õ¼ê 4 á ëY5 §9.2 õ¼ê9ÙëY5 9.2.1 õ¼ê, þ¼ê ¼êÒ´þþm«'X (½«éA). ±ò¼êw¤´N. 1 ◦ N kü8Ü X, Y, 95K f. e?¿½ X ¥ x, ѱUì5K f, é Y ¥(½ y (P¤ y = f(x)) x éA, K¡ f ´ X Y N, P¤ f : X → Y. y ¡´ x 3 f e, x ¡ y . P f(X) = {f(x)| x ∈ X} ¡ X 8. w, f(X) ⊂ Y. 1/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 XJN´ü, =, k, @o±½Âl f(X) X Nµ f −1 : f(X) → X. §N5K f −1 ´: ? y ∈ f(X), 7k x ∈ X ¦ y = f(x). ù x Ò´ y 3 f −1 e. =´`XJ y = f(x), K x = f −1 (y). ù¡ f _N, f −1 ¡ f _N. §u¼ê¼ê. X Y f 2/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 2 ◦ õ¼ê D ⊂ Rn. Kl D ¢ê8 R N f : D → R ¡ n ¼ê. Ï Rn ¥:kI (x1, x2, · · · xn), ¤±±¤ z = f(x1, · · · , xn). Ï Rn :éA n þ ~x = x1~e1 + · · · + xn~en, ¤±õ¼ê´þêþN (éA'X), ùp ~e1, · · · , ~en Rn Iþ. AO, n = 2 , Ò´¼ê z = f(x, y), (x, y) ∈ D ⊂ R 2 . r z ¤3ê¶: (x, y) ²¡ R2 :3å, ¿Ru²¡, ù·Ò k nm, :8 E = {(x, y, f(x, y))| (x, y) ∈ D} Ò´Üm¡. ùÜ¡w¤kn gdÝnm¥: (x, y, z) É å z = f(x, y) ¤/¤. x y z D E 3/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 3 ◦ þ¼ê þ¼ê´l n m Rn m m Rm N f~ : D → R m, D ⊂ R n = ~x = x1~e1 + · · · + xn~en 7−→ ~y = f~(~x) = y1~e1 + · · · + ym~em ½öP (x1, x2, · · · , xn) 7−→ (y1, y2, · · · , ym), ùp y1 = f1(x1, · · · , xn), · · · ym = fm(x1, · · · , xn) Ñ´ n ¼ê, yi ¡þ¼ê1 i ©þ¼ê. m = n , þ ¼ê¡C. 4/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 XJ©þ¼êÑ´5, =, y1 = a11x1 + a12x2 + · · · + a1nxn, y2 = a21x1 + a22x2 + · · · + a2nxn, · · · · · · ym = am1x1 + am2x2 + · · · + amnxn, Ù¥ aij Ñ´~ê, KN´5, A = (aij)m×n = a11 a12 · · · a1n a21 a22 · · · a2n . . . . . . . . . . . . am1 am2 · · · amn Ò´5NÝ . P x = (x1, x2, · · · , xn) T , y = (y1, y2, · · · , ym) T , (ù p T L«=), K5N±{üL y = Ax. 5/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 9.2.2 õ¼ê4 z = f(x, y) ´½Â3²¡:8 D þ¼ê, M0 = (x0, y0) ´ D à:, XJ3¢ê a, ¦é?Û ε > 0, 3 δ > 0, 0 < ρ(M, M0) < δ, M = (x, y) ∈ D , k |f(M) − a| < ε, Ò` M ªu M0 f(M) ± a 4, P¤ lim M→M0 f(M) = a. ±¤ lim (x,y)→(x0,y0) f(x, y) = a ½ lim x→x0 y→y0 f(x, y) = a. ùÒ´¼ê4½Â. l¥wÑ, ´r¼ê4½Â¥ ýé£þ:ål¤¤²¡þ:ål=. n ¼ê4 ½Â´aq. 6/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 ¼ê4±¡4. XJ^/Oþ0ó (= ∆x, ∆y ´üOþ), K¼ê3 M0 = (x0, y0) 4Ò´ lim ∆x→0 ∆y→0 f(x0 + ∆x, y0 + ∆y) = a Ï ρ = p ∆x2 + ∆y2 ÷v |∆x|, |∆y| 6 ρ 6 |∆x| + |∆y| ¤±, 4±L«¤ lim ρ→0 f(x0 + ∆x, y0 + ∆y) = a. ,¼ê4½Â3/ªþ¼ê, ´ÏêO \ , ¼ê4?Øå5E, . ~X, ØÐ?ؼêü N5£Ï²¡¥:vk/S0¤±9¼êm4¯K. k3¼ ê4L§¥, ügCþCzªäkgdÝ, §3 M0 NC±^?¿ª£!Ú^! ¤C M0. 7/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 ~ 1 ¦y lim (x,y)→(0,0) x 2y 2 x4+y2 = 0. y² |^Ø ª |2x 2y| 6 x 4 + y 2 0 6 x 2y 2 x4 + y2 6 1 2 |y|, ¤±, é?¿ ε > 0, δ = 2ε, 0 < |x| < δ, 0 < |y| < δ , k x 2y 2 x4+y2 < 1 2 δ < ε. â4½Â lim (x,y)→(0,0) x 2y 2 x4 + y2 = 0. 8/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 ~ 2 ¦y lim (x,y)→(0,0) x 2y x4+y2 Ø3. y² y = kx2£(x, y) ²¡¥Ô¤, Kk lim x→0 y=kx2 x 2y x4 + y2 = lim x→0 kx4 x4 + k2x4 = k 1 + k2 , = (x, y) ÷XØÓÔª (0, 0) , ¼êkØÓ4, ¤±4Ø 3. ´ lim y→0 lim x→0 x 2y x4 + y2 = 0, lim x→0 lim y→0 x 2y x4 + y2 = 0. ù«k4, 2,4{\g4. 4Ú\g4´ üØÓVg, §35vk7,%¹'X, ´4Ú\g4 Ñ3, §7, . 9/15 kJ Ik J I £ ¶ '4 òÑ
õ¼ê 4 á ëY5 9.2.3 á¼ê XJ x → x0, y → y0 , ¼ê f(x, y) 4´", K¡¼ê f(x, y) ´ x → x0, y → y0 áþ. d, XJéu¿© ρ = p (x − x0) 2 + (y − y0) 2 , k f(x, y) ρ 6 C K¡ f(x, y) x → x0, y → y0 ( du ρ → 0) ρ kÓ , P f(x, y) = O(ρ). XJ lim ρ→0 f(x, y) ρk = C£"~ê¤ K¡ f(x, y) ρ → 0 ´ ρ k Ó Ã¡þ, AO C = 1 , P f(x, y) ∼ ρ k . XJ4´", K` f(x, y) ρ → 0 ´' ρ k p à ¡þ, P f(x, y) = o(ρ k ). Ó, ·±?Øü4"¼êm ', ù Vg¼êaq, Ø2Kã. 10/15 kJ Ik J I £ ¶ '4 òÑ