,.,. 2, R 2, ( )., I R. c : I R 2, : (1) c C -, (2) t I, c (t) (0, 0). c(i). c (t)., c(t) = (x(t), y(t)) c (t) = (x (t), y (t)) : (1)

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1 ( ) 1., : ;, ;, ; =. ( ).,.,,,., 2.,.,,.,.,,., y = f(x), f ( ).,,.,.,., U R m, F : U R n, M, f : M R p M, p,, R m,,, R m A tamaru math.sci.hiroshima-u.ac.jp 1

2 ,.,. 2, R 2, ( )., I R. c : I R 2, : (1) c C -, (2) t I, c (t) (0, 0). c(i). c (t)., c(t) = (x(t), y(t)) c (t) = (x (t), y (t)) : (1) c : R R 2 : t (cos(t), sin(t)), (2) c : R R 2 : t t a + b ( a 0) : (1) c : R R 2 : x (x, x ), (2) c : R R 2 : t (t 2, t 3 ) ( t > 0, ) I R. C f : I R, (1) {(x, f(x)) x I} y = f(x), (2) {(f(y), y) y I} x = f(y) C f : I R,., y = f(x) ( x = f(y)). 2.5,.,, c : I R 2,,, : t I, I 0 (t ) : c(i 0 ). 2

3 , f. f,., c : R R 2 : t (a 1 t + b 1, a 2 t + b 2 ) ( (a 1, a 2 ) (0, 0)), : (1) a 1 0, y = (a 2 /a 1 )x + b 1 (a 2 b 1 /a 1 ), (2) a 2 0, x = (a 1 /a 2 )y + b 2 (a 1 b 2 /a 2 ) U R 2, F : U R., F (x, y) = 0, : (1) F C -, (2) F (x, y) = 0 (JF ) (x,y) (0, 0). (JF ) (x,y) Jacobi : 2.9. F (x, y) = x 2 + y 2 1. (JF ) (x,y) := ( F F (x, y), (x, y)). x y a > 0, F (x, y) = y 2 x 2 (x + a) C - f : I R y = f(x), F : I R R : (x, y) y f(x), F (x, y) = 0.,, F : U R F (x, y) = 0,., : (x 0, y 0 ) (F (x 0, y 0 ) = 0), U U ((x 0, y 0 ) ) : {(x, y) U F (x, y) = 0}., f. f,., F (x, y) = 0 y = f(x) x = f(y). 2.4, c : I R 2 I,,., c. 3

4 2.13. M. C - γ : ( ε, ε) M t = 0 γ (0), M γ(0) M. γ : ( ε, ε) M C -, R 2 γ : ( ε, ε) R 2 C -., p x 2 + y 2 = 1 : a R, (0, a) p = (1, 0) T p M := {p + u u p M }, M p M., p,.,.,, (, I,, ) M, c : I R 2, F (x, y) = 0., p = c(t 0 ), : {p + sc (t 0 ) s R} = T p M = {p + u (JF ) p u = 0}. R 2, p R 2 T p R 2 := p + R 2 = {p + u u R 2 }, T p M T p R 2. ; T p R 2 p 2,, T p M 1 ( )., Jacobi (JF ) p, p y = f(x) (x 0, f(x 0 )) y = f (x 0 )(x x 0 ) + f(x 0 ) x-, C - c : R R 2, c(r) x-, M., M θ (a, b) M, a > 0. y 2 x 2 (x + a) = , 2.16 Jacobi. 4

5 3, R 3, ( ).,., D R 2. p : D R 3, : (1) p C -, (2) (u, v) D, rank(jp) (u,v) = 2. (Jp) (u,v), p (u, v) Jacobi. p(u, v) = (x(u, v), y(u, v), z(u, v)), p u, p v, Jacobi (Jp) (u,v) = (p u, p v ) (u,v) = x u x v., rank(jp) (u,v) = 2, {p u (u, v), p v (u, v)} : y u z u y v z v (u,v) (1) p : R 2 R 3 : (u, v) u a + v b + c ( a b ). (2) p : R 2 R 3 : (u, v) (cos u, sin u, v).,., xy- z-, c : I R 2 : t c(t) = (x(t), y(t))., p : I R R 3 : (u, v) (x(u), y(u), v).,, xz- z-, c : I R 2 : t c(t) = (x(t), z(t)). x(t) > 0 ( t I), p : R I R 3 : (u, v) (cos(u)x(v), sin(u)x(v), z(v)). p(u, v), xz- c, z u : cos(u) sin(u) 0 p(u, v) = sin(u) cos(u) 0 x(v) z(v) p(u, v). 5

6 3.5. : (1) ( ) p : R (0, π) R 3 : (u, v) (cos(u) cos(v), sin(u) cos(v), sin(v)), (2) p : R 2 R 3 : (u, v) (cos(u)(2 + cos(v)), sin(u)(2 + cos(v)), sin(v)). p : D R 3, p(d) (, x 2 + y 2 + z 2 = 1 )., ( ). 3.2, C f : D R, p : D R 3 : (u, v) (u, v, f(u, v)) C f : D R, {(x, y, f(x, y)) R 3 (x, y) D} z = f(x, y)., x = f(y, z) y = f(x, z)., z = f(x, y). 3.6,.,, p : D R 3,,, : (u, v) D, D 0 ((u, v) ) : p(d 0 ).,, (2, ) p : R 2 R 3 : (u, v) (a 1 u + b 1 v + c 1, a 2 u + b 2 v + c 2, a 3 u + b 3 v + c 3 ) ( ) a1 b rank 1 = 2 a 2 b 2, z = αx + βy + γ. 3.3, U R 3, F : U R., F (x, y, z) = 0, : (1) F C -, (2) F (x, y, z) = 0 (JF ) (x,y,z) (0, 0, 0). 6

7 3.11. F (x, y, z) = x 2 /a 2 + y 2 /b 2 + z 2 /c 2 1, F (x, y, z) = 0 ( a, b, c > 0) C - f : D R z = f(x, y), F : D R R : (x, y, z) z f(x, y), F (x, y, z) = 0.,, F (x, y, z) = 0,.,, : (x 0, y 0, z 0 ) (F (x 0, y 0, z 0 ) = 0), U ((x 0, y 0, z 0 ) ) : {(x, y, z) U F (x, y, z) = 0},.,., F (x, y, z) = 0, z = f(x, y) F (x, y, z) = ax + by + cz + d. c 0, : z = (1/c)(ax + by + d). 3.4, M = p(d) p : D R 3 D,,., p M. C - γ : ( ε, ε) M t = 0 γ (0), M γ(0) M. γ : ( ε, ε) M C -, R 2 γ : ( ε, ε) R 2 C x 2 + y 2 + z 2 = 1, : a, b R, (0, a, b) p = (1, 0, 0) T p M := {p + u u p M }, M p M., p,. M ( ),,. 7

8 3.18. M, p : D R 3, F (x, y, z) = 0., p 0 = p(u 0, v 0 ), : {p 0 + ap u (u 0, v 0 ) + bp v (u 0, v 0 ) a, b R} = T p0 M = {p 0 + w (JF ) p0 w = 0}., T p0 M p 0 2. Jacobi (JF ) p0,, ( )., Jacobi (JF ) p0, p z = f(x, y) (x 0, y 0, f(x 0, y 0 )) : z = f u (x 0, y 0 )(x x 0 ) + f v (x 0, y 0 )(y y 0 ) + f(x 0, y 0 ). 4, ( ).,. 4.1,, M = (M, O), 2,, : x, y M (x y), O x, O y O : x O x, y O y, O x O y =., R n, M, {(U α, ϕ α )} n : (1) {U α } M, (2) α, ϕ α : U α ϕ α (U α ) R n, (3) U α U β, ϕ β ϕ 1 α : ϕ α (U α U β ) ϕ β (U α U β ) C -., U α R n ( (2)), M ( (1)). (3),. (U α, ϕ α ), (3) ϕ β ϕ 1 α M n, {(U α, ϕ α )}., C -. 8

9 (U α, ϕ α ),.,,., S 1 := {(x, y) R 2 x 2 + y 2 = 1},, S 1 1., c : R R 2 : t (cos(t), sin(t)) {(U 1, c 1 1 ), (U 2, c 1 2 )} : c 1 := c (0,2π), c 2 := c (π,3π), U 1 := Im (c 1 ), U 2 := Im (c 2 ). 4.2, D R m, f : D R n C -. f : graph(f) := {x = (x i ) D R n f(x 1,..., x m ) = (x m+1,..., x m+n )}. m = n = 1, graph(f) y = f(x) C - f : D R m+n, ϕ : graph(f) D : ϕ(x 1,..., x m+n ) := (x 1,..., x m ). ϕ f,., ϕ ϕ 1, D R m, f : D R m+n C -, graph(f) m. 4.6 ϕ, (graph(f), ϕ).,.,, {(U x>0, ϕ x>0 ), (U x<0, ϕ x<0 ), (U y>0, ϕ y>0 ), (U y<0, ϕ y<0 )}, S 1 : U x>0 := {(x, y) S 1 x > 0}, ϕ x>0 : U x>0 ( 1, 1) : (x, y) y, U x<0 := {(x, y) S 1 x < 0}, ϕ x<0 : U x<0 ( 1, 1) : (x, y) y, U y>0 := {(x, y) S 1 y > 0}, ϕ y>0 : U y>0 ( 1, 1) : (x, y) x, U y<0 := {(x, y) S 1 y < 0}, ϕ y<0 : U y<0 ( 1, 1) : (x, y) x. (, U x>0 x = 1 y 2 ), 4.6,. C -, S 2 := {(x, y, z) R 2 x 2 + y 2 + z 2 = 1} 2., S

10 4.3,,., n, U R 2, F : U R, F (x, y) = 0., M := {(x, y) U F (x, y) = 0} 1. ; 2.12,., p M, p U p. U p, ϕ p : U p I p (I p R ). {(U p, ϕ p ) p M}, M U R 3, F : U R, F (x, y, z) = 0., M := {(x, y, z) U F (x, y, z) = 0} 2.,. 3.13,,.,, U R m, F : U R n C -, M := {p U F (p) = 0}. rank(jf ) p = k ( p M), M (m k) (m, n, k) = (2, 1, 1), 4.11 (m, n, k) = (3, 1, 1),., S n := {(x 1,..., x n+1 ) R n+1 x x 2 n+1 = 1} n. S n n., F (x 1,..., x n+1 ) = x x 2 n+1 1 F : R n+1 R SL 2 (R) := {X M 2 (R) det(x) = 1} 3. M n (R) n n., M 2 (R) R 4, 4.12., : (1) SL n (R) := {X M n (R) det(x) = 1} n 2 1. (2) O(n) := {X M n (R) t XX = I n } n(n 1)/2. SL n (R) n, O(n) n., ( ). 10

11 4.4 R n,, R n.,., RP n := {l R n+1 l 1 } n. RP n, RP n X/., X := R n+1 \ {0}, : x y λ R \ {0} s.t. x = λy.,. RP n, X = R n+1 \ {0} R n+1, X/ π : X X/ RP n., x = (x 0, x 1,..., x n ) X π π(x) = [x 0 : x 1 : : x n ] RP n n. {(U i, ϕ i ) i = 0,..., n} : U i := {[x 0 : : x n ] RP n x i 0}, ϕ i : U i R n : [x 0 : : x n ] (1/x i )(x 0,..., x i,..., x n )., x i x i. 4.5, (, ).,.,, M, N, f : M N. (1) f p M C -, : (U, ϕ) : p M, (V, ψ) : f(p) N s.t. ψ f ϕ 1 ϕ(p) C -. (2) f C -, : p M, f p C -. ψ f ϕ 1 ϕ(p) (ϕ(u) ). 11

12 4.21. C - : (1) f : R S 1 : t (cos(t), sin(t)), (2) f : S n RP n : x Rx. C -, C, C M, N, f : M N. f p M C -, : (U, ϕ) : p M, (V, ψ) : f(p) N, ψ f ϕ 1 ϕ(p) C M N, f : M N., f : M N, : f, f f 1 C -. ( ), r > , M m, U M, U m., M m, N n, M N m + n. M N : (1) R 2, R R, (2) {(x, y, z) R 3 x 2 + y 2 = 1}, S 1 R., (covering map) E, X. π : E X, : x X, U (x ) s.t. U λ (π 1 (U) ), π Uλ : U λ U., E X. 12

13 4.29. R x y x y Z, R/Z := R/., π : R R/Z E m, X, π : E X. U π 1 (U), X m., π : E X. X. π : E X, : p E, V (p ), U (π(p) ) s.t. π : V U , : (1) R/Z S 1, (2) R 2 /Z {0} S 1 R, (3) R 2 /Z 2 S 1 S ,,.,,. M C - C (M) C - c : ( ε, ε) M t = 0 ċ(0) : ċ(0) : C (M) R : f d (f c)(0). dt ċ(0) d c(0). dt p M, T p M,, T p M := {ċ(0) c : ( ε, ε) M : C, c(0) = p}. M n, T p M n., : M n, (U, ϕ) p M, ϕ = (x 1,..., x n )., : ( ) ( ) span{,..., } = T p M = {v : C (M) R : }. x 1 p x n p,, C (M) (, )., v(fg) = v(f)g(p) + f(p)v(g) ( f, g C (M))., (av 1 + bv 2 )(f) = av 1 (f) + bv 2 (f),. 13

14 4.35. M n, T p M n., n, : ( ) ( ) {,..., }. x 1 p x n p, M = R m, T p R m R m : T p R m ( ) a i (a 1,..., a m ) R m. x 1 p 4.8 F : M N C -., C - F : M N p M : (df ) p : T p M T F (p) N : ċ(0) d (F c)(0). dt., C - c : ( ε, ε) M, (U, ϕ) p = c(0), ϕ = (x 1,..., x n ). Jacobi, : ċ(0) = ( ) a i ( ) a 1,..., a n = t J(ϕ c) 0. x i., f C (M), : ċ(0)f = ( ( ) a i )f. x i p, : ( ( ) a i )f = a i (f ϕ 1 )(ϕ(p)). x i p x i,,, p ċ(0)f = d dt (f c)(0) = J(f ϕ 1 ϕ c) 0 = J(f ϕ 1 ) ϕ(p) J(ϕ c) 0. J(f ϕ 1 ) =,. ( ) (f ϕ 1 )(ϕ(p)),..., (f ϕ 1 )(ϕ(p)) x 1 x n 14

15 , c., (df ) p., F : M N C -, p M, (U, ϕ) p, (V, ψ) F (p). ϕ = (x 1,..., x m ), ψ = (y 1,..., y n ), : (df ) p ( ( ) a i ) = ( ) b j x i p y j F (p). M c, : ċ(0) = a i ( x i,, ( ) bj = (df ) p ( ( a i y j x i F (p) ) 1 b. = J(ψ F ϕ 1 ) ϕ(p) b n ). p ) = (df ) p (ċ(0)) = d (F c)(0). p dt a 1. a m. c F c 4.37, ( a1,..., a m ) = t J(ϕ c) 0, ( b1,..., b n ) = t J(ψ F c) 0.,, J(ψ F c) 0 = J(ψ F ϕ 1 ϕ c) 0 = J(ψ F ϕ 1 ) ϕ(p) J(ϕ c) 0.,. M = R m, N = R n, F : R m R n ( ). 15

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) =

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) = 1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,