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

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1 1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC r 1 A B B C C A (1),(2),, (8) A, B, C A,B,C

2 2 1 ABC A BC (1)+(3) A A 2 A 1 ABC AB C (1)+(4) B B ABC ABC (1)+(2) C C 2 (1) = (8) 2 (2), (3), (4) (1) + (2) + (3) + (4) + (5) + (6) + (7) + (8) = 4π ABC (1) A + B + C π 1 ( ) r ABC r 2 ( A + B + C π) t p(t) = (x(t), y(t)) ẋ = dx, ẏ = dy dt dt ṗ = (ẋ(t), ẏ(t)); ṗ = (ẋ(t)) 2 + (ẏ(t)) 2 ; s(t) = t 0 ṗ(t) dt; 3 s p(s);

3 3 e 1 (s) p(s) e 1 (s) = d p (= p (s)) ds e 2 (s) e 1 (s) 90 e 1, e 2 e 1 = p (s) e 1 e 1 = 1 e 2 e 2 = 1 e 1 e 2 = 0 5 d dt (a b) = a b + a b 6 κ(s) { e 1 = κe 2 e 2 = κe 1 3 κ(s) p(s) (curvature) ( ) t 3 p(t) = 2t 3 r { x(t) = r cos t y(t) = r sin t 1 r 4 7 e 2 (s) p(s), p(s) κ(s), κ(s) p(s) p(s) s κ(s) = κ(s) ( ) { x = a cos t y = b sin t (a > b > 0) ab κ = (a 2 sin 2 t + b 2 cos 2 t) 3 2

4 4 4 ( ) (x(t), y(t)) = (cos 3 t, sin 3 t) (0 t π) 2 ( t t ). ( ) (x(t), y(t)) = cos au2 0 2 du, sin au2 0 2 du. ẋÿ ẍẏ ẋ = dx dt, ẍ = d2 x dt 2 5 b a κ(s) ds (ẋ 2 + ẏ 2 ) p + 1 e κ 2 c l c l x(s) p(s) = y(s) (a s b); z(s) s p (s) = dp (s) = 1) ds p (s) = dp ds = x (s) y (s) e 1 (s) z (s) 7 9 e 1 e 1 e 1(s) κ(s) p(s). 3.

5 5 κ(s) { 0 e 2 = 1 8 κ e 1 e 3 = e 1 e 2 e 2, e 3 e 1, e 2, e 3 e 1, e 2, e 3 Frenet (Frenet frame) 9 e 2 e e 2 + κe 1 e 3 e 2 + κe 1 = τe 3 τ τ (torsion) e 1 e 3, e 2 e 3, e 3 e 3 e 3 e 1 e 3 e 2 e 3 e 3 11 e 3 = τe 2 (Frenet-Serret ). e 1 e 2 e 3 p = e 1 e 1 = κe 2 e 2 = κe 1 + τe 3 e 3 = τe 2 0 κ 0 = κ 0 τ 0 τ 0. x = a cos t 5 y = a sin t κ (torsion) Frenet z = b t a cos t a sin t d p(t) = a sin t, p(t) = dt a cos t bt b t s = d 0 dt p(t) t dt = a2 + b 2 dt = a 2 + b 2 t 0 ds = a dt 2 + b 2 e 1 = dp ds = dt dp ds dt = 1 ( a sin t, a cos t, b) a2 + b2 e 1 = de 1 ds = dt de 1 ds dt = 1 ( a cos t, a sin t, 0) a 2 + b2 κ = e a 1 = a 2 + b 2 e 2 = 1 κ e 1 = ( cos t, sin t, 0) e 1 e 2 e 3

6 ( ) b b a e 3 = e 1 e 2 = sin t, cos t, a2 + b2 a2 + b2 a2 + b 2 e 2 = de 2 ds = dt ( ) de 2 ds dt = 1 a2 + b sin t, 1 2 a2 + b cos t, 0 ( 2 ) e b κe 1 = ( a 2 + b 2 ) sin t, b 2 3 ( a 2 + b 2 ) cos t, ab 3 ( a 2 + b 2 ) 3 τ = e b 2 + κe 1 = a 2 + b 2. t s x = a 2 t 6 a (a > 0) y = at2 κ 2 z = t3 6 4a. (2a 2 + t 2 ) 2 12 κ τ 0 p(s) 13 p(s) p(s) κ(s), τ(s), κ(s), τ(s) p(s) p(s) κ(s) = κ(s) τ(s) = τ(s) 1 p ṗ = dp dt, p = d2 p dt,... p = d3 p 2 dt 3 τ = κ = ṗ p ṗ 3 det(ṗ, p,... p) ṗ p a > 0 x 2 + z 2 = a 2, y = 0 p(s) (a s b) b a κ(s) ds 4.

7 7 12 ( ) u, v (x(u, v), y(u, v), z(u, v)) ) ( x u x v y u y v 2 z u z v (1) z = f(x, y) x = u y = v z = f(u, v) 6 (u, v, f(u, v)) F (x, y, z) = 0 z = f(x, y) (2) x 2 + y 2 + z 2 = a 2 (a > 0) (a) z = f(x, y) x = u 8 z > 0 y = v z = a 2 u 2 v 2 x = u z < 0 y = v z = a 2 u 2 v 2 z = 0 x = a 2 u 2 v 2 x = a 2 u 2 v 2 x > 0 y = u, x < 0 y = u, z = v z = v x = u y > 0 y = x = u a 2 u 2 v 2, y < 0 y = a z = v 2 u 2 v 2 z = v (b) x = a cos u cos v y = a cos u sin v z = a sin u

8 8 (3) (x, z) z x = f(u), z = g(u) z x cos v sin v 0 f(u) y = sin v cos v 0 0 z g(u) x = f(u) cos v y = f(u) sin v z = g(u) (a) x 2 + y 2 + z 2 = a 2 (a > 0) f(u) = a cos u, g(u) = a sin u (b) (torus) r < R f(u) = R + r cos u, g(u) = r sin u (4) x = f(u) cos v y = g(u) sin v z = h(u) (a) x2 + y2 + z2 = 1 a 2 b 2 c 2 f(u) = a cos u, g(u) = b cos u, h(u) = c sin u (b) x2 + y2 z2 = 1 a 2 b 2 c 2 f(u) = a cosh u, g(u) = b cosh u, h(u) = c sinh u (cosh x = ex +e x, sinh x = ex e x, cosh 2 x sinh 2 x = 1) 2 2 (c) x2 + y2 z2 = 1 a 2 b 2 c 2 f(u) = a sinh u, g(u) = b sinh u, h(u) = c cosh u

9 9 5. p = (x(u, v), y(u, v), z(u, v)) p u = ( x, y, ) z u u u, pv = ( x, y, z v v v) p u, p v. x, y, z 13 E, F, G E = p u p u ( ) F = p u p v G = p v p v ( E F ) F G 14 p(u, v) I = Edudu + 2F dudv + Gdvdv dp dt = p du u dt + p dv v dt = p du u dt + p dv v dt dt dp = p u du + p v dv. I = dp dp = (p u du + p v dv) (p u du + p v dv) = p u p u dudu + p u p v dudv + p v p u dvdu + p v p v dvdv = p u p u dudu + 2p u p v dudv + p v p v dvdv 15 e = p u p v p u p v 1 p uu = 2 p u, p 2 vv = 2 p v, p 2 uv = p u v. p u p v 2 = p u 2 p v 2 (p u p v ) 2, p uv = p vu 16 L, M, N L = p uu e M = p uv e

10 10 N = p vv e 14 II = Ldudu + 2Mdudv + Ndvdv L = p u e u M = p u e v = p v e u N = p v e v Proof. e p u p v p u e = 0 p v e = 0 p uu e + p u e u = 0 p uv e + p u e v = 0 p vu e + p v e u = 0 p vv e + p v e v = 0 L = p u e u, M = p u e v = p v e u, N = p v e v II = dp de = (p u du+p v dv) (e u du+e v dv) = Ldudu+2Mdudv +Ndvdv 15 λ L λe M λf M λf N λg = 0 17 κ 1, κ 2 K = κ 1 κ 2 Gauss H = 1 2 (κ 1 + κ 2 ) K = κ 2 1, H = κ 1 L λe M λf. M λf N λg

11 11 (EG F 2 )λ 2 (EN + GL 2F M)λ + LN M 2 = 0 λ 2 2Hλ + K = 0 16 K = κ 1 κ 2 = LN M 2 EG F 2 H = 1 2 (κ 1 + κ 2 ) = EN + GL 2F M 2(EG F 2 ) 7 : x2 + y2 + z2 a 2 b 2 c 2 = 1 p(u, v) = (a cos u cos v, b cos u sin v, c sin u) Gauss K H p u = ( a sin u cos v, b sin u sin v, c cos u) p v = ( a cos u sin v, b cos u cos v, 0) p uu = ( a cos u cos v, b cos u sin v, c sin u) p uv = p vu = (a sin u sin v, b sin u cos v, 0) p vv = ( a cos u cos v, b cos u sin v, 0) E = p u p u = a 2 sin 2 u cos 2 v + b 2 sin 2 u sin 2 v + c 2 cos 2 u F = p u p v = (a 2 b 2 ) sin u sin v cos u cos v G = p v p v = a 2 cos 2 u sin 2 v + b 2 cos 2 u cos 2 v p u p v = ( bc cos 2 u cos v, ac cos 2 u sin v, ab sin u cos u) = b 2 c 2 cos 2 u cos 2 v + a 2 c 2 cos 2 u sin 2 v + a 2 b 2 sin 2 u e = 1 ( bc cos u cos v, ca cos u sin v, ab sin u) L = abc M = 0 N = abc cos2 u K = LN M 2 EG F = a2 b 2 c EN + GL 2F M H = 2(EG F 2 ) = abc{(a2 + b 2 + c 2 ) (a 2 cos 2 u cos 2 v + b 2 cos 2 u sin 2 v + c 2 sin 2 u)} 2 3

12 12 K, H x, y, z K = a 1 a 2 b 2 c 2 ( x2 + y2 + z2 ) a 4 b 4 c 2 4 H = (a2 + b 2 + c 2 ) (x 2 + y 2 + z 2 ) 2a 2 b 2 c 2 ( x2 + y2 + z2 ) 3 a 4 b 4 c 4 2 K = 1 a 2, H = 1 a 9 x = cos u y = sin u z = v Gauss K, H 18 K > 0 (elliptic point) K < 0 (hyperbolic point), K = 0 (parabolic point) 17 z = f(x, y) x = u y = v z = f(u, v)

13 13 K H. p = (u, v, f) p u = (1, 0, f u ), p v = (0, 1, f v ) p uu = (0, 0, f uu ), p uv = (0, 0, f uv ), p vv = (0, 0, f vv ) p u p v = ( f u, f v, 1) e = ( f u, f v, 1) 1 + f 2 u + f 2 v E = p u p u = 1 + f 2 u F = p u p v = f u f v G = p v p v = 1 + f 2 v EG F 2 = 1 + fu 2 + fv 2 f uu L = p uu e = 1 + f 2 u + fv 2 M = p uv e = N = p vv e = f uv 1 + f 2 u + f 2 v f vv 1 + f 2 u + f 2 v LN M 2 = f uuf vv f 2 uv 1 + f 2 u + f 2 v EN + GL 2F M = f vv(1 + f 2 u) + f uu (1 + f 2 v ) 2f u f v f uv 1 + f 2 u + f 2 v K = f uuf vv f 2 uv (1 + f 2 u + f 2 v ) 2 H = f uu(1 + f 2 v ) 2f u f v f uv + f vv (1 + f 2 u) 2(1 + f 2 u + f 2 v ) 3 2 f uu f vv fuv 2 > 0 K > 0 f uu f vv fuv 2 < 0 K < 0 ( I )z = f(x, y) f x (0, 0) = 0, f y (0, 0) = 0 (1) f xx f yy f 2 xy > 0 (2) f xx f yy f 2 xy < 0 10 f(x, y) = x 3 3xy + y 3.

14 14 3 上の例は z = f (x, y) が極値を取る場合についてその近くでの曲面の形を表している更に曲率が回転や平行移動により変わらないことを用いると定理 17 を証明することができる 6. 主方向定義 19 Gauss 曲率や平均曲率を求めたときの κ1, κ2 を主曲率という κ1, κ2 は λ に関する方程式 L λe M λf =0 M λf N λg (1) の解であったこの解 λ に対して ( )( ) L λe M λf ξ =0 M λf N λg η ( ) ξ = 0 が存在するいま w = ξpu + ηpv とするとき w 方向を満たすベクトル η w を主方向 (principal direction) というの単位ベクトルすなわち ± w )( ) ( )( ) ( ξ E F ξ L M =λ M N η F G η 場合１方程式 (1) が重解を持たない場合 κ1 = κ2 のときそれぞれの κi について (i = 1, 2) ( )( ) L κi E M κi F ξi =0 ηi M κi F N κi G ( ) ξi i が存在する wi = ξi pu + ηi pv について ± w を満たす wi を主方向とした ηi

15 15 18 κ 1 κ 2 w 1 w 2 8 z = f(x, y), f(x, y) = x 3 3xy + y 3 (1, 1, 1) (1) κ 1 = κ 2 19 (umbilic point) 11 x2 + y2 + x2 = 1 (0, 0, c) a 2 a 2 c 2 7. p(u, v) e 1, e 2 ( e 1 (u, v), e 2 (u, v) u, v ) e 1 e 1 = e 2 e 2 = 1, e 1 e 2 = 0 20 e 3 = e 1 e ( 2 ) 1 1 a 1 a 2 21 A = 2 2 a 1 a 2 e 1 = p u p u, e 2 = p v (p v e 1 )e 1 p v (p v e 1 )e 1 { p u = a 1 1 e 1 + a 2 1 e 2 p v = a 1 2 e 1 + a 2 2 e 2. det A > 0 det A < 0 e 1 e 2 20 p u p v = (det A) e θ 1, θ 2 θ 1 = a 1 1 du + a 1 2 dv θ 2 = a 2 1 du + a 2 2 dv (*)

16 16 () dp := p u du + p v dv = (a 1 1 e 1 + a 1 2 e 2 )du + (a 2 1 e 1 + a 2 2 e 2 )dv = (a 1 1 du + a 2 1 dv)e 1 + (a 1 2 du + a 2 2 dv)e 2 = θ 1 e 1 + θ 2 e 2. e 1, e 2, e 3 e 1 u = k 1e 1 + k 2 e 2 + k 3 e 3 k 1, k 2, k 3 e 1 = h v 1e 1 + h 2 e 2 + h 3 e 3 h 1, h 2, h 3 de 1 := e 1 du + e 1 dv u v de 1 = (k 1 e 1 + k 2 e 2 + k 3 e 3 )du + (h 1 e 1 + h 2 e 2 + h 3 e 3 )dv = (k 1 du + h 1 dv)e 1 + (k 2 du + h 2 dv)e 2 + (k 3 du + h 3 dv)e 3. ω 1 1 = k 1 du + h 1 dv, ω 2 1 = k 2 du + h 2 dv, ω 3 1 = k 3 du + h 3 dv de 1 = ω 1 1 e 1 + ω 2 1 e 2 + ω 3 1 e 3 de 1 = ω 1 1 e 1 + ω 2 1 e 2 + ω 3 1 e 3 du dv ω 1 1, ω 1 2, ω 1 3 e 2, e 3 du dv ω 1 1, ω 1 2,, ω 3 3 de 1 = ω 1 1 e 1 + ω 2 1 e 2 + ω 3 1 e 3 de 2 = ω 1 2 e 1 + ω 2 2 e 2 + ω 3 2 e 3 de 3 = ω 1 3 e 1 + ω 2 3 e 2 + ω 3 3 e 3 21 d(a b) = da b + a db 22 ω i j = ω j i (ω 1 1 = ω 2 2 = ω 3 3 = 0) (*) { du = 1 a 1 1 a 2 2 a 2 1 a 1 2 (a 2 2 θ 1 a 1 2 θ 2 ) dv = 1 a 1 1 a 2 2 a 2 1 a 1 2 ( a 2 1 θ 1 + a 1 1 θ 2 ) ω 1 3 ω 2 3 du, dv { ω 1 3 = b 11 θ 1 + b 12 θ 2 b 11, b 12, b 21, b 22. ( ) b 11 b 12 B = b 21 b 22 ω 2 3 = b 21 θ 1 + b 22 θ 2

17 17 23 B ( ) L M L = p u e u, M = p u e v, N = p v e v S = M N S = t ABA 13 x = cos u, y = sin u, z = v e 1 = ( sin u, cos u, 0), e 2 = (0, 0, 1), e 3 = (cos u, sin u, 0) B I = dp dp = (θ 1 e 1 + θ 2 e 2 ) (θ 1 e 1 + θ 2 e 2 ) = (θ 1 ) 2 + (θ 2 ) 2 II = dp de 3 = θ 1 ω θ 2 ω 3 2 B κ 1, κ 2 Gauss det B = κ 1 κ 2, 1 2 trace B = 1 2 (κ 1 + κ 2 ) 14 x = cos u, y = sin u, z = v Gauss Gauss K 0 (flat) Gauss 0 25 H 0 ( (right helicoid)) x = u cos v y = u sin v z = av + b (a, b )

18 18 9. (1) R 2 26 R 2 1 (1 form) R 2 p X ω : R 2 R X, Y k, l ω(kx + ly ) = k ω(x) + l ω(y ) ( ) a 27 p X = a b dx, dx ( ) ( ) a a dx( ) = a, dy( ) = b b b 28 R 2 f, g f dx + g dy ( ) ( ( )) ( ( )) a a a (fdx + gdy) = f dx + g dy b b b = f a + g b 15 1 ω = x 2 dx + xy dy ( ) v = 2 1 ω(v) ( ) R 2 2 (2 form) R 2 p X, Y ω : R 2 R 2 R ω(ax + by, X 2 ) = a ω(x, X 2 ) + b ω(y, X 2 ) ω(x 1, cx + dy ) = c ω(x 1, X) + d ω(x 1, Y ) ω(x 1, X 2 ) = ω(x 2, X 1 ) 30 R 2 0 form R 2

19 19 (2) 31 1 form ω 1, ω 2 2 form ω 1 ω 2 ω 1 ω 2 (X 1, X 2 ) = ω 1 (X 1 )ω 2 (X 2 ) ω 1 (X 2 )ω 2 (X 1 ) 26 ω 1 ω 2 = ω 2 ω 1 ω ω = 0, dx dx = 0, dy dy = 0, dx dy = dy dx (3) 32 0 form f df df = f f dx + x y dy 1 form ω = f dx + g dy dω = df dx + dg dy = ( g x f ) dx dy y 2 form ω = f dx dy 0 dω = form f, g d(fg) = df g + f dg 27 f d d f = 0 28 ( ) 1 form φ = f(u, v)du + g(u, v)dv a u b, c v d dφ = 0 a u b, c v d h φ = dh

20 20 (4) form 33 R I = [a, b] 1 form ω = f(x)dx ω I a I b ω = f(x)dx x = g(t) dx = dg = dg dt dt I ω = g 1 (b) g 1 (a) f(g(t)) dg dt dt 34 a x b, c y d 2 form f(x, y)dxdy f(x, y) dx dy { x = φ(s, t) y = ψ(s, t) ( x y f(φ(s, t), ψ(s, t)) s t x t ) y s ds dt 29 ( ) S κ θ 1 θ 2 = A + B + C π S S r κ = 1 r 2, 1 (S ) = A + B + C π r2 S θ 1 θ 2 = (S )

( ) ( )

( ) ( ) 20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))