I f x, g x I f x + g x I I I f x f x + f x I I {f x + g x} I f x + g x I c cf x I cf x c f x I 3 f x f x I f x > x f x 4

5 x f x 4 f x f x S, f S, f sup f x inf f x w I k sup f x inf f x w I k 5 f x, g x I f x g x f, g C x, y I f x, g x C f x g x f y g y f x g x f y g x + f y g x f y g y g x f x f y + f y g x g y C f x f y + g x g y 6 f x f x, I sup f x g x inf f x g x w I k x I k x I k C sup f x inf f x w I k + C sup g x inf g x w I k fx c > fx, foudmentl theorem of clculs f x: I [, b] x b < x < b, I [, x] [x, b] f x [, x] F x x f x F x [, b] I { F x f x F G x: I [, b], b G x f x G x F x + G f x f x F x + h F x x+h x+h x f x + f t dt x f x 5

6 inf f t f t sup f t t [x,x+h] t [x,x+h] α inf t [x,x+h] f t, β sup t [x,x+h] f t α h f t h α, β f x x+h x f t dt β h α h F x + h F x β h α α f x, β f x f x F G x F x F x + h F x h F x f x β H x G x F x H c H c H y c H c H G x F x + f x: I G x f x, b G x f x b f x G b G [G x] b, integrl by substitution f x, g x: C b b f x g x [f x g x] b f x g x f x g x f x g x + f x g x b f x g x b f x g x + b b [f x g x] b f x g x b f x g x f x g x 6

7 , integrl by prts φ x: J [α, β] [, b] f x: I [, b] x φ sf x x f x b f x β α f φ s dφ ds ds G β F b b F φ s G s f x G s F φ s F φ s φ s G β β α G s ds β α f φ s dφ ds ds b b log x x log x [x log x] b [x log x] b x log x b log b log b + x x sin s ds ds ds rcsin b rcsin < c < b f x: I [, b] F x x f x, b F b F b F c c b f x f c b F x f x F x F x x f x + x f x + C 7

8 x x { +x x+α + C α log x + C α x e x ex + C x log x x log x x + C 3 cos t sin tdt cos tdt sin t x tn x x sin x cos x cos x cos x log cos x 4 x sin s x s rcsin x cos s ds cos s x tn s + x s cos s rctn x cos s ds 5 b x b s b x s x s x s b s b b x b s s b s ds b b s s s ds ± s b b s s ds s t t 8

9 x 3 p x x px x < x < x sin s ds cos s p x p sin s cos s ds x cos s p sin s ds sin s sin m s sin ns px x ± x + x sinh s {sinh x ex e x cosh x ex +e x x x cosh s d sinh s ds cosh s sinh s + cosh s d cosh s ds sinh s cosh s + sinh s p x x ± sinh s p cosh s ds e ±mx x sinh s t e s > x es e s t t t xt t x + x + s log x + x + x + x tn s ds + tn s x + + tn s 9

10 x b x b x p x x s x + ds rctn x s x s b x b p x x x s b s s b s s s ds s + mα α, x α + r + x x + b r p X, Y X, Y f x p cos x, sin x s tn x cos x X s + s sin x Y s + s ds ds + tn x + s f x p cos x, sin x s p + s, s ds + s + s s b b f x g x [f x g x] b f x g x Tylor f x [, b] C n+ f n+ x < M, x [, b] f x f + + f n n! x n + x n+ M n +!

11 x x t n f n+ t dt n! [ x t n+ f n t n! ] x x n f n + n! x + [ n x t x t n f n t dt n! n! f n t ] x x + x t n f n t dt n! f x f x x n f n n! x x t n M dt n! x n+ M n +! x t n f n+ t dt n! f x [, b] C f x [, b] N,, N h b N f x h f + f + + f N + f N T N f f x T N f M 6 h b i+ i [ ] x i x i+ f x [ ] i+ x i x i+ f x [ i {x i + x i+ } f x i+ i ] i+ i+ + i i i+ i+ i f i+ i+ i f + hf i+ + i+ hf + h f i + f i+ + i i+ i f x f x f x T N f f x x i x i+ {x i + x i+ } f x i f x f x f x M M i+ M i i+ i 3 x i x i+ M Nh h M h b M b 3 N

12 optiml estimte p x x p x log x p x log x p x x p x e x p x e x [p x e x ] p x e x e x R R {x, y x b, c y d} R f x, y: R : R {R ij R ij {x, y i x i, c j y c j }} S, f inf f x, y Are R ij x,y R ij Are R ij i i c j c j S, f S, f S, f: inf sup,, : S, f S, f S, f S n, f S n, f S, f f x, y dy lim S, f lim S, f f x, y dy R R S S f R integrble f x, y dy [] [double] integrl R f, g R α, β: αf + βg dy α R liniuty f monotonicity 3 R R k : f R k R fdy R R fdy k fdy + β gdy R R k fdy

13 didturity 4 f { sup f inf f sup f inf f S, f S, f sup f inf f Are 5 fg f x, y g x, y f, g M sup fg inf g M sup f inf f + M sup inf g D D R f x, y: R f x, y dy D f:r x, y R φ D x, y { x, y D x, y / D chrcteristic function R φ D x, y dy inf φ D x, y AreR ij x,y R ij AreR ij R ij D φ D x, y sup φ D x, y AreR IJ R AreR ij R IJ D ϕ sup φd inf φ D AreR ij AreR ij R ij D ϕ R ij D R ij D AreR ij R φ D x, y dy φ D x, y dy R D D is of denite re R φ D x, y dy D re φ D x, y dy inner re R R φ D x, y dy outer re f, g D {x, y x b, f x y g x} 3

14 f x, g x: [, b] ε > δ > x x < δ f x f x < ε δ ε b ε D: f: R f D φ D : R f: R fφ D D fdy fφ D dy R D f x, y R D R g, h x D { x b, g x y h x} f x, y dy b hx D gx f x, y dy t b D t { x t, g x y h x} F t lim h> F x f x, y dy D t D t+h fdy h D t fdy fdy D t+h fdy D t fdy D t+h D t h t x t + h f x, y f t, y < ε f x, y dy f t, y dy ε h t g t + ε h D t+h D t D t+h D t F x iterted integrl x hxfx,ydy gx dy 4

15 y r x r r x r dy r r r π r x π r π πr π cos tdt cos t + dt π π r cos t r cos t dt x y dy D x x y dy x x φ t, φ α, φ β b b f x β α f φ t φ t dt,,, N : [, b] α, α,, α N : [α, β] k k φ α k α k α k f k k k f φ α k φ α k α k α k fdt f φ t φ t dt { x φ s, t y ξ s, t φ, ξ: C s, t: R {α s α + δ, β t β + ε} 4 φ α, β, ξ α, β,φ α, β + δφ s α, β, ξ α, β + δξ s α, β,φ α, β + εφ t α, β, ξ α, β + εξ t α, β,φ α, β + δφ x φ α, β + φ s α, β s α + φ t α, β t β y ξ α, β + ξ s α, β s α + ξ t α, β t β x, y det δφs εφ t δε δξ s εξ t det φs φ t ξ s ξ t s, t, u Γ x x s, t, u y y s, t, u z z s, t, u 5

16 x, y, z D f x, y, z dydz f x s, t, u, y s, t, u, z s, t, u D Γ det x s x t x u y s y t y u z s z t z u dsdtdu { x r cos θ y r sin θ r, θ: D f x, y dy G x r cos θ x θ r sin θ y r sin θ y θ r cos θ xr x det θ y r t θ r f x s, t, y s, t det xs x t dsdt y s y t D R e x +y dy G π π π e r rdrdθ R e r rdr dθ R R e r rdr e s ds π [e s ] R π e R r s Γ r R, θ π A A e x e y dy A A D e x +y dy π e A dy D dy D dy D π e A A π e A e x π e A A 6

17 A e x lim e x π A A f x e x π t e x t f t, x π f x f, x t f t, tx e x f x π g t, x x e t πt g t, x g t, x N σ, x e x σ πσ σ t σ N µ, σ, x e x µ σ πσ µ σ F t F xx e x 4t F t, x π t t x F d dt t t t F t x + t 4t F F x x t F F xx x t + F F t t t F t, x t t + t F x t, x + x t F x t, x t F x t, x + x F x t, x t F xx t, x x t F xx t, x t F t x, y F xx x, y 7

18 rndom wlk t Brownin motion ε F t, n F t, n + F t, n F t, n F t, n + + F t, n F t, n F t, n F t, x x n F t, n + + F t, n F t, n F xx t, n F t t, x F xx t, x σ G R Γ π x e x σ πσ x y e x +y σ πσ dy r 4 cos θ sin θe r σ πσ cos θ sin θdθ rdrdθ r 4 e r πσ σ r dr σ 4 t t πt e x t N x σ, x x N σ, x σ xσ x N xx σ, x + σ 4 N σ, x N xx x ±σ [, b] f x, b, ], ε > x ε x [ x ] ε ε 8

19 , b] I [, b, b f x. I I f I. h, k > b lim f x h +h lim k lim k,h b k b k +h f x f x inproper integrl b f x b α < x α +α b [ x α [log x] b ] b x +α +α x α α k h [rcsin x] k x h π π k, h Cuchy [, b b k lim k f x ε > δ > < k < k < δ b k b k b k f x f x f x < ε b k f x [, b b f x b f x b f x b k f x b k b k f x b k b k b k f x π sin x π x sin x x 9

20 [, b f x x b f x < b x α b, α > f x b b k b k f x b k [ b k b x α b x+α + α ] b k b k k+α k +α + α [, + x f x x α f x M N [ x +α f x + α ] M N M +α + N +α + α ], π π 4 4 log sin x < β < x β log sin x < x β x > sin x x sin x x log sin x log x + log sin x x x β log x x β log sin x x x β log x f x, y: D D D D f x f x OK Are D < Are D D D D f x, y D m > n D m f x, y dy D n Are D k Are D D k f x, y dy lim f x, y D k f x, y dy D m \D n D m \D n D m f x, y dy f x, y dy f dy D n f dy

21 R e x y dy π x >, Γ x e t t x dt e t t x dt t e t, t x t e t t x < t Γ x Gmm function x, y > B x, y Bet function t x t y dt. Γ, Γ π. Γ x > 3. Γ x + xγ x, Γ n + n!n 4. Γ x e r r x dr. : 3:. integrnd 4: t r Γ x + Γ e t dt [ e t] e t t x dt [ e t t x] + x e t t x dt xγ x Γ x e t t x dt e r r x rdr e r r x dr. : Γ π e r dr e r dr. B x, y π sinx θ cos y θdθ. B x, y ΓxΓy Γx+y

22 : B x, y t x t y dt t sin θ t cos θ B x, y π π sin x θ cos y θ sin θ cos θdθ sin x θ cos y θdθ : D { s r cos θ t r sin θ Γ x Γ y 4 4 D D e s s x ds e t t y dt e s +t s x t y dsdt { < r < < θ < π Γ x Γ y e r r x+y rdr r Γ x + y B x, y π cos x θ sin y θdθ B x, y Γ x Γ y Γ x + y. π Γ n n! Γ n + n n 3 Γ sin x θ cos y θdθ Γ x Γ y B x, y Γ x + y π sin n θdθ n + B, π n sin n θdθ { m!! m!! m!! m+!! π n m πγ n+ Γ n + m!! m m m 4, m +!! m + m n Br n { x,, x n x + x + + x n r } r n n-dimentionl bll of rdius r : r x r r : r πr

23 B n r V n,r V n,r r n V n, r n V n Br n { r x n r, x + + x n r x } n B n r {x n } B n r r B n V n Vn V n n y Vn dy y x n sin θ, dy cos θdθ V n V n π cos n θ cos θdθ n + V n B, n 3 n + V n V n B, n V n B, n + B, Γ n Γ V n Γ Γ n+ Γ n+ Γ n + π n V n Wllis' formul V 3 π 3 V 4 3 π, V 5 π 5 4π 3 8π 5, V 4 π 4 V π, V 6 π 6 V 4 π3 6, π lim n 4 k k π log log k k n k π log log k k 3

24 kn+ k N B x +, y Γ x + Γ y Γ x + y + x B x, y x + y xγ x Γ y x + y Γ x + y n B n +, n n + B n,. n n + n! n B, Γ Γ Γ 3 B n +, n n n n! n n! B, Γ Γ Γ π B n +, B n +, n + n 3 π n! n + n n + n n n n k π k k + k k 3 π lim Bn+, Bn+, t t x x B x +, y 3 Stirling x x+y B x, y B x, y t x t y dt B x, y B x +, y B x +, y x B x, y x + y B n, B n +, B n, B n +, n + 3 n + 4

25 n! πn n+ e n πe n+ log n n Wllis' lim n π lim n n! n n e n n π n! n + n + π lim n lim n + n n! n + lim n n! n! n! n + n + n + log x log x log + log + + log log + n n n n log n n log log + log n n log n log + n lim n + n + n n + n n e n n n + n + n en n n n n!e n n n n n! Stirling n! n! Fourier V V V f, g α, β αf + βg V f f f O αβ f α βf γ αf + βg γαf + γβg 5

26 αf + βf α + β f I R I αf x α f x f + g x f x + g x I C I I C I C I f, g g, f f, f, f, f g 3 αf + βf, g α f, g + β f, g V f, g f, g R αf + βf, g α + βf g α f g + β f g α f, g + β f, g I [ π, π] cos nx, sin nx m + n π cos m + n x π m n π cos m n x π cos mx, cos nx π π π π cos mx cos nx cos m + n x + cos m n x π m n π m n > m n m, n > cos mx, cos nx π π cos mx cos nx m, n > sin mx, sin nx sin m x sin n x π cos m n x cos m + n x π { π m n > m x, π 6

27 , π π cos nx, π π { cos nx cos mx n m, π π n m { sin mx sin nx m n, π π m n sin mx, π π cos mx sin nx, π π, cos x cos x cos mx,,, π π π π sin x sin x,, π π onyhonorml vector, k, b k [ π, π] N N f x + k cos kx + b k sin kx k k f x N b b N f x sin kx π b k f x f x cos kx k π V V v, v, v 3, V α R, v α N. α v α v α v α i v i v k k N i N α i v i v k k N V v,, v N V N α i v i i i α v 7

28 α v, v α v, α v, α, v + α v, v, α, α, v v α v, v, v α v, v v,, v N V N α i v i i α k, v k k v k, v k k { cos mx, π π } sin nx π Fourier { π R } { [ π, π] f f π f π } π f x π + m π m π π π f x m cos mx π + b m m b m sin mx π f x f, π π cos mx cos mx f, π π sin mx f, π f N f f, π π N m f, cos mx cos mx π π f N, f N N m f, sin mx sin mx π π f, f + m + m m b m 8

29 x π,, π x π f t, x f t,, f t, π t, g x f t, x, g t x g t x f τ f x f, x h x f t, x g t x t + cos mx m t + sin mx b m t π π π f τ f x Left-side t + cos mx m t + m sin mx π π m t π Right-side m m t cos mx π α constnt m m m m α m e m t b m m b m b m β m e m t m b m t sin mx π t h x α + α m cos mx + β m sin mx π π π α k cos mx h x, π [ π, π] Fourier Fourier series f t c f x f t, f t, π f t, x t + cos mx m t + b m t π π f, x h x t m c t m α m cos cmt + β m sin cmt Both-edge α m m β m sin cmt + m cos mx + b m sin mx π π π sin mx x π 9

30 I [, ] V n { n } f, g f g V n d k P k x x k k! k k { p k, p l n+ k l k l p k x k Legendre Legendre porinomil of degree k l m d l p l, p m x l+m l!m! l l d m x m m [ d l constnt x l l d m x m ] m [ d l+ constnt x l l d m x m ] + constnt. ±constnt [ x ] ± constnt { l m ±constnt ml x m m m x t x t, x t l m constnt d l+ l+ x l d l+ l+ x l d l+m l+m x l x m x x + x 4t t d m m x d m m x x m 4 m t m t m 4 m B m +, m + m Γ m + Γ m + 4 Γ m + 4 m m! m +! { p l, p l 4 m m! m! m m! m+! m+ k+ p k k,,, n p p d x x p 8 d x 4 x + 3 x p 3 d 3 x x 4 + 3x 5 x3 3 x 3

31 weight functionφ x > constnt Chebyshev {f n x} n,, : f n x f x pointwise convergent x f n x f x x n n N ε, x f n x f x < ε f n x f x imiformly convergent n n N ε x f n x f x < ε f n x f n x f x f x y x y x < δ x f y f x < ε n f n x f x < ε 3 x, y f n x y x < δ δ x f n y f y < ε 3 f n x f n y < ε 3 f y f x f y f n y + f n y f n x + f n x f x < ε I [, ], f n x n x f n, x > f n x f x lim n f n x { x x > I [, b]: f n x: I f x lim n F n x F x x x lim f n x f x f n lim f n D n F x F n x x f x n f f n < ε D n x x ε ε x ε b x f n f f n 3

32 f n x: I [, b] C c I f n c f n x f n x f x f C f n x f x f n x g n x g x G n x x g n x G x x g x G n x f n x G n c f n c f n x G n x + f n c G n c G x + e G c f x G x G c + e f n x {,, } n n x c n k f n c + r r n n r n < s < r s x [c s, c + s] f n x k r R r n n! e x x n f n x n k n! x k k! x s < r e x n r n n r n n n constnt r n A r n x c s < r n x c nn+ nn+ n s n A A nn+ n x c n n s r N+ s r s r x 3

33 f x C f n c n f x Tylor n x c n n f n x n k x c k f x f n x r f r c n nr n f n x x c < s < r f x k F x x c f x n x c n f x n n x cn+ n + x n x n log x x x x n+ n + n + x x + x 4 x 6 + rctn x x + x x x3 3 + x5 5 x7 7 + rcsin x x x f y y y f n y 3 n y n n n 3 y n y n n!! y n n! n!! n!! x n!! x n rcsin x x n x n!! n!! x n+ n + 33

34 rcsin x x x rcsin x x n!! n!! n + x x n+ x sin θ x θ π π x x n+ θ x sin n+ θdθ sin n+ θdθ B, n + Γ n! Γ n + 3 Γ n! n + n n!! n +!! π 8 rcsin n n + Γ n n n s s 4 + π 8 s n s π 8 n n n π 6 n n s S s Riemnn zet Euler S n π 6 I, J: R t, x I J, f t, x I J F t f t, x I I J {t, x t I, x J} I J R : J 34

35 f t, x I J ε > δ > d t, x, t, x < δ f t, x f t, x < ε F t F t < J J f t, x f t, x ε ε J f t, x I J C f t g t J f t C G t g t I [, b], J [c, d] G t f J G t G t g t H t t t d c d c G t g t dt d c f t, x t f t, x dt t t f t, x f t, x dt t dt C curve surfce length re R 3 prmeter representtion {x t, y t, z t t [α, β]} x, y, z t motion trjectony x t, y t, z t x t t + b y t ct + d z t et + f x t r cos t y t r sin t z t 35

36 3 cycloid 4 helicoid x t r t + cos t y t r sin t z t x t r cos t y t r sin t z t t x, y, z t C t t x t y t z t x t x t + x t t y t y t + y t t z t z t + z t t vritim x t y t z t x t + y t + z t t t t t t t x + y + z dt t C: {x t, y t, z t t [α, β]}, x, y, z C C β α x + y + z dt cycloid α t β, r x sin t y cos t x + y sin t t t π s x + y + cos s cos s 4 β+ π α+ π [ sin s cos s ds ] β+ π α+ π 36

37 helicoid α t β, r x sin t y cos t z x + y + z + β α + velocity v t x + y + z v t speed v t v t dt s f t t t C t s t g s g f g s C g s dt ds ds dt f s d dt x t s ds dt ds d d d x t s + y t s + z t s + ds ds ds dt dy + dt dz dt dt ds + ds dy + ds dz ds ds dy dz dt + + dt dt dt ds ds dy dz + + dt dt dt dt rc length prmeter t x t x t y t z t ds dt s t t + dt x + y + z dt dy + dt d dt x t x ds dt ds ds dt ds dt dt ds dt dt dt dz dt dt s: ds e s 37

38 de ds s f s e s.f s e s s de ds f f f f 3, g g g g 3 e, de ds d ds f, g df d ds d f, g ds e s e s, e s e s, e s df ds g + f dg ds f, g f g + f g + f 3 g 3 ds g dg df + f ds + ds, g + e, de e, de ds ds de ds, e + f, dg ds e, e d e, e ds x s f t curvture s s f s, β f s x s C 3 x s s + y s β s + 3 z s 3 β x, y θ β sin θ, β cos θ β sin θ s θ β rcsin s βs + cos θ cos βs βs κ de ds f s 38

39 de ds curvture rdius de ds κe s κ, e s e e e 3 e, e de s ds e τ torsion γ γ de ds γe + τe 3 de ds, e d ds e, e κ e e s de ds κe de ds κe + τe 3 e, de ds e 3 δ, ε κ, τ δ de 3 ds δe + εe de3 ds, e d ds e 3, e e 3, de ds ε de3 ds, e e 3, de τ ds κ s, τ s Frenet-Serret- κ s, τ s R 3 ++ x t e t e t e s e s, e s : x + by + cz + d {x, y, z x + by + cz + d }, b, c,, 39

40 x + y b + z c r α x + β y b + γ z c r α, β, γ, r z f x, y s, t x x s, t y y s, t z z s, t x r cos θ cos φ + y r cos θ sin φ + b z r sin θ + c π θ π, π φ π x, y, z s, t C s, t x x, y y, z z, x s, x t x y s, t z x s, t x + x s y s, t y + y s z s, t z + z s x, s + t, t y, s + t, t, s + z t, t x x y z, x x y z, x s x s y s z s, x t x t y t z t x x x x s s + x t t x tngent plne t x x tngent vector x s x t x s, t x s, x t s, t, s + δs, t, s + δs, t + δt, s, t + δt x s, t, x s + δs, t, x s + δs, t + δt, x s, t + δt x s + δs, t x s, t + x s s, t δs x s, t + δt x s, t + x t s, t δt 4

41 exterior product x s s, t δs x t s, t δt sin θ α β γ δ ε ζ x s s, t x t s, t sin θ δsδt βζ γε γδ αζ αε βδ α β γ, δ ε ζ θ α β δ ε γ ζ sin θ α β γ δ ε ζ α β γ α, δ ε ζ δ, α, δ α δ cos θ α δ sin θ α δ cos θ α, α δ, δ α, δ α + β + γ δ + ε + ζ αδ + βε + γζ βζ γε + γδ αζ + αε βδ α δ x s x t dsdt r, x x s x t cos s cos t cos s sin t sin s sin s cos t sin s sin t cos s cos s sin t cos s cos t x s, x t x x s x t cos s cos t cos s sin t sin s cos s cos t sin s cos s sin t cos s cos s cos t cos s sin t sin s 4

42 cos s π s π, π t π π π cos sds π [sin s] π π π 4π x + y b + z c x cos s cos t y b cos s sin t z c sin s cf. b > { x cos t y b sin t { ẋ sin t ẏ b cos t ẋ + ẏ sin t + b cos t + b cos t s tn t, ds dt cos t + s dt t + b cos tdt α + b + s ds + s rotting surfce f s x f s cos t y f s sin t z s π b + f s ds x t dt x s x t dsdt 4

43 s, k m, k n x y z cos kπ m sin kπ m k n m n x y z cos k+ m π sin k+ m π k+ n n m, m + n m t: x t x m t: x k, t, x k, x k ordinry dierentil eqution prtil dierentil eqution. Newton x t f x, x, t. f t c f x + f y + f z t, x, y, z liner ordinry dierntil eqution x x t x t. x n t x t A t x t x t x t principle of superposition x t, x t x + bx R x + f t x + g t x y x x y { y + f t y + g t x x y { x y y f t y g t x g t f t x y 43

44 { x A t x x x t exp B t B t A t B A t ij t t B t ij t dt exp B k B k k! x y x λ x { x y y λ x x y B B λ t λ t λ t λ t λ t λ λ exp B B k X t X t k k! B k k! + k B k B k +! cos λt sin λt λ λ cos λt exp B λ sin λt λ cos t sin λt cos λt λ sin λt λ cos t sin λt X t X cos λt + X λ x x t exp B t x x x x + tx X X sin λt 44

45 exp B t B k k! exp B exp B exp B t B B k k! A exp B exp B x y x exp B y y exp B x exp B x + exp B x A exp B x + A exp B x y y t y B x x x t exp B t y exp B t complete dierentil eqution F x, t: C F x x, t x + F t x, t F x t, t C x t d dt F x t, t F x x, t x + F t x, t eqution with seprble vribles x q t p x p dp x, P p x q dq t dt, Q q x dt p x x q t d P Q dt P Q C x + x t x + x t rctn x t C rctn x C + t x tn t + C 45

46 mx + f x x mx x + f x x mx + F x F x x f x m: mx + F x E kinetic energy potentil conservtion lw for energy x E F x x E F x m E F x m m t C y ex+ x y e x, x y e x x x x + e x e + x dt e + x e + t + C x ex+ x y dy y e e > log e y dy y 4 3y 3 y 3 t + C x t c 3 x t c

47 θ θ t sin θ θ E + cos θ θ E + cos θ dθ E + cos θ t + C E > θ θ θ t E < θ t E θ t θ x x 4x x x x x x t + C generl solution x t + c c x t, t specil solution x singulr solution f : t f x f t f x *' dt * dt A t x, A t B t x t exp B t x A t x f t * V x *' x t y t x t * x t + V x y t { dy dt A t y f t A t x f t * dt d x y dt A t x y t * x t exp B t t x A t exp B t t + exp B t t A t x t + exp B t 47

48 x A t x t f t exp B f t exp B f t t t exp B f t dt t x t exp B t exp B f t dt f x,, x n x k y x + ct, z x ct f t, x f t y, z, x y, z g y, z t c x f t k f x f t c f x t + c x x x,, x k, x k+,, x n { x x y, z y+z z t t y, z y z c t y x + ct c t z t c x y x + ct x z x t c y z x y + z x c c t z x + c c t y f 48

49 g y z g y z g z z k z K z g K z z g K z g H g g y, z H y + K z f t, x H x + ct + K x ct + f t, ±π cos nct cos nx cos nct sin nx etc f t, x n t cos nx + b n t sin nx c f t n cos n x ± ct, sin n x ± ct n n t cos nx + b n t sin nx c f x c n n t cos nx n b n sin nx n n c n n α cos nct + β sin nct f t c x + y f r, θ t f T t R r A θ T RA c T R A + r T R + r T RA x + y r + r x + r θ T RA f T T t c R R + r R R + A r A t r, θ c k T T c k T c kt 3 R R + R r R + A r A k 49

50 3 T t k T e ± c t k k > c T cos k t sin c k t k < k < k < sin θ, cos θ A θ cos nθ 3 r αr s s A θ A θ + π R R + R r R n r k < 4 r α α k s R k R R k k n r R r k R + r R + n R 5 Bessel R s n ns n 5 s n + n + n+ + n + n+ + n n n+ n+ s n + n n n 4 n + R r A θ g x, y x, y C R r cos nθ x, y C k k n, k n k n R s s n + sn+ 4 n + + s n+4 6 n + n s n+k 4 k n + n + k + J n s n Bessel Eulr q, p n L q, p q p x t n t < t E x t t L x t, ẋ t dt x E x x t L q, p p + p L x, ẋ dt x t x t x L qi x t, ẋ t d dt L p i x t, ẋ t Eulr x t t t, t 5

51 d dz ε L x + ε t, ẋ + εȧ t dt L qi x, ẋ i t + L pi x, ẋ ȧi dt ε d dz ε L q x, ẋ ȧdt [L p x, ẋ t] L x + ε t, ẋ + εȧ t dt d dt L p x, ẋ dt L qi x, ẋ d dt L p i x, ẋ i dt t i L q d dt L p i Eulr 5

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f 22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )