Fubini

Similar documents

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -

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

DVIOUT


v er.1/ c /(21)

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

1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), 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

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y


y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =

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



x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

B 38 1 (x, y), (x, y, z) (x 1, x 2 ) (x 1, x 2, x 3 ) 2 : x 2 + y 2 = 1. (parameter) x = cos t, y = sin t. y = f(x) r(t) = (x(t), y(t), z(t)), a t b.

i

= M + M + M + M M + =.,. f = < ρ, > ρ ρ. ρ f. = ρ = = ± = log 4 = = = ± f = k k ρ. k

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x


II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (

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


( ) ( )

II 2 II

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

mugensho.dvi

(1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

Chap11.dvi

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

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,

x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v

2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n

K E N Z OU

A

y = f(x) y = f( + h) f(), x = h dy dx f () f (derivtive) (differentition) (velocity) p(t) =(x(t),y(t),z(t)) ( dp dx dt = dt, dy dt, dz ) dt f () > f x

6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4


i

2011de.dvi

7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a


1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1

pdf


1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,

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

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

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

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)

Untitled

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x

untitled

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

29


Z: Q: R: C: sin 6 5 ζ a, b

notekiso1_09.dvi

I 1

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F

meiji_resume_1.PDF

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

2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A

untitled

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

webkaitou.dvi

A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.

untitled

F S S S S S S S 32 S S S 32: S S rot F ds = F d l (63) S S S 0 F rot F ds = 0 S (63) S rot F S S S S S rot F F (63)

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

2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta

04.dvi

III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y


2014 S hara/lectures/lectures-j.html r 1 S phone: ,

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

入試の軌跡

d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,

2009 IA I 22, 23, 24, 25, 26, a h f(x) x x a h

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f

D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j

Note.tex 2008/09/19( )

120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2


KENZOU

Transcription:

3............................... 3................................ 5.3 Fubini........................... 7.4.............................5..........................6.............................. 3.7.............................. 5.8 Γ................................9 B................................9. e x.........................9........................ 7.9.3.............. 3.9.4 sin x x....................... 3.9.5 Clothoid....................... 33 35........................... 35................................ 36.3...................... 4.4........................... 4.5.............................. 45.6............................ 47.6. Lgrnge.................. 49.7............................ 5.8.............................. 57.9........................... 6.9. (grd........................ 6.9. (div......................... 6.9.3 (rot......................... 6................................ 63............................. 63.......................... 66

.. Green....................... 68..3.......................... 7..4 (Frenet Serret.......... 74..5............................ 76................................ 78.. D dω C ω....................... 83 3 86 3............................... 86 3................ 86 3........................ 86 3..3.............. 87 3..4...................... 87 3..5... 87 3..6....................... 88 3..7 (div......................... 88 3..8 (rot......................... 9 3..4 grd, div, rot................ 9

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 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 3 + 4 cos θ + cos 4θ 8

.. 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 3 4 3

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 + 4 3 5 6n + n n (n k + k n 3 k 3. 8 n n(n 4 3 6n + n 8 n 3 n(n + (n + 6 8 n 3 n(n (n 6 + n 3 n(n + + n 3 n(n x x x x 4. (x x dx 4 3

.3. Fubini 7 x x x x dx 4 3.3 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 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

.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

:. 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

.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 4 8 3 4 3

.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/

.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

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

.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

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

.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

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

.7. 9 4 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

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 5 4 3 5.8 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

.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

.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

.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

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

.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 +

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 +

.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

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 +

.9. B 9 + x 4 dx 4 rctn (x + 4 rctn (x + + 4 log x x + x + x + + x 4 dx π 4 4 (π + π 4 + 4 (π π 4 R R +z 4 z e πi/4, e 3πi/4 lim z (i±/ C z i z 4 + 4 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

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

.9. B 3.9.3 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

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

.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.75.5.5 - -.5.5 -.5 -.5 -.75.: Fresnel t ( π, π (x (t, y (t (cos t, sin t, (x (t, y (t ( t sin t, t cos t

34 v(t, (t t (cos t, sin t v(t ( sin t, cos t, (t ( cos t, sin t (t...8.6.4...4.6.8...4

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 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 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 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

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

.. 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

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

.4. 43 8 (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 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

.5. 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 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

.6. 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. 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

.6. 49.6. 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

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(,, (,, (,,

.6. 5 5 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

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

.7. 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.. 3.. 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. 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 3 3 3 3 f(x, y x y

.7. 55.3: f(x x y 3 3 3 3 3 3 3 3.: f(x x y 3 3 3 3 f(x, y x y f(x, y x 3 y 3 3x + 3y

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

.8. 57.8 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 f(x, y y ± x 3 x ( 3x + x grd f y ( 6x + H(x, y (, ( (, ( 3, ± ( (,.5.5 - - 4 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

.8. 59 (, ± ( + (, ( 6, 3± 5 ( + 3± 5 x y ±x 3/ f x ( 9, 4 7 f y ( 4, 8.3.. -...4.6.8 -. -. 5 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

6 f x ( /3, /3 f y ( /3, /3.5.5 - -.5.5.5 -.5-6 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 (±,

.9. 6.4. - -.5.5 -. -.4.9.9. (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

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 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 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 9 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 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 ( 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. 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 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

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 3 3 4 3 4 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 3 4 3 8 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

.. 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

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 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 ( (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..5 3 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 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 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 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 8 3 3 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

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

.. 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

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 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 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 3 3. 3.. 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

3.. 87 3..3 ( 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 3 3..6 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

3.. 89 3 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

9 3 3 3 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

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,

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

3.. 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

2024 © docsplayer.net プライバシー | 利用規約 | フィードバック