K E N Z OU

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1 K E N Z OU

2 y = ax + b y = ax + bx + c x + y = r y = fx 1 x y t { x = xt a < = t < = b y = yt 1.1 xt, yt t t, 3 x + y a = { x = a cost y = b sint 3 x = xt, y = yt, z = zt n x i = x i t, x = x t,, x n = x n t x i i p 1. p = pt p = pt 1.3 t pt = xt, yt 1.4 ṗt dp dxt dt =, dyt dt dt ṗt p = pt t ṗt ṗt ṗt ṗt = ẋt + ẏt 1.5 t p p t = pt + t pt t 1 C C 3

3 t pt dpt pt pt + t pt lim = lim = dpt = ṗt t t t t dt... dpt = ṗtdt t a < = t < = b s s = b a dpt = b a ṗt dt = b a ẋt + ẏt dt 1.6 y pt + t dp dxt dt =, dyt dt dt pb pt + t pt B p = pt + t pt A pt x pa s pt t = a t = b a b t s = t ṗt dt 1.7 s t s = st t s t = ts pt t s s p = ps = xs, ys 1.8 dp = lim p s s = e 1, e 1 = 1 e 1 = x s, y s 1.9 e 1 P s 1 t = s e t = t P ps e 1 p Q s s ps ps = ps + e 1 s s s e 1 ps O O 3

4 s e 1 s ps = ps + e 1 s s s 1.1 { x = xs + x s s s y = ys + y s s s 1.11 e 1 e 1 π/ e k e 1 e e 1 e 1 = 1, e e = 1, e 1 e = 1.1 e 1, e s e 1 e 1 = 1 s d e 1 e 1 = e 1 e 1 = e 1 e 1 + e 1 e 1 = e 1 e 1 = e 1 e 1 e 1 e κs 3 e 1s = κse 1.13 e e e e 1 e 1 e = s e 1 e = e 1 e + e 1 e = κ + e 1 e = e = κe e e 1 κ 4 e 1 e = κ κ e 1 e 1.15 κ ps e 1 s, e s ps 1 ps e 1, e frame s P s + s Q s P e P 1 Q eq 1 θ P θ lim θ s = dθ s e 1 e 1s 4 - Frenet-Serret 4

5 e = e Q 1 ep 1 θ e 1.16 θ lim θ s = dθ = lim θ κ e s = de P 1 = κ e P = κ 1.17 s P e P ρ e P 1 sq p ρ θ e Q 1 κ = dθ e P 1 θ e Q 1 e = e Q 1 ep 1 C C C d p = κe 1.18 ρ p ρ θ dp = ρdθ = ρκ = e 1 = 1 ρ = 1 κ ps ps Cs e 1 s x θ 1.11, Cs = ps + ρe s 1. e 1 s = x 1s, y 1s = cos θ, sin θ tan θ = y 1 s x 1 s θ = tan 1 y 1 s x 1 s e s = sin θ, cos θ = y 1s, x 1s e 1s = κe s 1.1 e 1s = de 1 = de 1 dθ dθ = d = dθ tan 1 y 1 s x 1 s x 1 x 1 + y 1 x 1 y 1 x 1 y 1 x 1 = d dq tan 1 q dq = x 1 y 1 x 1 y 1 x 1 + y 1 q = y 1 s x 1 s = x 1y 1 x 1y 1... x 1 + y 1 = e 1 = 1... κs = x 1sy 1s x 1sy 1s, ρ = 1 κ = 1 x 1 sy 1 s x 1 sy 1 s

6 κ = dθ y = fx C θ ρ θ s θ + θ C ρ ps e e1 1: O Cc 1, c Cs 1. Cs = ps + ρe s 1.3 Cs 1.3 s Cs C s = p s + ρ se s + ρse s = ρ se s c 1 s = x 1 s c s = y 1 s + y 1 s x 1 sy 1 s x 1 sy 1 s x 1 s x 1 sy 1 s x 1 sy 1 s 1.5 y = fx x, y 1.17 κ = dθ 1.6 dy dx = y = tan θ θ = tan 1 y = dx + dy = dx 1 + y dx = 1 + y dθ = dθ dx dx = dθ dy dx dy dx = y y 1 + y = y 1 + y 3/... κ = y 1 + y 3/ ρ = 1 κ = 1 + y 3/ y e 1 e 1 e 1 =, y y, e =, y 1 + y 1 + y 1 + y Cc 1, c y c 1 = x ρ = x y 1 + y 1 + y y 1 c = y + ρ = y y 1 + y y

7 1. xt = r cost, 1.7 s = yt = r sint t ẋt + ẏt dt = rt t = s/r 1.3 xs = r cos s r ys = r sin s r ps = r cos s r, r sin s r dp = e 1s = sin s r, cos s r e s e 1 s 9 s e s = sin r + π s, cos r + π = cos s r, sin s r de 1 = 1 cos s r r, sin s = κe s κ = r r κ ρ = 1/κ = r 1.5,. y = ax a e 1 =, y 1 = 1 + y 1 + y 1 + 4a x, ax 1 + 4a x y 1.35 e =, 1 ax = 1 + y 1 + y 1 + 4a x, a x κ = y 1 + y 3/ = a 1 + 4a x 3/ ρ = 1 κ = 1 + 4a x 3/ a c 1 = x y 1 + y y = c = y y y = 1 + 4a x 3/ a a 1 + 4a x 3/ = 4a x 3 = 1 a + 3ax y y = x x 7

8 y = ±x e C 1 C 1 y = x κ =, ρ = 1/ C 1 c 1, c =, 1/ e C C y = x κ =, ρ = 1/ C c 1, c =, 1/. xt = t, yt = at a t s = ẋ + ẏ dt dt = ẋ + ẏ = 1 + 4a t dp dt = d dt t, at = 1, at e e 1 = dp = dp dt dt = a t, at 1 + 4a t e e 1 e =, e = 1 at e = 1 + 4a t, a t κ ρ 1.13 de 1 = de 1 dt dt = 4a t 1 + 4a t 3/, 8a 3 t 1 + 4a t 3/ + a a t 1 + 4a t a at = 1 + 4a t 3/ 1 + 4a t, a t... κ = = κe a 1 + 4a t 3/, ρ = 1 κ = 1 + 4a t 3/ a 3. x = a cos t, y = b sin t a > b > t e e 1 = dp = dp dt dt = s = ẋ + ẏ dt dt = a sin t + b cos t dp dt = d a cos t, b sin t = a sin t, b cos t dt a sin t a sin t + b cos t, b cos t a sin t + b cos t

9 e e 1 e =, e = 1 b cos t e = a sin t + b cos t, a sin t a sin t + b cos t κ ρ de 1 = de 1 dt dt = ab cos t a sin t + b cos t 3/, ab sin t a sin t + b cos t 3/ = ab a sin t + b cos t 3/ e 1 a sin t + b cos t... κ = = κe ab a sin t + b cos t 3/, ρ = 1 κ = a sin t + b cos t 3/ ab 1.45 C = p + ρe = a cos t, b sint a sin t + b cos t 3/ b cos t ab a sin t + b cos t, a b = cos 3 t, b a sin 3 t a b a sin t a sin t + b cos t t.. c 1 t = a b cos 3 t. a c t = b a sin 3 t b t 1.46 ac 1 /3 + bc /3 = a b / y x + y 1 = 1 O x 4. x = xt, y = yt y = dy dx = dy dt dt dx = ẏ ẋ, y = d dx ẏ = d ẋ dt ẏ dt ẋÿ ẍẏ = ẋ dx ẋ 3 9

10 1.7 κ = y ẋÿ ẍẏ 1 + y = 3/ ẋ + ẏ 3/ κ = 1 ẋ + ẏ 3/ ẋ ẍ ẏ ÿ = 1 ẋ + ẏ 3/ ṗt pt r = fθ θ = t 1.48 xθ = rθ cos θ, yθ = rθ sin θ xt = rt cos t, yt = rt sin t ẋ = ṙ cos t r sin t, ẏ = ṙ sin t + r cos t, ẍ = r cos t ṙ sin t r cos t ẍ = r sin t + ṙ cos t r sin t 1.48 κ = ẋÿ ẍẏ ẋ + ẏ 3/ = r + ṙ r r r + ṙ 3/ = dr r + r d r dθ dθ { } dr 3/ 1.5 r + dθ t p = pt, pt = xt, yt, zt 1.51 a < = t < = b s 1.6 s = b a ẋt b + ẏt + żt dt = ṗt dt s = a t ṗt dt 1.5 t s p = ps 1.9 e 1 dp = e 1, e 1 s e 1 s = e 1 = x s + y s + z s = e 1 s e 1 s = e 1se 1 s + e 1 se 1s = e 1se 1 s = 1.54 e 1 s e 1s e 1 e 1 π/ e e 1 e 1 s 1

11 e s e 1 s e 1 s e s e 1 s κs κs = e 1s = e 1 s e 1 s = x s + y s + z s 1.55 κs κs > = e s e 1 s = κse s e s = 1 κs e 1s, κs 1.56 κs = e s κs > e 1, e e 3 e 3 = e 1 e, e 3 = e e 3 s = 1 κ y z z y, z x x z, x y y x 1.58 e 1, e, e 3 Frenet Frame e i e j = δ ij = { 1 i = j i j s e i e j + e i e j = 1.6 e 1 e + e 1 e = κ + e 1 e = 1.61 i =, j = e e = 1.6 e e e e 1 e 3 e = αe 1 + βe α = κ e = κe 1 + βe i = 1, j = 3 e 1 e 3 + e 1 e 3 = e 1 e 3 = 1.64 i =, j = 3 e e 3 + e e 3 = κe 1 + βe 3 e 3 + e e 3 = β + e e 3 =... e 3 = βe 1.65 i = j = e 3 e 3 =

12 e x z e e 3 e 3 e 1 z e y e 1 e 3 e 3 e x e e 1 e β τ e 1 = κe e = κe 1 + τe 3 e 3 = τe 1.67 d e 1 e e 3 = e 1 e e 3 = κ k τ τ e 1 e e τ ρ = 1/τ 1 κs τs e 1 s, e s, e 3 s κs τs ps κ, τ 1.67 e 1s = κ + τ e 1 + κτc C : 9 e 1 s = C 1 e is κ +τ + C e is κ +τ + C κτ κ + τ = A cos κ + τ s B sin κ + τ s + C κτ κ + τ... ps = e 1 s = A κ + τ sin κ + τ s + B κ + τ cos κ + τ s + C κτ κ + τ + D A, B, C, D e i = 1 ps τ 1.65 e 3 { 9 e 1 = κe e = κe 1 e 1 = κe e = κe 1 + τe 3 e 3 = τe

13 1 ρ e 1 e e 3 τ e 1 e ps a a a ps = ps ps O a ps = c 1.7 s s a p s = a e 1 = 1.71 a e 1 = a κe = 1.7 κ a e = a e 1, e a e = s a e = a κe 1 + τe 3 = a τe 3 = 1.73 a e 1, e e 3 τ = ps τ 1.69 e 3 = e 3 s e 3 ps s e 3 ps = e 3 p s = e 3 e 1 s =... e 3 ps = c ps e 3 ps = c 1 6. helix pt = a cos t, a sin t, bt a, b > 1.75 t t s = ṗt dt = a + b dt = a + b t,... t =... ps = a cos sc, a sin sc, bc s s a + b = s c c = a + b

14 s p s = e 1s = a c sin s c, a c cos s c, b = e 1 s c a c cos s c, a c sin s c, = κe s, κ = cos s c, sin s c, e s = b e 3 s = e 1 e = c sin s c, b c cos s c, a c 1 e s = c sin s c, 1 c cos s c, e 1 e 1 = a c = a c e 1 + b c e 3 = κe 1 + τe κ = a a + b, τ = b a + b 1.78 κ τ p = pt κ τ ṗ p p κ = ṗ p ṗ ṗ p p τ = ṗ p ṗ, p, p 3 ṗ p p = ẋt ẏt żt ẍt ÿt zt xt 1.68 yt zt

15 ṗ = dp = p dt dt, ṗ = e 1 dt = dt p = p + p d s dt dt ṗ p = p dt { p dt p p = e 1 κe = κe 3,... κ = ṗ p ṗ 3 ṗ = p dt = dt e 1, { d 3 s p = dt 3 κ dt = αe 1 + βe + κτ ṗ p p } + p d s dt = 3 p p = ṗ 3 p p dt p p = κ,... ṗ p = κ ṗ 3 p = d s dt e 1 + κ e... e = κe dt } { 3 } 3 e 1 + κ + 3κ d s dt dt dt e + κτ = 3 e 3... e = κe 1 + τe 3 dt dt d s dt κ dt 3 dt α β κτ ṗ p = κ ṗ 6 = κ dt.. ṗ p p. τ = ṗ p 6 = κ τ 6 dt 3 e 3 dt p = ps 1.8 τ = 1 κ p p p 1.8 p = e 1, p = κe, p = κe = κ e + κe = κ e 1 + κ e + κτe p p p = κ = κ τ κ κ κτ... τ = 1 κ p p p 15

16 1..3 ps s = Taylor ps = ps + sp + s! p + s3 3! p p s = e 1, p s = κe, p s = κ e 1 + κ e + κτe s 3 ps = p + s 1 6 κ s 3 + e 1 + p + xse 1 + yse + zse 3 1 κs κ s e + 6 κτs3 + e 3 + xs = s 1 6 κ s 3, ys = 1 κs κ s 3, zs = 1 6 κτs κ τ.1 P u, v Qx, y, z x = xu, v, y = yu, v, z = zu, v.1 P u, v Qx, y, z u, v pu, v = xu, v, yu, v, zu, v. v v u pu, v u u u v v u, v pu, v 1 z O v u, v u = a z u = a v = b u y O x v = b x p u O v y 1 uv 3 16

17 x + y + z = a x = a sin u cos v, y = a sin u sin v, z = a cos u x a + y b + z c = 1 x a + y b + z c = 1 z = x a + y b x = a sin u cos v, y = b sin u sin v, z = c cos u x = a sinh u cos v, y = b sinh u sin v, z = c cosh u x = u, y = v, z = u a + v b. pu, v v = b u pu, b u = a v pa, v pa, b pa, b xa, b p u a, b = =, u u pa, b xa, b p v a, b = =, v v 13 p v a, b ya, b, u ya, b, v za, b u za, b v.3 u = a p u a, b v = b p u p v 1 14 p u, p v pa, b p u a, b p v a, b X p u p v 1 pa, b p u p v 1 {X} {X} = {ξp u a, b + ηp v a, b ξ, η }.4 15 X ξ X = ξp u a, b + ηp v a, b = p u p v η.5 13 u, v 14 c 1a 1+c a +, c na n = c 1 = c =, c n = a 1, a,, a n 1 a 1 a 1 p p = c 1a 1 + c a c 1, c

18 .3 1 X X X t X 16 X X = X = X t p X = ξ η u ξ p p u p v v η p = ξ η u p u p u p v ξ E F = ξ η p v p u p v p v η F G = Eξ + F ξη + Gη E, F, G 17 ξ η.6 E = p u p u, F = p u p v, G = p v p v.7 E, F, G pu, v 1 E = p u >, G = p v > 1 18 E F F G = EG F = p u p v p u p v > = p u / p v p u p v 1 E F F G >.8 X 1 = ξ 1 p u + η 1 p v, X = ξ p u + η p v E F ξ X 1 X = ξ 1 η 1 = Eξ 1 ξ + F ξ 1 η + ξ η 1 + Gη 1 η.9 F G X 1, X θ η X 1 X = X 1 X cos θ... cos θ = X 1 X X 1 X = Eξ 1 ξ + F ξ 1 η + ξ η 1 + Gη 1 η Eξ 1 + F ξ 1 η 1 + Gη1 Eξ + F ξ η + Gη.1 Eξ 1 ξ + F ξ 1 η + ξ η 1 + Gη 1 η = X 1 X θ 1 pu, v pu + du, v + dv dp dp = p du + u p dv = p v u du + p v dv = p u p v du dv a b a b = a t b, a b t = b t a t, t : transpose 17 n x 1, x,, x n P n i,j aijxixj A xt Ax A a ij = a ji x t = ` x 1 x x n 18 a b < = a b 18

19 19.5 ξ = du, η = dv u, v dp I = dp = dp dp = = Edu + F dudv + Gdv = du dv E F F G du dv.1 pu, v 1 pu, v 4 P, Q, R, T 4 ds u P u + du Q R T v v + dv P u, v Qu + du, v Ru, v + dv T u + du, v + dv : P Q p u du, P R p v dv ds = p u p v dudv.13 A B C D = A CB D B CA D p u p v = p u p u p v p v p u p v = EG F... p u p v = EG F ds = EG F dudv.15 ds 1.7 S pu, v = r sin u cos v, r sin u sin v, r cos u, p u = r cos u cos v, r cos u sin v, r sin u, p v = r sin u sin v, r sin u cos v, E = p u p u = r, F = p u p v =, G = p v p v = r sin u, EG F = r sin u π π S = ds = r sin ududv = 4πr.8 S xz C x R + z = r z x + y R + z = r 19 pu + du, v + dv pu, v = du u + dv «p + 1 du v! u + dv «p + 1 v 19

20 θ, φ < = θ < = π, < = φ < = π xθ, φ = R cos θ + rcosφ cos θ yθ, φ = R sin θ + r cos φ sin θ zθ, φ = r sin φ pθ, φ pθ, φ = R cos θ + rcosφ cos θ, R sin θ + r cos φ sin θ, r sin φ p θ = R + rcosφ sin θ, R + r cos φ cos θ, p φ = r cos θ sin φ, r sin θ sin φ, r cos φ E = p θ p θ = R + r cos φ, F = p θ p φ =, G = p φ p φ = r... EG F = r R + r cos φ S = π π EG F dθdφ = π π rr + r cos φdθdφ = 4π rr P u, v p u, p v 1 e = p u p v p u p v.16 e P α pu, v Q d P u, v Qu + du, v + dv P u, v Qu + du, v + dv P Q d d e P Q d = e P Q.17 Q d P d e P Q α d > d < P Q P Q = pu + du, v + dv pu, v = du u + dv p + 1 du v! u + dv p +.18 v p u du + p v dv + 1 { puu du + p uv dudv + p vv dv }

21 .17 e p u, p v d = 1 { puu edu + p uv edudv + p vv edv } = 1 { Ldu + Mdudv + Ndv }.19 L = p uu e, M = p uv e, N = p vv e. L, M, N.19 { } II = Ldu + Mdudv + Ndv.1 e p u, p v p u e =, p v e = u, v p uu e + p u e u = p uv e + p u e v = p vu e + p v e u = p vv e + p v e v = L M N L = p uu e = p u e u M = p uv e = p u e v M = p vu e = p v e u N = p vv e = p v e v..3 1 P a, b C s C u, v s u = us, v = vs C C X p = pus, vs.4 X = dp = p du u + p dv v = p du u + p dv v X = Eξ + F ξη + Gη = 1 ξ = du, η = dv.5 X P a, b e dx C X = dx/ e κx d du p u = 1

22 du p uu + p dv uv κx = X e = p uu e = = L du du du du + p uv e du dv du p uu e p uv e p uv e p vv e dv + M du dv + N dv + p vv e du du du = = Lξ + Mξη + Nη L M M N du dv.6 κx X = ξp u + ηp v κ X e P, 1 = dp dp II = dp de.7 II = dp de = p u du + p v dv e u du + e v dv = Ldu + Mdudv + Ndv.8 L M du = du dv M N dv X 1 X = 1 X e = X e = X e + X e = X e = X e.9 κ = dx de e = X = dp de de = dp = Ldu + Mdudv + Ndv Edu + F dudv + Gdv = II I P a, b e hu, v P a, b hu, v = {pu, v pa, b} ea, b.31 dha, b = h u a, bdu + h v a, bdv = p u a, b ea, bdu + p v a, b ea, bdv = P a, b C e X 1 1

23 P a, b h h H H =... Ha, b = h uu h vu h uv h vv = La, b Ma, b Ma, b Na, b p uu e p vu e p uv e p vv e, det H = LN M.3 det Ha, b h 1. det H > II LN M > p P a, b e. det H < II LN M < p P a, b e e II e P e P P e II II II Eξ + F ξη + Gη = κx = Lξ + Mξη + Nη.34 Gξ, η, λ = Lξ + Mξη + Nη λeξ + F ξη + Gη

24 ξ, η.33 G = Lξ + Mη λeξ + F η = ξ G = Mξ + Nη λf ξ + Gη = η.36 X ξ, η X = ξp u + ηp v.36 L λe M λf M λf N λg ξ η =.37 ξ, η, λ L λe M λf M λf N λg = EG F λ EN + GL F Mλ + LN M =.38 EG F > D = EN F M + GL 4EG F LN M EG F = 4 EM F L + EN GL F E EM F L >= E.39 λ λ 1, λ, λ 1 < = λ 1 λ ξ, η ξ 1, η 1 ξ 1, η 1 Eξ1 + F ξ 1 η 1 + Gη1 = 1 { L λ 1 Eξ 1 + M λ 1 F η 1 = L λ 1 Eξ1 + M λ 1F ξ 1 η 1 =.4 M λ 1 F ξ 1 + N λ 1 Gη 1 = M λ 1 F ξ 1 η 1 + N λ 1 Gη1 = Lξ 1 + Mξ 1 η 1 + Nη 1 λ 1 Eξ 1 + F ξ 1 η 1 + Gη 1 =... λ 1 = Lξ 1 + Mξ 1 η 1 + Nη 1 = κ 1 ξ 1, η 1.41 λ = κ ξ, η.38 κ 1, κ K κ 1 κ = LN M EG F.4 H 1 κ 1 + κ = EN + GL F M EG F.43 K H κ 1, κ κ 1, κ X 1 = ξ 1 p u + η 1 p v, X = ξ p u + η p v.8 EG F > K LN M 3 3 ax + bxy + cy + d = a, b, c 8 < ac b > ac b < : ac b =

25 K > 4 K < K =.4,.43 κ 1, = H ± H K D = EG = F EM = F L L E = M F = N G EN = GL.45 κ 1 = κ κ = L = M = N =.36 Lξ 1 + Mη 1 κ 1 Eξ 1 + F η = L κ 1 Eξ 1 + M κ 1 F η 1 = Mξ 1 + Nη 1 κ 1 F ξ 1 + Gη 1 = M κ 1 F ξ 1 + N κ 1 Gη 1 = Lξ + Mη κ Eξ + F η = L κ Eξ + M κ F η = Mξ + Nη κ F ξ + Gη = M κ F ξ + N κ Gη = 1 ξ, η.46 L κ 1 Eξ 1 ξ + M κ 1 F ξ 1 η + ξ η 1 + N κ 1 Gη 1 η = ξ 1, η 1 κ 1 κ.9 L κ Eξ 1 ξ + M κ F ξ 1 η + ξ η 1 + N κ Gη 1 η =.48 κ 1 κ Eξ 1 ξ + F ξ 1 η + ξ η 1 + Gη 1 η =.49 Eξ 1 ξ + F ξ 1 η + ξ η 1 + Gη 1 η = X 1 X =.5.9 a p = a sin u cos v, a sin u sin v, a cos u, p u = a cos u cos v, a cos u sin v, a sin u p uu = a sin u cos v, a sin u sin v, a cos u, p v = a sin u sin v, a sin u cos v, p vv = a sin u cos v, a sin u sin v,, { e = p/ p = p/a E = p u p u = a, F = p u p v =, G = p v p v = a sin u L = p uu e = a, M = p uv e =, N = p vv e = a sin u 4 P P P http : //ameblo.jp/scitamehtam/entry html 5

26 K H K = κ 1 κ = LN M EG F = 1 a, H = 1 κ EN + GL F M 1 + κ = EG F = 1 a κ 1, κ EG F λ EN F M + GLλ + LN M = a λ + aλ + 1 =... λ= κ 1, κ = 1 a 5.1 xy y = fx z pu, v = u, fu, v x = xu, y = fu, z = v z x y = fu y p u = 1, f,, p uu =, f,, p v =,, 1, p vv =,, p uv =,,, e = p u p v p u p v = f, 1, 1 + f E = 1 + f, F =, G = 1 L = f / 1 + f, M =, N = K = κ 1 κ = LN M EG F = H = 1 κ 1 + κ = EN + GL F M EG F f = 1 + f 3/ κ 1 = κ = H κ 1 =.37 λ = L M ξ f / 1 + f = ξ = M N η η f = H = κ 1 = κ = f ξ = η κ 1 X 1 = p u +ηp v = η,, 1 z κ = H.37 λ = H f /1 + f 3/ 5 1 II = κi ξ η = 6

27 ξ η = κ X = ξ p u + p v = ξ1, f, x, y xy 6 κ = f /1 + f 3/ xy.11 pu, v = R cos u + r cos v cos u, R sin u + r cos v sin u, r sin v < = u, v < = π p u = R + r cos v sin u, R + r cos v cos u, p v = r sin v cos u, r sin v sin u, r cos v p uu = R + r cos v cos u, R + r cos v sin u, p uv = r sin u sin v, r cos u sin v, E = R + r cos v, F =, G = r L = cos vr + r cos v, M =, N = r K = cos v rr + r cos v, R + r cos v H = rr + r cos v u R =, r = 1 det H = LN M = + cos v cos v < v < π/ K > π/ < v < 3π/ K < 3π/ < v < π K > v = π/, 3π/ K =.1 z = x + y xu = u, yv = v, zu, v = u + v, pu, v = u, v, u + v p u = 1,, u, p uu =,,, p v =, 1, v p vv =,,, p uv =,,, e =, 1, E = 1 + 4u, F = 4uv, G = 1 + 4v L = / 1 + 4u + 4v, M =, N = / 1 + 4u + 4v K = κ 1 κ = u + 4v, H = 1 κ 1+κ = 1 + u + v 1 + 4u + 4v 3/ 4 z κ 1, κ.38 λ κ 1 = 1 + 4u + 4v 3/ κ = 1 + 4u + 4v x - y - 6 e X 1 =, e X =, X 1 X = 7

28 7 u, v =, κ 1 = κ u, v =, det H > h uu, = L, >,.13 pu, v = u, v, u v p u = 1,, u, p uu =,,, p v =, 1, v p vv =,,, p uv =,, e = u/ 1 + 4u + 4v, v/ 1 + 4u + 4v, 1/ 1 + 4u + 4v E = 1 + 4u, F = 4uv, G = 1 + 4v L = / 1 + 4u + 4v, M =, N = / 1 + 4u + 4v 4 K = 1 + 4u + 4v, H = 4u v 1 + 4u + 4v 3/ u, v =, det H <,.14 pu, v = coshu cosv, coshu sinv, u p u = cosv sinhu, sinv sinhu, 1, p uu = cosv coshu, coshu sinv, p v = coshu sinv, cosv coshu,, p vv = cosv coshu, coshu sinv, p uv = sinv sinhu, cosv sinhu,, e = cosv/ coshu, sinv/ coshu, tanhu E = cosh u, F =, G = cosh u, L = 1/ cosh u, M =, N = 1/ cosh u... K = 1/ cosh 4 u, H = 8 gx 1, x,, x n = z = fx 1, x,, x n x 1, x,, x n Gx 1, x,, x n, λ Gx 1, x,, x n, λ = fx 1, x,, x n λgx 1, x,, x n 7 κ 1 8 http : // furuhata/ed/hokkyodai/ 8

29 G x 1, x,, x n G = f λ g = x 1 x 1 x 1 G = f λ g = x x x. G = f λ g = x n x n x n gx 1, x,, x n = x 1, x,, x n C ξ, η.36 λ { Lξ + Mη λeξ + F η = Mξ + Nη λf ξ + Gη = Lξ + Mη Eξ + F η Mξ + Nη F ξ + Gη =.53 EM LF ξ + EN LGξη + F N MGη =.54 ξ = du/, η = dv/ EM LF du + EN LG du dv + F N MG dv =.55 C F = p u p v =.54 ξ = 1, η = 9 EM LF = M = 3 F =, M =.54 LG EN ξ = η = C LG ENξη =.56 F = M = F = M = pu, v = R + r cos v cos u, R + r cos v sin u, r sin v u,v u, v 9 ξ =, η = 1 3 E = p u p u 9

30 1 X = ξp u + ηp v.3 F = M = EGλ EN + GLλ + LN = κx = Lξ + Nη Eξ + Gη.58 κ 1 = L E, κ = N G E ξ G ξ κx = κ 1 + κ.6 Eξ + Gη Eξ + Gη X p u θ.1 E ξ G η cos θ = Eξ + Gη, sin θ =.61 Eξ + Gη κx = κ 1 cos θ + κ sin θ.6 R, R 1, R.6 1 R = cos θ + sin θ.63 R 1 R.15 pu, v = u, v, u + v p u = 1,, u, p v =, 1, v 1 e = p u p v / p u p v = u, v, 1 4u + 4v E = p u p u = 1 + 4u, G = p v p v = 1 + 4v F = p u p v = 4uv L = p uu e = 4u + 4v + 1, M = p uv e =, N = p vv e = 4u + 4v + 1 EG F λ EN + GL F Mλ + LN M = κ 1 = 4u + 4v + 1 1/, κ = 4u + 4v + 1 3/ 3

31 EM LF ξ + EN LGξη + F N MGη = uvξ u v ξη uvη = uξ + vηuη vξ =... uξ + vη =, vξ uη = ξ = vk, η = uk, ξ = uk, η = vk k k = 1 1 X 1 = v, u, X = ξp u + ηp v = u + v 1 X = u + v + 4u + v u, v, u + v, X 1 X = EM LF du... u du + v dv =, + EN LG du dv + F N MG v du udv = u + v dv =, u + v = C 1, u = C v C 1, C dv = d ln u u, v =,.16 pu, v = u + v, u v, uv 1 p u = 1, 1, v, p v = 1, 1, u 1 e = u + v, u + v, u + v + 4 = d ln v E = + v, F = uv G = + u 1 L =, M = u + v + 4, N = 5 K = LN M EG F = 4 4u + 4v + 1 H = EN GL GM EG F 4u v = 4u + 4v + 1 3/

32 du EM LF + EN LG du dv dv + F N MG =.. dv + v. du = ± du + u u + = ± dv v + d du lnu + u + = ± d dv lnv + v + u + u + v + v + = C 1, u + u + v + v + = C, C 1, C u, v G OOD L U C K! S E E Y OU A G A I N! by K E N Z OU