Fubini

Transcription

1 Fubini Γ B e x sin x x Clothoid Lgrnge (grd (div (rot

2 .. Green (Frenet Serret D dω C ω (div (rot grd, div, rot

3 3. g(y b f(x, y dx d g(y dy c d c g(y dy ( d b f(x, y dx dy c, b y D {(x, y: x, y, x + y } f(x, y dxdy D f(x, y [, b] [c, d] b f(x, y dx. ε > δ > d((x, y, (x, y < δ f(x, y f(x, y < ε y y < δ g(y g(y b f(x, y f(x, y dx < ε(b x x ( x dy dx x dx x π π/ π/ cos θ dθ (x sin θ

4 4 π/ π/ cos θ dθ π/ π/ π + [ sin θ 4 π + cos θ dθ ] π/ π/ x x ( ( x ( x y x x π 4 3 π x y dz ( π/ ( x cos θ dθ π/ ( x dx π dy ] [x x3 3 dx dx (y x sin θ x x y dy dx 3 4 ( ( x ( x y x y z dw dz dy dx π π 4π 3 4π 3 x x y ( ( x x y x ( x x y z x y z dz x y x ( x y dy dx (( x 3/ 3 ( x 3/ dx π/ π/ ( x 3/ dx ( + cos θ cos 4 θ cos 4 θ dθ 4π 3 3π 8 π + cos θ + ( + cos 4θ/ 4 dy dx cos θ + cos 4θ 8

5 .. 5. D [, b] [c, d] f(x, y D < < < n b, c c < c < < c m d S S n m i j n m i j sup i x i c j y c j f(x, y( i i (b j b j inf i x i c j y c j f(x, y( i i (b j b j f(x, y dxdy D D D f(x, y dxdy inf S f(x, y dxdy sup S 4 (. [, ] [, ] z x z y [, ] [, ] n n [ k n, k+ n [ l n, l+ n ] min{ k+ n, l+ n } min{ k n, l n } x [ k n, k+ n,s ( { k } l n n n + k k(k + k(n k n n 3 + n 3 k n k n 3 + k n 3 l lk+ k S n(n + n(n + (n + n(n + n n 3 + 4n 3 n

6 6. S ( n (n + (n 4(n (b n ((n + (n 4(3n 3 (c k n ((n k + + (n k 4(n k + n (n k + k n 3 k S 8 n n(n n + n n (n k + k n 3 k 3. 8 n n(n 4 3 6n + n 8 n 3 n(n + (n n 3 n(n (n 6 + n 3 n(n + + n 3 n(n x x x x 4. (x x dx 4 3

7 .3. Fubini 7 x x x x dx Fubini (Fubini D [, b] [c, d] f(x, y dxdy D ( b d ( d b f(x, y dxdy f(x, y dy dx f(x, y dx dy D c c. : < < n b : c c < < c m d D x y n m inf ( i i (c j c j f(x, y i x i i j c j y c j n m ( inf inf f(x, y (c j c j ( i i i x i c j y c j i j ( m n ( inf inf f(x, y ( i i (c j c j i x i c j y c j j i

8 8 f(x, y n m sup ( i i (b j b j i j i x i c j y c j ( n m ( sup sup f(x, y (c j c j ( i i i i x c j i i x i ( m n ( sup sup f(x, y ( i i (c j c j c j y c j j c j y i i f(x, y dxdy D b ( d c f(x, y dy dx c c ( b f(x, y dx dy, ( b d ( b d f(x, y dxdy f(x, y] dy dx f(x, y] dy dx c D ( b d f(x, y dy dx b c ( d f(x, y dy dx ( b d f(x, y dy c c d c c b b f(x, y dy d c ( d f(x, y dy dx c ( d f(x, y dy dx c f(x, y dy dx d c b b b ( d f(x, y dy c f(x, y dy ( d f(x, y dy dx c ( d f(x, y dy dx c d c f(x, y dy f(x, y y f(x, y dxdy D dx 5 (, ( x y (x + y dx dy

9 .3. Fubini 9 ( x y (x + y dy dx ( x y (x + y dx dy [rctn x] π 4 [ x x + y ] dy ( x y (x + y dy dx [ y x + y ] dx [rctn y] π 4 ( f [, b] [c, d] C. d dy b f(x, y dx b f (x, y dx G(y g(y b b f(x, y dx f f(x, y dx Fubini d c g(y dy b d b c f (x, y dydx (f(x, d f(x, c dx G(d G(c d c G (c g(c π e x dx

10 :. f(t g(t ( t e x dx e (+x t dx + x f (t g (t t t u tx g (t e x dx e t ( t( + x e (+x t dx + x t te (+x t dx e t u du f (t f(t + g(t dx f( + g( + + x π 4 g(t e (+x t + x e t e t dx e t (t lim t f(t π 4 6 (log x n x y dx ( n n! (y + n+ (y > f(x, y x y y + x y dx

11 .4. y (y + (log xx y dx.4 A R (x, y A A (x, y otherwise A R 3 7 ( n k (k n π k n π π n k k n π n(n + n n 8 ( n n 3 ( 4n+4(n 8n 4 4(n + 4(n 8n k k 8n 8k + 4 n k n(n + n(n + (n + k(8n 8k + 4 8n 8 + n(n + 6 n 3 n

12 .5 A f(x, y A D [, b] [c, d] f(x, y (x, y A g(x, y otherwise f(x, y dxdy A A D A φ(x ψ(x A {(x, y: x b, φ(x y ψ(x} D g Fubini f(x, y dxdy A D b g(x, y dxdy b ( ψ(x f(x, y dy dx φ(x ( d g(x, y dy dx c 9 ( log x (x + e y dy dx D {(x, y: y log x, x } {(x, y: e y x, y log } Fubini log ( (x + e y dx dy e y log [ ] + e y x x dy log log e y + e y ( ey + ey dy e y + e y dy 4 log e y + e y dy + t dt + t dt 4 [ t + t + log( ] + t + t [( + t 3/] 4 6 ( 5 + log( 5 + log( + 6 (53/ 3/

13 .6. 3 f(x, y dxdy D. f(x, y x + y, D {(x, y: y, x, y x}. f(x, y x + y, D {(x, y: x y x} 3. f(x, y x, D {(x, y: x + y } 4. f(x, y e y/x, D {(x, y: y, x, y x} 5. f(x, y y e x, D {(x, y: y x } 3. ( x 3 f(x, y dy dx x. ( x x f(x, y dy dx 3. ( y f(x, y dx dy y.6 A A A n A A f(x, y A n f(x, y dxdy ( f + (x, y mx{f(x, y, }, f (x, y mx{ f(x, y, } f(x, y dxdy A A f + (x, y dxdy f (x, y dxdy A D {(x, y: x + y } x y dxdy D

14 4 D δ > D δ D δ {(x, y: x + y ( δ } x y dxdy δ π rcsin( δ ( π δ ( π cos π π r dθ dr r sin t dt π( cos rcsin( δ δ δ D f {K n } K n f(x, y dxdy I ε >, n s.t. n n f dxdy I < ε K n M sup (x,y Kn f(x {L n } n s.t. m n K n \L m < ε f dxdy L m f dxdy L m K n f dxdy Mε I (M + ε K n lim f dxdy f dxdy Mε I (M + ε m L m K n lim f dxdy I m L m K L

15 .7. 5 I N n s.t. n n f dxdy > N K n lim f dxdy > N m L m f f + f D f +(x, y dxdy D f (x, y dxdy f(x, y dxdy D 4. D. D 3. D dxdydz x y z D {(x, y, z: x, y, z >, x + y + z } dxdy (x+y r D {(x, y: x, y } dxdy D {(x, y: < x y } x +y 4. R 3 dxdydz (x +y +z +.7 t x x F (t F {t i } [, b] {x i } F [F (, F (b] t i+ t i x i+ x i F (t i F (b F ( f(x dx f(x i F (t i f(x i F (t i f(f (t i F (t i b f(f (tf (t dt F (b f(x dx b F ( f(f (tf (t dt F < [, b] x F (t i F ( f(x dx b F (b F (b f(x dx b F ( f(t F (t dt f(f (tf (t dt

16 6 f x F (b F ( b t f F.: te t dt f(x e x x F (t t te t dt 4 e t t dt e x dx ( e 4 f [F (, F (b] F C f F f F ( n S (f sup f(x (F ( i F ( i i F ( i x F ( i ( n F ( i F ( i sup f(f (t ( i i i i t i i i ( n sup f(f (t F (c i ( i i ( i c i i i i t i n f(f (c i F (c i ( i i i b f(f (t F (t dt

17 .7. 7 F (b f(x dx b F ( f(f (t F (t dt, F (b f(x dx b F ( f(f (t F (t dt F (b F ( f(x dx F (b F ( b f(x dx f(f (t F (t dt b F (b F ( f(f (t F (t dt ( φ(u, v (x, y F (u, v ψ(u, v f(x dx xy D uv E (u, v (u, v (u + u, v + v F (u +, v (u, v ( ( φ(u + u, v φ(u, v ψ(u + u, v ψ(u, v ( φ u ψ v (u, v u (u, v 4 J(u, v ( φ u ψ u φ v ψ v det J(u, v u v det J(u, v f(x, y dxdy f(f (u, v det J(u, v dudv F (D f(x + y dxdy x + y u y v ( u f(x + y dxdy f(u dv du + D uf(u du + ( ( uf(u du f(u dv u du

18 8 3 x sin θ π/ π/ x dx cos + cos θ θ dθ dθ π π/ π/ D dxdy D 3 ( π r dθ dr π [ r ] π x r sin θ cos φ, y r sin θ sin φ, z r cos θ det J(u, v r sin θ 4 3 D x r cos θ, y r sin θ, r D x y dxdy 3 D ( π/ dxdydz D ( π π Γ(Γ(3/ Γ(5/ sin m θ cos n θ dθ B(m + ( π r r dθ dr 4π π/ / π π 3/ / π 4π 3, n + r sin θ dφ dθ dr 4π [ r 3 3 ] 4π 3 cos θ sin θ dθ πb(, 3

19 D x y z dxdydz 4 dr r sin θ 8π π/ 8π π π/ π/ π π 4 π 8π 8π sin θ cos θ dθ π/ dθ π dφ r r sin θ r r dr [ cos θ] π/ r r dr π/ sin cos 4θ θ dθ π dθ sin θ cos θ dθ 4πB( 3, 3 Γ(3/ 4π 4π ( π/ π Γ(3 5 (x + y x y x y r cos θ 4 D {(r, θ: π 4 θ π 4, r cos θ} D dxdy π/4 π/4 π/4 ( cos θ r dr dθ π/4 cos θ dθ 5. (x + y x 4 (x + y ( >. (steroid x /3 +y /3 ( x r cos 3 θ y r sin 3 θ 6 3 (x, y, z (r cos θ, r sin θ, z (x, y, z (r cos φ sin θ, r sin φ sin θ, r cos θ

20 6. D {(x, y, z: x + y, z xy } z dxdydz D. D {(x, y, z: x + y + z, x, y, z } xy dxdydz D.. z dxdydz D ( ( π r sin θ r z dz dθ dr ( π r 5 sin (θ dθ dr π 6 xy dxdydz D ( ( π π/ r sin θ cos φ r sin θ sin φ r sin θ dφ dθ dr r 4 dr π sin 3 θ dθ π/ sin φ cos φ dφ Γ Γ(x t x e t dt (x > t t x e t/ t t x e t e t/ for suffciently lrge t < x < t t x e t t x t x dt. Γ( [ ] x tx x

21 .9. B. x > Γ(x (x Γ(x Γ(x t x e t dt [ t x ( e t ] (x t x ( e t dt (x Γ(x 7. Γ(n (n!. ( Γ Γ t / e t dt e x dt ( π e x dx π (t x, dt xdx t / dx.9 B B(p, q x p ( x q dx (p, q > < p < x < q < < x < x p ( x q x p q x ( p q < q < x p dx p [xp ] p < q < x

22 .9. e x 3 π e x dx. e (x +y ( N e (x +y dxdy e dx x x +y N x +y N x r cos θ, y r sin θ ( ( x x r θ cos θ r sin θ sin θ r cos θ r θ r π/ N e (x +y rdr r dxdy dθ e x +y N π N e t dt π 4 ( e N e (x +y dxdy (t r, dt rdr e (x +y dxdy π 4 : ( e x(+y dy dx I e x ( ( x e x x dx I e xy dy dx e t dt dx ( y t e s ds I ( x s ( e x(+y dx dy + y dy [rctn y] π

23 .9. B 3 3 ( f [, b] [c, d] C. d dy b f(x, y dx b f (x, y dx Fubini G(y g(y b b f(x, y dx f f(x, y dx d c g(y dy b d b c f (x, y dydx (f(x, d f(x, c dx G(d G(c d c G (c g(c 4 : π e x dx. f(t g(t f (t g (t t t u tx g (t ( t e x dx e (+x t e x dx e t dx + x ( t( + x e (+x t dx + x t te (+x t dx e t u du f (t

24 4 f(t + g(t dx f( + g( + + x π 4 g(t e (+x t + x e t e t dx e t (t lim t f(t π 4 5 B(p, q Γ(pΓ(q Γ(p + q. N N Γ(pΓ(q lim e x y x p y q dxdy N c c c lim e x y x p y q dxdy D D (u x + y, v x x + y ( M lim M M lim M lim b b Γ(p + q B(p, q e u u p+q du lim b e u (uv p (u uv q u dv b x uv y u uv ( ( x u u v v u v u u v v v p ( v q dv u u du 6 (. B, π

25 .9. B 5 (. Γ π. ( ( π B, Γ ( Γ( ( ( Γ. ( B, dx x( x x x t x +t dx t (+t dt ( B, + t + t dt + t [ rctn t ] t π t ( + t dt. ( B, dx /4 (/ x dt /4 t /4 dt t [ rcsin t ] π ( π π (t x [ ] rccos t ( π π 7 π/ sin m x cos n x dx B ( m +, n +

26 6. π/ sin m x cos n x dx t m/ ( t n/ dt t( t B ( m + t (m / ( t (n / dt, n + (sin x t 8 n. I n (x x x dx x x x x I n ( n I (x x x x n dx n x x n x dx x I n (x c n ( x n/ c x I n (x I n (x + x dx x x c n ( x x (n / dx x π/ c n ( x (n / ( sin θ (n / x cos θ dθ (x x sin θ π/ π/ c n ( x n/ cos n θ dθ π/ ( c n ( x n/ B, n +

27 .9. B 7 ( c n c n B, n + ( c n B, n ( B, n + ( c B, 3 ( B, 4 ( B, n + Γ(/Γ(3/ Γ(/Γ(4/ Γ(/Γ((n + / Γ(4/ Γ(5/ Γ((n + / π (n / Γ(3/ Γ((n + / π n/ Γ((n + /.9. ( e xy sin x dx dy sin(x dx π sin x π dx x ( I e xy sin x dx e xy sin x dy dx (. I e xy cos x y cos x e xy dx e xy cos x y e xy cos x dx ( e xy cos x y e xy sin x + y e xy sin x dx e xy cos x y e xy sin x y 4 I I e xy cos x + y sin x + y 4

28 8 ( e xy sin x dx dy + y 4 dy [ e cos x + ] xy y sin x + y 4 dy 4 π (. t xy e xy sin x dy sin x e xy dy sin x e t dt x sin x x π ( π sin x e xy sin x dy dx dx (.3 x (. (.3 (. π sin x x dx 4 π sin x dx π x π 4 π 8 ( + x 4 dx / x/ x / + x/ + x + x + dx x + ( / x/ (x / + / + / + x/ (x + dx / + / y x / / x/ /4 y/ (x / + / dx y dy + / 4 rctn y 4 log(y + / 4 rctn (x 4 log(x x +

29 .9. B 9 + x 4 dx 4 rctn (x + 4 rctn (x log x x + x + x + + x 4 dx π 4 4 (π + π (π π 4 R R +z 4 z e πi/4, e 3πi/4 lim z (i±/ C z i z e 3πi/4, 4 e 9πi/4 i 4, i 4 dz i πi ( + z4 4 + i 4 π R R z 4 + dz + x dx z Re iθ π R 4 e 4iθ + Reiθ dθ R R 4 π R π + x 4 dx 4 x y e xy x, y x y e xy Fubini b ( e xy sin x dy e xy sin x dy dx π x sin x ( b e xy sin x dx dy

30 3 (, x, b ( π sin x e xy sin x dy dx dx x ( c b e xy sin x dx dy e xy sin x dx F (x, y c F (b, y dy c F (, y dy F (x, y +y +y 4 e xy sin x dx e xy + y 4 (cos x + y sin x +y +y c 4 ( b e xy sin x dx dy F (b, y dy F (, y dy F (, y dy e y + y 4 (cos + y sin dy sin e y lim ε > F (b, y dy F (, y dy ε ε + y 4 dy F (b, y dy + F (b, y dy + y + y 4 dy + ε e bε ε + y + y 4 dy ε 3 b b c ( e xy sin x dx dy + y 4 dy

31 .9. B f(z e z C [, R] 4 C e iπ/4 π 4 O (R, L f(z dx C R f(z, dz + f(z dz + f(z dz C L Cuchy : : f(z dz C R f(z dz R R π/4 π/4 π/ R e x dx π e x dx e (Reiθ ire iθ dθ (z Re iθ π/ e R cos θ R dθ e R cos φ dφ e R ( φ/π dφ (cos φ φ π R π/ e e R Rφ/π dφ R π e R R (er π ( e R 4R

32 3 3 : f(z dz L R/ R/ R/ e ix ( + i dx (z ( + ix (cos(x i sin(x ( + i dx (cos(x + sin(x dx i (cos(x + sin(x dx i (cos(x + sin(x dx (cos(x sin(x dx sin(t dt π R/ π sin(x dx (t x (cos(x sin(x dx (cos(x sin(x dx sin x.9.4 x sin x x dx π C r R R R r r r R r C r e ix + R x dx e iz z dz R i r R r r + R sin x x cos x + i sin x x dx dx

33 .9. B 33 r R i R r π π ir cos θ R sin θ e Re iθ ire iθ dθ (R ir cos θ r sin θ e re iθ πi (r ire iθ dθ i π e R sin θ dθ π e ir cos θ r sin θ dθ sin x x dx π.9.5 Clothoid x(t t cos t dt, t y(t sin t dt (x(t, y(t : Fresnel t ( π, π (x (t, y (t (cos t, sin t, (x (t, y (t ( t sin t, t cos t

34 34 v(t, (t t (cos t, sin t v(t ( sin t, cos t, (t ( cos t, sin t (t

35 35. f(x, y (, b ε > δ > s.t. d((x, y, (, b < δ f(x, y f(, b < ε d 9 xy x f(x, y +y (x, y (, (x, y (, (x, y (, f x x y (x + y, f x xy (x + y (x, y (, f(h, f(, k y x f(x, y (x, x (, x ( + x + xy x y x f(x, y +y (x, y (, (x, y (,

36 36 f(x, y x f(x, y y x4 +4x y y 4 (x +y (x, y (, (x, y (, x x4 4x y y 4 (x +y (x, y (, (x, y (, x, y f(, y x y f(x, x f x f(, f f(,, x f(,, f(, x f(, f(, x x. f f(x + h, y f(x, y (x, y lim x h h f C k k f (, b ( f(x, y f(, b + f f (, b(x + (, b(y b x h d((x, y, (, b o(h 4 f C

37 C θ, θ f(x + h, y + k f(x, y + k f x (x + θ h, y + kh f(x, y + k f(x, y f x (x, y + θ kk f(x + h, y + k f(x, y ( ( f f (x, y + o( h + (x, y + o( k x f f (x, yh + x (x, yk + o( h + k ε > f f x δ > s.t. h, k < δ f x f (x + h, y + k f x (x + h, y + k f (x, y < ε (x, y < ε f f(x + h, y + k f(x, y h f (z, y k (z, y x (h + kε h + k < h + k f f(x + h, y + k f(x, y h f (z, y k (z, y x h 3 + k ε h + k ε h, k. f(x, y (xy /3 (x, y f x 3 x /3 y /3 f 3 x/3 y /3 f f(h, f(, (, lim x h h

38 38 f (, ( R(x, y f(x, y f(, f x f (, x + (, y f(x, y y x R(x, y x + y (xy/3 x + y /3 x /3 + x x. f(x, y (xy /3 f f(h, f(, (, lim x h h R(x, y f(x, y R(x, y x + y (xy/3 x + y z mx{ x, y } x + y z (xy z 4 R(x, y z4/3 z z /3 x, y z 3. f(x, y e x+y (, f(,, f f (,. x (, e x R e x+y ( + x + y ec (x + y x r cos θ, y r sin θ R r 5 f C f x f x k. φ(x, y f(x, y + k f(x, y ψ(x, y f(x + h, y f(x, y

39 .. 39 f(x + h, y + k f(x + h, y f(x, y + k + f(x, y φ(x + h, y φ(x, y ψ(x, y + k ψ(x, y θ, θ φ x (x + θ h, yh ( f x (x + θ, y + k f (x + θ, y h f x (x + θ h, y + θ khk θ, θ f x (x + θ h, y + θ khk f x (x + θ h, y + θ k f x (x + θ 3h, y + θ 4 k f C h, k 6 ( f(x, y x x(t y y(t. df f (x(t, y(t dt x (x(t, y(t x (t + f (x(t, y(t y (t f(x(t+h, y(t+h f(x(t, y(t+ f (x(t, y(t(x(t+h x(t+ f (x(t, y(t(y(t+h y(t+r x f(x(t + h, y(t + h f(x(t, y(t h f x (x(t, y(tx (t + θ h + f (x(t, y(ty (t + θ h + R h R h R (x(t + h x(t + (y(t + h y(t (x(t + h x(t + (y(t + h y(t h

40 4 (x(t + h x(t + (y(t + h y(t h (x(t ( + h x(t y(t + h y(t + h h (x (t + (y (t (h 9 z x + z z r + z r θ + z r r z r z θ z z cos θ + x sin θ z z r sin θ + x r cos θ z C z r z x cos θ + z x cos θ sin θ + z sin θ z θ z x r sin θ z x r cos θ z x r cos θ sin θ + z r cos θ z r sin θ z r + z r θ z x z r x cos θ + z sin θ z r sin θ z x + z z r r H(p, q( H p dq dt, H q dp dt dh dt H C dh dt H dp H dq (t + p dt q dt (t dq dt p m, F md(p/m dt dp dt

41 .. 4 potentil U ( T p m F U q H T + U Newton H p p m dq dt, H q U q F m dp dt L T U Lgrngen Lgrnge d L dt q L q q dq dt L q q L(q q L(q, q + L q (q, q (q q + L q q, q ( q q + R R p L q L q d L dt q dp dt L(q q L(q, q + dp dt (q q + p ( q q + R L(q q L(q, q, q q q, q q q, p p(q, q p(q, q L dp q + p q + R dt p q L H (p q p q + p q H(q q H(q, q + q p dp dt q + R Newton

42 4.3 f C r f(x + h, y + k r i i! i i f ic j x j i j (x, y hi k i j + R n (x, y j θ R n (x, y r! r r f rc j x j r j (x + θh, y + θk hj k r j j. g(t f(x + th, y + tk g(t r i t g (i (t g (i ( i! t i + g(r (θt t r r! i i f ic j x j i j (x + th, y + tk hj k i j j r i ( h i! x + k i f(x, y + ( h r! x + k r f(x + θh, y + θk.4 7 ( f [, b]. < c < b f(b f( b. < c < h > c > h f (c f(x + h f(x + f (ch 3. < θ < f(x + h f(x + f (x + θh, f(b f( + f ( + θ(b

43 (Rolle f [, b] C f( f(b f (c < c < b. f f (, b < c < b x < c < x f(x f(c x c > > f(x f(c x c f x c x, x c f (c f (c f (c 9 ( F, G [, b] C < c < b F (b F ( G(b G( F (c G (c. Q(x F (x F ( F (b F ( (G(x G( G(b G( Q( Q(b < c < b Q (c Q (c F (c F (b F ( G(b G( G (c ( f x C n f(x n k R n (x f (k ( (x k + R n (x k! R n (x f (n (c (x n n! x c

44 44 f(x n k r n f (n (c n!. n F (x f(b f(x G(x (b x n k f (k ( (x k + r n (x n k! f (k n (x (b x k f(b k! k f (k (x (b x k, k! < c < b n F (x k n k F (b F ( G(b G( F (c G (c f (k+ k! n (b x k f (k (x + (b xk (k! k f (k+ n (x (b x k + k! k f n (x (b xn (n! n F (b F ( F ( f(b k f (k+ (x (b x k k! f (k ( (b k k! (G(b G( F (c G (c (b n f ( n(c(b c n (n!n(b c n (b n f (n (c n! ( R n (x x f (n (t (n! (x tn dt

45 f(x f( x R n (x [f (n (x tn (t (n! f (t dt R (x ] x + x f (n ( (n! (x n + R n (x f (n (x tn (t dt (n!.5 ( v b f(x + h, y + bh f(x, y Dvf(x, y lim h h ( ( e, e De f(x, y f (x, y, x D e f(x, y f (x, y ( < θ <, < θ < f(x + h, y + bh f(x, y f(x + h, y + bh f(x, y + bh + f(x, y + bh f(x, y h f f (x + θh, y + bh + bh x (x, y + bθ h f C grd Dvf(x, y f (x, y + b f (x, y x (De f + bde f (x, y (De f, De fv grd f (De f, De f ( f x, f

46 46 n f : R n R Df ( f x,..., f x n U X. U x. X n x n V. U v. V n v n N x k u ik X k k f U f(u(, X j + h, f(u(, X j, lim X j h h n f( k lim u ikx k + u ij h f(( n k u ikx k h h n f u ij x i i ( f U,..., f U X X n V. V n n j n j i n i f U V j X j n f x i u ij V j f x i v i Df v n f x i i n u ij V j f : R n R n ( Df n n φ(x, y n f(x, y ψ(x, y Df(x, y ( φ x ψ x φ ψ j

47 Jcobi ( x(u, v F (u, v y(u, v F (D J ( x u u f(x, y dxdy D x v v f(f (u, v det J dudv.6 f(x, y y φ(x : x + y +, e y + x : x + y (y + ± x sin y x (y rcsin x + nπ tn y x (y rctn x + nπ ( e y x (y log x + i rg x ( y n x (y x n x /n e i rg x/n ( f (, b C f (, b, f(, b U φ C f(x, φ(x, φ( b. φ (x f x f (x, φ(x (x, φ(x. φ

48 48. φ 3. φ C f (, b > step. b < b < b b < y < b f (, y > f(, b f(, b < < f(, b ε > s.t. ε < x < + ε f(x, b < < f(x, b f (x, y > (b < y < b U ( ε, + ε x U b < c < b f(x, c c φ(x step c φ δ > {x n } s.t. x n c nd φ(x n φ(c > δ φ(x n d f lim f(x n, φ(x n f(c, d n d φ(c step 3. f(x+h, y+k f(x, y h f x (x+θh, y+θk+k f(x+θh, y+θk ( < θ < y φ(x, k φ(x + h φ(x h f x (x + θh, φ(x + θk + (φ(x + h φ(x f(x + θh, φ(x + θk h h

49 Lgrnge g(x, y (, b g(, b, g g (, b, x (, b (Lgrnge f, g C g(x, y f (, b. g(, b (, b g. F α (x, y f(x, y αg(x, y ( g(, b (b α F α F (, b, x α (, b g. (, b x y φ(x s.t. g(x, φ(x φ( b F (x f(x, φ(x x g(x, φ(x F ( f f (, b + x (, bφ ( g g (, b + x (, bφ ( α (, φ ( ( f x f g x g (, b (, ( ( f g x x f (, b α g (, b

50 5 x + y xy g (, g(, F α (x, y xy α(x + y F x y αx, F x αy α x + y x y ± g(x, y x 3 + y 3 3xy f(x, y xy F α x F α y α(3x 3y x α(3y 3x g(x, y x 3 + y 3 3xy Lgrnge x y 3 g (, ( g(, (, f Hessé ± g(x, y y 3 x + o f(x, y 3 x3 + o x 3 y + o f(x, y 3 y3 + o 3 g(x, y (x + y (x y f(x, y xy ( g (, Hessé ± (y(x y + y 3 + y x(x 3 + xy x x y + f y + + x ±, y ± 4 g(x, y, z xy + yz + zx 3 f(x, y, z xyz (,, g(,, (,, (,,

51 n H(p,..., p n p i log p i p p n n i 6 s +b+c s(s (s b(s c F s(s (s b(s c λ( + b + c 3 s 3, b, c, b θ F b sin θ λ( + b + c µ( + b b cos θ c F F b F c F θ b sin θ λ ( b cos θµ sin θ λ (b cos θµ λ + cµ b cos θ b sin θµ µ cos θ sin θ b sin θ ( + c cos θ sin θ (b + c cos θ + b b cos θ c + b + c 3 b c, b, c, B AC x,c y +b+c 3, x +y,x +(b y c F bx λ( + b + c 3 µ(x + y ν(x + (b y c 8

52 5.7 f z f(x, y f f (, b (, b x xy 7 f C v D v f(, b (, b. f( + hp, b + hq f(, b + hd v f(, b + o(h Hessé H(, b ( f x f x f x f f C A A A t A 3. x, y (Ax, y (x, Ay x λ λ(x, x (λx, x (Ax, x (x, Ax (x, λx λ(x, x x (x, x > λ λ λ µ λ y λ(x, y (Ax, y (x, Ay (x, µy µ(x, y

53 µ (x, y A λ x y Ax λx, Ay λy + x λ(x, y (Ax, y (x, Ay (x, λy + x λ(x, y + (x, x x y 8 Hessé λ, µ u, v w αu + βv h w (t f((, b + tw f(, b h w (t t(αu + βv f x (, b + t(αu + βv f (, b + t (αu + βv +o(t f x (, b + t (αu + βv (αu + βv f x (, b + t (αu + βv t(w, grd f(, b + t t whw + o(t f (, b t (α λ + β µ + o(t grd f(, b ( f x f (, b 9 ( p q q r

54 54. p + r > pr q >. pr q < 3. p + r < pr q >. p + r pr q grd f(x, y ( f x f (x, y f R f(x, y x + y.: f(x x + y f(x, y x y

55 : f(x x y : f(x x y f(x, y x y f(x, y x 3 y 3 3x + 3y

56 56 第 章 偏微分 の図は grd f は (±, ± で (, になり そこでは 6 (x, y (, 6 6 (x, y (, 6 H(x, y 6 (x, y (, 6 6 (x, y (, 6 となります これから想像がつくように grd f のベクトル場 等高線のベ クトル場は次のようになります

57 f C f(x, y f (x, y (x, y f x (x, y y (x, y f(x, y f(, b + f f (, b(x + (, b(y b + x y b f(,b/ x f(,b/ (x Hessé det H(, b > f(x, y u λ v µ t xhx u, v f((, b + αu + βv f(, b + f f (, b(x + (, b(y b x + t (αu + βvh(αu + βv + (α λ + β µ + λµ < f(x, y (x, y (, b (, b + α(u ± v λ/µ (α 3 f(x, y y x 3 + x

58 58 f(x, y y ± x 3 x ( 3x + x grd f y ( 6x + H(x, y (, ( (, ( 3, ± ( (, f(x, y f(x, y y + xy x 3 y x ± x + 4x 3 grd f H(x, y ( y 3x y + x ( 6x

59 (, ± ( + (, ( 6, 3± 5 ( + 3± 5 x y ±x 3/ f x ( 9, 4 7 f y ( 4, f(x, y x 3 + y 3 3xy grd f H(x, y ( 3x 3y 3y 3x ( 6x 3 ( ( (, ±3 ( ( (, 3, 9 y x 3 6y

60 6 f x ( /3, /3 f y ( /3, / f(x, y (x + y (x y grd f H(x, y ( 4x + 4x(x + y 4y + 4y(x + y ( 4 + 8x + 4(x + y 8xy 8xy 4 + 8y + 4(x + y ( ( (, ±4 ( ( (±, 8 f x (± 3, ± f y (±,

61 (grd f R grd f ( f x, f f v (v x, v y D v f(x, y lim h f(x + hv x, y + hv y f(x, y h f D v f(x, y f x v x + f v y (grd f, v v grd f grd f f grd f 3 ( x, grd f f (f + bg f + b g (fg f g + g f

62 6.9. (div F (f, g R div F f x + g h (x, y f(x, y h f(x+ h, y h (f(x + h, yh f(x, yhh f x h div F h 4 div F 3 ( x, div F (, F f F, G (, (F + bg (, F + b(, G (, (fg f(, G + ( f, G div grd x + n.9.3 (rot 3 grd div F (f, g, h R 3 rot F ( h g z, f z h x, g x f xy (x, y, z h 4 (x, y, z (x + h, y, z

63 .. 63 f(x, y, z h (x, y + h, z f(x, y + h, z h f(x, y + h, z h + f(x, y, z h f h y g(x + h, y, z h g(x, y, z h g x h g x f xy yz zx rot rot(grd f div(rot f. C C. C (x(t, y(t (t [, b] b x (t + y (t dt ds x (t + y (t dt

64 64 R f f ds C ds C (x(t, y(t (x(t + h, y(t + h (x(t + h x(t + (y(t + h y(t x (t + θ h + y (t + θ h h ( θ, θ t x y f(x (, f( (b, f(b y (t dy dt f (x b + f (x dx 7 Asteroid x cos 3 θ, y sin 3 θ π 6 6 ( 3 cos θ sin θ + (3 sin θ cos θ dθ 3 4 π/ sin θ dθ 6 [ ] π/ cos θ π/ cos θ sin θ dθ 8 Crdioid r ( + cos θ x cos θ( + cos θ, y sin θ( + cos θ ( dx + dt ( dy ( + cos θ dt π ( + cos θ dθ π θ cos dθ 8

65 f n f [, b] C (f n (x f(x f n(x f (x f n ( f n (b f( f(b. f n f b b + f n (x dx + f (x dx 7 ( ABC AB BC AB,BC AC M,N,P AM,MP,PN,NB AB BC C AC x(u, v Φ Φ(u, v y(u, v L(t (u(t, v(t ( t z(u, v dφ dt dt dφ dt Φ du u dt + Φ v dv dt ( Φ u, Φ v ( du dt dv dt ( E F F G dφ dt dt (( Φ u, Φ ( u Φ u, Φ v ( du dt, dv dt ( du dt, dv dt E ( du dt ( Φ ( Φ u, Φ v v, Φ v ( Φ u ( Φ Φ u, Φ v v ( E F F G ( du dt dv dt dt + F du dv dt dt + G E, F, G ( du dt dt dv dt ( dv dt dt

66 66 Φ 3 u Φ(u, v v Φ u, Φ v ( E F ( F G.. C U ( ω f(x, ydx + g(x, ydy C ω ω ( dx C dt, dy dt dt ( ( f(x, y x (t dt g(x, y y (t (f(x, y x (t + g(x, y y (t dt ω f(x, y dx + g(x, y dy C C C Y (s s [, b] τ : [, ] [, b] Y (τ(t X(t τ b ω(y (s dy ds ds ω(y (τ(t dy τ dt ω(x(t dx dt dt dt C y x (, 4 (, C ω ω(x, y (x xy dx + (y xy dy

67 .. 67 ( x x (X(x C ω x ( ( x x 3 dx x 4 x 3 x (x 5 4x 4 x 3 + x dx 369 C. ω(r, θ r dr + θ dθ. y ω(x, y ( x + y dx ( x x + y dy C ω ( θ [, π (X(θ cos θ sin θ ( ( π r ( π π ( ( cos θ sin θ sin θ cos θ ( dθ sin θ sin θ cos θ cos θ dθ dθ π ω U φ( ω φ φ dx + x dy φ F ω grd φ 4 3. ω φ. U X Y C ω X Y C 3. U C C ω

68 68. φ ( φ dx ω x dt + φ C dy dt dt dφ(x(t, y(t dt dt φ(x(, y( φ(x(, y( ( U X X X C φ(x ( C ω h X (x, y C X X + ( ( h ( ( h f(x + x, y φ X + φ(x ω dx g(x + x, y C h f(x + x, y dx φ f(x, y x.. Green d dxf (x f(x b f(x dx F (b F ( 5 (Green ω f(x, y dx + g(x, y dy U C D U D C. C ω D ( g x f dxdy g(x, y D φ(x ψ(x x [, b] (X(x (x, φ(x X(x (x, ψ(x g ( ( b f(x, φ(x ω dx + g(x, φ(x φ (x C b (f(x, φ(x f(x, ψ(x dx b ( ( f(x, ψ(x dx g(x, ψ(x ψ (x

69 .. 69 D f dxdy b b ( ψ(x φ(x f dy dx (f(x, ψ(x f(x, φ(x dx Green f(x, y φ(y ψ(y (y [c, d] D D C ω d c g dxdy g(ψ(y, y g(φ(y, y dy d c d c ( ψ(y φ(y g dx dy (g(ψ(y, y g(φ(y, y dy ω (f(x, y dx + dy + ( dx + g(x, y dy D C D dx dx dy dy dy dx dx dy d(f dx + g dy f f dx dx + x ( f + g x dy dx + g x dx dy ω f dx + g dy ω D D dω D x dy y dx D Green f(x, y y g(x, y x x dy y dx ( + dxdy C C D g dx dy + dy dy

70 7 ω y dx + x dy dω dy dx + dx dy dx dy 3 Asteroid x(θ cos 3 θ, y(θ sin 3 θ C x dy y dx π π π π cos 3 θ 3 sin θ cos θ sin 3 θ 3 cos θ( sin θ dθ cos θ sin θ dθ sin θ dθ cos 4θ 3 8 π 3π 4 dθ 3π 8 x r cos 3 θ, y r sin 3 θ ( r, θ π cos 3 θ sin 3 θ 3r cos θ sin θ 3r sin θ cos θ 3r 4 sin θ D dxdy π π π ( 3r 4 sin θ dr dθ [ ] r sin θ dθ + cos 4θ dθ 3π 8 ω f(x, y dx + g(x, y dy C : (x(t, y(t N (y (t, x (t (x (t, y (t (y (t + (x (t

71 .. 7 ( f(x, y F N g(x, y C F N ds C f(x, y y (t g(x, y x (t dt Green ( f F N ds x + g dxdy F N ds Guss. C C D D C div F dxdy g(x, y dx + f(x, y dy C C t ( t z(t h(z h(z dz C h(z(t z (t dt R z(t x(t + iy(t h(z u(x, y + iv(x, y h(z dz C h(z(tz (t dt { } u(x(t, y(t + iv(x(t, y(t (x (t + y (t dt u(x, y dx v(x, y dy + i v(x, y dx + u(x, y dy (. (u, v (v, u C C D ( (. v D x u dxdy (. ( u (. x v dxdy (.3 D

72 7 h u x v u v x (. (.3, h(z dz C..3 y f(x (, f( f ( y f (x + f( ( ( + h, f( + h y f (x h + f( + h ( + h (, f( ( + h, f( + h (, f( (f ( + h f ((x f ( + hf ((f( + h f( f (h x f (( + f ( f ( f ( x + f ( f (( + f ( f ( + f ( f (( + f ( 3/ f ( θ s ds dθ dθ..4 (Frenet Serret 4 y Ax x ds

73 f ( x x (h, Ah Ah y (x h + Ah Ah x y A + Ah h A (, f( ( + h, f( + h +h + f ( dx f ( f ( + h d tn θ dx f ( tn θ f (x x d tn θ dx dθ ds dθ cos θ dx ( + f (x dθ dx dθ dx f ( + f ( dθ dx dx ds f ( + f + f ( ( f ( ( + f ( 3/ 3 (, f(, ( ± h, f( ± h ( h (x, y x ( h( + h f( + f( h h + f( f( h(h f( + f( h + f( + h f( + f( h + f( + h y h f( + f( h + f( + h ( f( + f( h + f( + h f( + h + f( h f( h f ( f( + h + f( h f( h (f ( + f(f (

74 74 h x f (( + f ( f ( y f( + + f ( f (..4 (Frenet Serret (x(t, y(t (x (t + (y (t e (t e (t (e i (t, e i (t (i, (e i (t, e i(t κ(t λ(t e (t κ(t e (t, e 3(t λ(t e (t (e (t, e (t (e (t, e (t + (e (t, e (t λ(t κ(t e (t κ(t e (t, e (t κ(t e (t (Frenet Serret κ(t ( ( ( d e (t κ(t e (t dt e (t κ(t e (t

75 .. 75 ( (t v(t b(t v(t (te (t + b(te (t d dt v(t (te (t + (t e (t + b (te (t + b(te (t ((t κ(t b(te (t + ((t κ(t + b(te (t ( ( ( ( d (t (t κ(t (t + dt b(t b (t κ(t b(t ( C C(t x(t y(t ( C (t x (t y (t e (t ( C x (t (t e (t κ(t e y (t (t C(t C(s + C (s(t s + C (s (t s + O((t s 3 C(s + (t se (s + (t s κ(se (s + O((t s 3 s e (, e ( ( ( x(t x( + t + O(t 3 y(t y( + κ(t / t x x( y y( + κ((x x( + O((x x( 3 4 y Ax A κ(

76 C(t (x(t, y(t, z(t x (t + y (t + z (t e (t (x (t, y (t, z (t (e (t, e (t ( e (t κ(t e (t κ e (t C (t e (t C (t e 3(t e (t e (t e (t e (t e (t e (t (te (t + τ(te 3 (t τ(t (t (e (t, e (t (e (t, e (t (t (e (t, e (t + (e (t, e (t κ(t + (t (t κ(t e (t κ(te (t + τ(te 3 (t e 3(t e 3 (t (e (t, e 3 (t (e (t, e 3 (t + (e (t, e 3(t (e (t, e 3(t

77 .. 77 e 3(t e (t (e (t, e 3 (t (e (t, e 3 (t + (e (t, e 3(t κ(e (t, e 3 (t + τ(t(e 3 (t, e 3 (t + (e (t, e 3(t τ(t(e 3 (t, e 3 (t + (e (t, e 3(t e (t τ(t e (t κ(te (t e (t κ(te (t + τ(te 3 (t e 3(t τ(te (t e (t κ(t e (t d e dt (t κ(t τ(t e (t e 3 (t τ(t e 3 (t 3 e 3 (t (e 3(t C (t e (t κ(te (t κ(t C (t e (t κ(te (t + τ(te 3 (t 3 3 (t τ(t (e (t, e 3 (t (e (t, e (t e (t det(e (t, e (t, e (t det(c (t, C (t, C det(e (t, κ(te (t, κ (te (t + κ(te (t τ(t κ(t det(e (t, κ(te (t, e (t C (t det(c (t, C (t, C

78 78 det(e, e, e 3 (e, e e 3 C(t (x(t, y(t, z(t C (t e (t C (t e (t κ(te (t, C (t κ(te (t κ(t e (t + κ(tτ(te 3 (t + κ (te (t C(t C(s + C (s(t s + C (s(t s + 3! C (s(t s 3 + O((t s 4 C(s + (t se (s + (t s κ(se (s + 3! { κ(se (t + κ (se (s + κ(sτ(se 3 (s}(t s 3 + O((t s 4. U R R 3 (x(u, v, y(u, v, z(u, v Φ(u, v ((u, v U S S Φ u x u u z u, Φ v x v v z v u v ( R C : (u(t, v(t ( dx dt, dy dt, dz Φ(u(t + h, v(t + h Φ(u(t, v(t lim dt h h Φ u u (t + Φ v v (t C ( Φ u, Φ ( Φ E, u u, Φ ( Φ F, v v, Φ G v

79 .. 79 E ( du dt C S + F du dv dt dt + G U Φ u Φ dudv v ( dv dt dt u, v u x, v y Φ u u 3 Φ v v 3 3 ds ds Φ u Φ dudv v θ π, φ π x r cos θ sin φ, y r sin θ sin φ, z r cos φ r cos θ cos φ sin θ cos φ Φ Φ sin θ cos φ, θ cos θ cos φ, sin φ Φ φ cos θ sin φ sin θ sin φ cos φ Φ θ Φ φ cos θ cos φ sin θ cos φ sin θ cos φ sin φ cos θ cos φ sin φ Φ θ Φ φ cos θ cos 4 φ + sin θ cos 4 φ + cos φ sin φ cos θ cos φ sin θ cos φ cos φ sin φ cos 4 φ + cos φ sin φ cos φ S ds 4π π/ ( π π/ cos φ dθ dφ cos φ dφ 4π [ ] π/ sin φ 4π

80 8 S f(x, y, z f(x, y, z ds f(φ(u, v Φ u Φ dudv v S U Φ u Φ v Φ Φ u v ( Φ u, Φ EG F u E F F G EG F dudv d e f d b e c f bf ce cd f e bd (bf ce + (cd f + (e bd d b, e ( + b + c (d + e + f (d + be + cf c f 5 (x, y, f(x, y (D {(x, y: x + y } f(x, y x y x y f x f x f f x f f ( f + x ( f + x y

81 .. 8 D x y dxdy 4π π ( π r dθ r dr [ r ] π S ω ω f(x, y, z [dy, dz] + g(x, y, z [dz, dx] + h(x, y, z [dx, dy] ( S ω U f(u, v ( Φ g(u, v, u Φ dudv v h(u, v N S Φ u Φ v Φ u Φ v ω (ω, N ds ω f(x, y, z dy dz + g(x, y, z dz dx + h(x, y, z dx dy S S dy dz du dv Φ u Φ z x v u v z v u S Möbius 6 (Stokes S S rot F F S S rot F x z f(x, y, z g(x, y, z h(x, y, z

82 8. S rot F U D y h D z g D z f D x h D x g D y f z u v z u v z x u v x z u v x u v x u v dudv (.4 S η(t (t [, ] S F F (x(u(t, v(t, y(u(t, v(t, z(u(t, v(t dη dt dt (f(x, y, z( x du u dt + x dv v dt +h(x, y, z( z du u dt + z v du + g(x, y, z( u dt + dv v dt dv dt dt (f(x, y, z x z + g(x, y, z + h(x, y, z u u u du dt +(f(x, y, z x z + g(x, y, z + h(x, y, z v v v dv dt dt (f(x, y, z x z + g(x, y, z + h(x, y, z u u u du +(f(x, y, z x z + g(x, y, z + h(x, y, z v v v dv C C S U C D Green (f(x, y, z x + g(x, y, z + h(x, y, z z u v v v z (f(x, y, z x + g(x, y, z + h(x, y, z v u u u dudv D f ( f x x u + f u + f z ( f x x v + f v + f z z u x v + f(x, y, z x u v x u f(x, y, z x u v z ( z x u v z x v u z v f ( x u v x v u + f (.4 7 (Guss Ω R 3 Ω div F dxdydz F Ω Ω

83 Ω div F dxdydz Ω ( f x + g + h z dxdydz Ω D R φ(u, v ψ(u, v ( h ψ(u,v Ω z dxdydz h D φ(u,v z dz dudv (h(u, v, ψ(u, v h(u, v, φ(u, v dudv Ω D S(φ {(u, v, φ(u, v: (u, v D}, S(ψ {(u, v, ψ(u, v: (u, v D} S(ψ S(ψ u D u ψ, S(ψ v D v ψ u v S(φ, S(ψ D u φ D u ψ N(ψ D v φ, N(φ D v ψ F F N ds Ω Ω z h(u, v, φ(u, v dudv + h(u, v, ψ(u, v dudv D D.. D dω C ω dω D C ω

84 84 Green div F dxdy F N ds D C ( f(x, y F g(x, y C (x(t, y(t ds x (t + y (t dt ( x (t y (t ( y (t N x (t + y (t x (t F N ds x (t + y (t (f(x, yy (t g(x, yx (t ds ( f(x, y dy g(x, ydx dt dt dt g(x, y dx + f(x, y dy ω dx dx dy dy Green 3 (Guss dω g f dy dx + dx dy x ( f x + g dx dy div F dx dy C (x(t, s, y(t, s x(t + t, s x(t, s + s y(t + t, s y(t, s s z(t + t, s x t t z t x s s z s z(t, s s dtds z t s s t z x t s z x s t x t s x s t

85 .. 85 f(x, y, z F (x, y, z g(x, y, z h(x, y, z { ( z f t s s ( z x + g t t s z s x + h t ( x t F N ds s x s f(x, y, zdy dz + g(x, y, zdz dx + h(x, y, zdx dy ω dω ( f x + g + h dx dy dz z div F dx dy dz } dtds t Stokes F N ds ω 3 C (x(t, y(t, z(t f(x, y, z x (t g(x, y, z y (t dt h(x, y, z z (t f(x, y, z dx + g(x, y, z dy + h(x, y, z dz dω f f g g h h dy dx + dz dx + dx dy + dz dy + dx dz + dy dz z x z x ( f + g ( dx dy + g x z + h ( dy dz + h x + f dz dx z dx dy rot F dy dz dz dx

86 V φ: V R V V e i e j (j i φ i φ i t e i {φ i } V {e i } {e i } U x U x U U U UU E y V y x y UU x (y U(U x y y U 3.. M v u u Mv φ: V V R φ: V V R ψ : V V R φ φ(v, v (Mv, v M( M (i, j φ(e i, e j

87 ( φ(v, v φ(v, v φ(v, v φ(v, v 3..4 V φ: V k R k k k ( φ,..., φ k v,..., v k φ(v,..., v k φ (v φ k (v k k φ φ φ k ( φ, φ [φ, φ ] φ φ φ φ 3 4 φ(v, v, v 3, v 4 v i 3..5 V (, v φ(v ( ( v, v det(, v, v ( k k φ, ψ ω [φ, ψ] 3 3 φ, ψ, τ [φ, ψ, τ]

88 C f ( f grd f x, f, f z (grd f, v df(v ( df df f ω ω ω x dx + ω y dy + ω z dz v (v x, v y, v z ω(v ω x v x + ω y v y + ω z v z f df f f f dx + dy + x z dz (x, y, z x x dx dx, dy, dz 3..7 (div R v v(x, y C R C n (v, n C C (v, n ds C 9 v (, C C(t (cos t, sin t ds x (t + y (t dt dt (v, n ds π C cos t( sin t + cos t dt

89 C ( v(x, y x x + y, y x + y (x, y (cos t, sin t v r (cos t, sin t (cos t, sin t π ds π F (f, g R div F f x + g h (x, y ( 3. f(x, y f(x + h, y (x, y (x + h, y 3.: x f(x, y h f(x+h, y h f(x + h, yh f(x, yh f x h div F h div F ( x, div F (, F

90 F (f, g, h div F f x + g + h z div F (, F x z f, g h C V v (f, g ( ( f(x(t, y(t y (t (v, n ds dt C g(x(t, y(t x (t ( f(x(t, y(ty (t + g(x(t, y(tx (t dt g f(x, y dy g(x, y dx + f x dxdy div v dxdy V C Green V 3..8 (rot v v(x, y C C (x, y ds t v t ds C C v t ds 3 x v(x, y (, C (cos t, sin t t ( sin t, cos t 3 π v(x, y sin t dt [ ] π cos t ( x x + y, y x + y π cos t sin t r + sin t cos t r dt

91 3.. 9 Green v (f(x(t, y(t, g(x(t, y(t (f(tx v t ds (t+g(ty (t dt C C f dx+g dy D g x f dxdy C 3 grd div z. (x, y + h, z g(x, y, z f(x, y + h, z (x + h, y + h, z (x, y, z g(x + h, y, z f(x, y, z (x + h, y, z 3.: F (f, g, h R 3 ( h rot F g z, f z h x, g x f rot F F x z f g h xy (x, y, z h (x, y, z (x + h, y, z f(x, y, z h (x + h, y + h, z (x, y + h, z f(x, y + h, z h f(x, y + h, z h + f(x, y, z h f h,

92 9 3 y g(x + h, y, z h g(x, y, z h g x h g x f xy yz zx rot 3..4 grd, div, rot grd(f + bg grd f + b grd g grd(fg f grd g + g grd f, f F, G (, (F + bg (, F + b(, G (, (ff f(, F + ( f, F div(grd f f (3. rot(grd f (3. div(rot F (3.3 rot(rot F grd(div F F (3.4. (3. ( f div(grd f div x, f, f z ( f + x x ( f + z ( f f z

93 (3. x ( f rot(grd f rot x, f, f z ( f z z ( f f C (3.3 F (f, g, h ( h div(rot F div x g z, f z h x, g x f ( h g + ( f z z h x + z ( g x f f, g, h C (3.4 ( h rot(rot F rot g z, f z h x, g x f x ( g x f ( f z z h x g x + h z x f f z f x + g x + h z x f x f f z div F f x