9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x

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Download "9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x"

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2 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin z z = x 2 y 3 g(x, y) = sin x 2 y 3 x y 3 x 2 y x 2 y 3 g x (x, y) = 2xy3 cos x 2 y 3 g y (x, y) = 3x2 y 2 cos x 2 y C or 1 1 f(x, y) g(x, y) y b x 1 φ(x) = 1 z = f(x, y) (x, y) = (g 1 (z), g 2 (z)) f(g 1 (z), g 2 (z)) = z g 1 (f(x, y)) = x, g 2 (f(x, y)) = y (x, y) = (g 1 (z), g 2 (z)) z f(x, y)

3 9 3 f(x, b), ψ(x) = g(x, b) f g fg (a, b) x (fg) x (a, b) = d(φψ) dφ (a) = dx dx = x (x)ψ(a) + φ(a)dψ dx (a) g (a, b)g(a, b) + f(a, b) (a, b) x 1 x f x, g x (a, b) 1 (fg) x = f x g + fg x 1/f (a, b) x φ(x) = f(x, b) x ( 1 f ) (a, b) = d dx ( 1 φ ) (a) = 1 (φ(a)) 2 dφ dx (a) = 1 (a, b) (f(a, b)) 2 x 1 ( ) 1 = f x f f 2 x

4 9 4 1 f(x) g(x) g(x) f(x) g(x) f(x) x x g g y = g(x) f(y) (x, y), (u, v), (s, t) (u, v) (s, t) s t 2 1 s x ξ y η x ξ(s) ξ(s, t) y η(s) η(s, t) ξ η f(x) 1 ξ(s) 1 s f(ξ(s)) s ξ ξ(s) f f(ξ(s)) 1: 1 f(ξ(s))? f(ξ(s)) f(x) (x) f f f(x) x = ξ(s) f(ξ(s)) fξ(s) f ξ

5 9 5 f ξ f ξ(s) = f(ξ(s)) ( 2) f φ f ξ s ξ f f(ξ(s)) 2: sin f ξ f ξ f ξ f ξ R R f ξ : R ξ f R R 1 f ξ f(ξ(s)) f ξ d(f ξ) (s) = df ds dx (ξ(s))dξ ds (s) (f ξ) (s) = f (ξ(s))ξ (s) s ( ) d(f ξ) df = ds dx ξ dξ (f ξ) = (f ξ) ξ ds 2 2 (s) (s)

6 9 6. ξ(s) x d(f ξ) (s) = df ds dx (x)dξ ds (s) (f ξ) (s) = f (x)ξ (s) s x x = ξ(s) d(f ξ) ds = df dξ dx ds (f ξ) = f ξ f x f x ξ s ξ s f ξ f f x df/ds f ξ ξ x ξ x x x(s) df ds = df dx dx ds df dx 31. φ(s) = sin(e s ) f ξ η 2 f(x) x = ξ(s, t) 1 1 f ξ(s, t) = f(ξ(s, t)) s, t f ξ : R 2 ξ f R R

7 9 7 s t ξ ξ(s, t) f f(ξ(s, t)) 3: 2 1 s t ξ f ξ f f(ξ(s, t)) 4: (f ξ) (s, t) = df (ξ(s, t)) ξ s dx s (s, t) (f ξ) t (s, t) = df dx (ξ(s, t)) ξ (s, t) t (f ξ) s (s, t) = f (ξ(s, t))ξ s (s, t) (f ξ) t (s, t) = f (ξ(s, t))ξ t (s, t) ( ) (f ξ) df = s dx ξ ξ (f ξ) s t = ( ) df dx ξ ξ t (f ξ) s = (f ξ) ξ s (f ξ) t = (f ξ) ξ t x = ξ(s, t) x s, t (f ξ) (s, t) = df s dx (x) ξ s (s, t) (f ξ) (s, t) = df t dx (x) ξ (s, t) t (f ξ) s (s, t) = f (x)ξ s (s, t) (f ξ) t (s, t) = f (x)ξ t (s, t)

8 9 8 (f ξ) s = df dx ξ s (f ξ) t = df dx ξ t (f ξ) s = f ξ s (f ξ) t = f ξ t 1 f 1 df/dx f 1 2 s t s b ξ f = s b ξ f 5: t = b t = b. 1 f ξ f ξ x s s = df x dx s d 32. φ(s, t) = sin(s 2 t) (1) 1 (2) φ(s, t) f(x) = sin x ξ(s, t) = s 2 t (1)

9 f 2 f(x, y) 1 ξ(s), η(s) 6 s s ξ ξ(s) f f(ξ(s), η(s)) s η η(s) 6: 2 1. ξ(s), η(t) 1 f(ξ(s), η(t)) x, y s, t ξ(s) s 1 s t 2 t f(ξ(s), η(t)) s t x s x y R ξ η R 2

10 9 10 s ξ f f(ξ(s), η(s)) η 7: f ξ φ(s) = f(ξ(s), η(s)) φ f 3 dφ ds (s) = x (ξ(s), η(s))dξ ds (s) + (ξ(s), η(s))dη y ds (s) φ (s) = f x (ξ(s), η(s))ξ (s) + f y (ξ(s), η(s))η (s) ξ η 1 dξ/ds ξ 1 (s) x = ξ(s), y = η(s) dφ ds = dξ x ds + dη y ds φ = f x ξ + f y η x y s s x y 3 physics photograph ph

11 9 11 f x 1 f y 1. (x, y) f(x, y) t (ξ(t), η(t)) t φ(t) φ f ξ η x, y df dt = dx x dt + dy y dt f = xẋ + y ẏ ẋ, ẏ, f x y 33. φ(s) = e s sin s (1) s 1 (2) φ(s) f(x, y) = xy ξ(s) = e s η(s) = sin s (1) s s f ξ η φ φ(s, t) = f(ξ(s, t), η(s, t))

12 9 12 s s ξ s t ξ(s, t) η(s, t) f f(ξ(s, t), η(s, t)) η t t 8: 2 φ s φ t (s, t) = x (s, t) = x (ξ(s, t), η(s, t)) ξ s (ξ(s, t), η(s, t)) ξ t (s, t) + y (s, t) + y (ξ(s, t), η(s, t)) η (s, t) s (ξ(s, t), η(s, t)) η (s, t) t φ s (s, t) = f x (ξ(s, t), η(s, t))ξ s (s, t) + f y (ξ(s, t), η(s, t))η s (s, t) φ t (s, t) = f x (ξ(s, t), η(s, t))ξ t (s, t) + f y (ξ(s, t), η(s, t))η t (s, t) x = ξ(s), y = η(s) φ s = ξ x s + η y s φ t = ξ x t + η y t φ s = f x ξ s + f y η s φ t = f x ξ t + f y η t s

13 9 13 s ξ f f(ξ(s, t), η(s, t)) η t 9: 2 1 t t s x y f s x y. f s, t s 1 f x ξ x s 1 d f y η y s + ξ x s + η y s

14 9 14 f s f f x x x ξ s y y η s y ξ x s + η y s 34. φ(s, t) = ( sin(st) )( cos(s 2 + t 2 ) ) (1) 1 (2) φ(s, t) f(x, y) = (sin x)(cos y) ξ(s, t) = st η(s, t) = s 2 + t 2 (1) f m m n f(x 1, x 2,..., x m ) x i = ξ i (s 1, s 2,..., s n ) s 1, s 2,..., s n φ φ(s 1, s 2,..., s n ) = f(ξ 1 (s 1, s 2,..., s n ), ξ 2 (s 1, s 2,..., s n ),..., ξ n (s 1,..., s n )) φ s i φ = ξ 1 + ξ ξ m m ξ j = (1) s i x 1 s i x 2 s i x m s i x j s i? j=1

15 chain rule chain (1) s f s f 1 1 (f ξ) = (f ξ) ξ (f 4 f 3 f 2 f 1 ) = (f 4 f 3 f 2 f 1 ) (f 3 f 2 f 1 ) (f 2 f 1 ) f 1? (f 4 f 3 f 2 f 1 ) = f 4 f 3 f 2 f 1 1 1

16 ! (4) 4 a f (x 0 ) 1 p(x) = f(x 0 ) + a(x x 0 ) f(x) p(x) = 0 (2) x x 0 x x

17 9 17 f(x) x = x 0 (2) 1 p(x) f x 0 p(x) = f(x 0 ) + f (x 0 )(x x 0 ) p(x) f(x) x = x f(x) x = ξ(s) f ξ(s) s = s 0 1 f(x) x 0 = ξ(s 0 ) ξ(s) s = s 0 f(x) x 0 1 ξ(s) s 0 1 p(x) = f (x 0 )(x x 0 ) + f(x 0 ) ϖ(s) = ξ (s 0 )(s s 0 ) + x 0 5 f ξ s 0 1 f x 0 = ξ(s 0 ) ξ s 0 f ξ s 0 f ξ s 0 1 P (s) P (s) = (f ξ)(s 0 ) + (f ξ) (s 0 )(s s 0 ) f ξ(s) = f(ξ(s)) p(x) x = ϖ(s) p ϖ(s) f ξ(s) s p(x) p polynomial ϖ p π ϖ π

18 9 18 p ϖ(s) p ϖ(s) = f (x 0 )(ϖ(s) x 0 ) + f(x 0 ) = f(x 0 ) + f (x 0 )(x 0 + ξ (s 0 )(s s 0 ) x 0 ) = f(x 0 ) + f (x 0 )ξ (s 0 )(s s 0 ) f ξ 1 P (s) f 1 ξ 1 p ξ(s) (f ξ) (s 0 ) = f (x 0 )ξ (s 0 ) 1 1 f ξ(s) p ϖ(s) = 0 (3) s s 0 s s 0 p ϖ(s) f ξ(s) s = s P (s) = p ϖ(s) f ξ s 0 (3) f ξ s 0 1 (3) (3) x = ξ(s) s = s 0 x = ϖ(s) x = ξ(s) y = f(x) y = f(ξ(s)) = f ξ(s) s = s 0 x = ϖ(s) y = p(x) 1 y = p(ϖ(s)) = p ϖ(s) x = ξ(s) 1 y = p(x) y = p(ξ(s)) = p ξ(s) 6 6 x = ξ(s) y = p(x) x = ϖ(s) y = f(x) ϖ ξ 1

19 9 19 y = f ξ(s) s = s 0 y = p ϖ(s) y = f ξ(s) s = s 0 y = p ξ(s) s = s 0 y = p ξ(s) s = s 0 y = p ϖ(s) x y x = ϖ(s) y = p(ϖ(s)) x = ξ(s) y = p(ξ(s)) y = f(ξ(s)) O s 0 s O s 0 s f ξ(s) p ϖ(s) s s 0 s s 0 f ξ(s) p ξ(s) + p ξ(s) p ϖ(s) = s s0 s s ( 0 ) f(ξ(s)) p(ξ(s)) p(ξ(s)) p(ϖ(s)) = + s s0 s s 0 s s f(ξ(s)) p(ξ(s)) s s 0 = f(ξ(s)) p(ξ(s)) ξ(s) ξ(s 0 ) ξ(s) ξ(s 0 ) s s 0 ξ s 0 ξ(s) ξ(s 0 ) s s 0 s s 0 = dξ ds (s 0) s s0 ξ(s) = ξ(s 0 ) = x 0 x = ξ(s) f(ξ(s)) p(ξ(s)) s s 0 ξ(s) ξ(s 0 ) = x x0 f(x) p(x) x x 0 = 0

20 p(x) f(x) x f(ξ(s)) p(ξ(s)) s s 0 s s 0 f(ξ(s)) p(ξ(s)) ξ(s) ξ(s 0 ) = s s0 ξ(s) ξ(s 0 ) s s ( 0 f(ξ(s)) p(ξ(s)) = s s 0 ξ(s) ξ(s 0 ) = 0 dξ ds (s 0) = 0 ) ( s s 0 ξ(s) ξ(s 0 ) s s 0 y = f ξ(s) s = s 0 y = p ξ(s) s = s 0 2 p(x) 1 p(x) = f(x 0 ) + f (x 0 )(x x 0 ) p(ξ(s)) p(ϖ(s)) = f(x 0) + f (x 0 )(ξ(s) x 0 ) f(x 0 ) f (x 0 )(ϖ(s) x 0 ) s s 0 s s 0 = f ξ(s) ϖ(s) (x 0 ) s s 0 ϖ(s) ξ(s) s 0 1 p(ξ(s)) p(ϖ(s)) s s 0 s s 0 = f (x 0 ) s s0 ξ(s) ϖ(s) s s 0 = f (x 0 ) 0 = 0 y = p ξ(s) s = s 0 y = p ϖ(s) (3) 1 (f ξ) (s 0 ) = f (x 0 ) ξ (s 0 ). 1 df ξ ds (s f ξ(s) f ξ(x 0 ) 0) = = s s0 s s 0 f(ξ(s)) f(ξ(s 0 )) s s0 ξ(s) ξ(s 0 ) f(x) f(x 0 ) ξ(s) ξ(s 0 ) = = df x x 0 x x 0 s s0 s s 0 dx (x 0) dξ ds (s 0) ) ξ(x) ξ(s 0 ) s s 0 7 s 0 ξ(s) = ξ(s 0 ) s ξ(s) ξ(s 0 ) ξ(s) s 0 ξ(s 0 ) f(ξ(s)) p(ξ(s)) 0 ξ(x) ξ(s 0 ) 0

21 f(ξ(s)) f(x 0 )) = f (x 0 )ξ (s 0 ) (4) s s 0 s s 0 1 f(ξ(s)) f(x 0 ) f (x 0 )ξ (s 0 )(s s 0 ) = 0 (5) s s 0 s s f(x, y) (x 0, y 0 ) f(x, y) (x 0, y 0 ) x y 1 p(x, y) p(x, y) = f(x 0, y 0 ) + x (x 0, y 0 )(x x 0 ) + y (x 0, y 0 )(y y 0 ) (a, b) (x 0, y 0 ) 1 f(x, y) (x 0, y 0 ) 2 1 p(x, y) f(x, y) p(x, y) (x,y) (x 0,y 0 ) (x x0 ) 2 + (y y 0 ) = 0 (6) 2

22 9 22 p(x, y) = f(x 0, y 0 ) + x (x 0, y 0 )(x x 0 ) + y (x 0, y 0 )(y y 0 ) (6) 1 p(x, y) 1 2 f(x) x = ξ(s, t) f ξ(s, t) (s 0, t 0 ) f(x) x 0 = ξ(s 0, t 0 ) ξ(s, t) (s 0, t 0 ) f(x) x 0 1 p(x) ξ(s, t) (s 0, t 0 ) 1 ϖ(s, t) p(x) = f(x 0 ) + df dx (x 0)(x x 0 ) (7) ϖ(s, t) = x 0 + ξ s (s 0, t 0 )(s s 0 ) + ξ t (s 0, t 0 )(t t 0 ) (8) ξ(s 0, t 0 ) = x 0 p(x) x = ϖ(s, t) 1 p ϖ(s, t) f ξ(s, t) (s 0, t 0 ) 1 p ϖ(s, t) = f(x 0 ) + f ξ(s, t) p ϖ(s, t) (s,t) (s 0,t 0 ) (s s0 ) 2 + (t t 0 ) = 0 (9) 2 (f ξ) (s 0, t 0 )(s s 0 ) + s (f ξ) (s 0, t 0 )(t t 0 ) t 1 p ϖ(s, t) (7) (8) p ϖ(s, t) = f(x 0 ) + df dx (x 0)(ϖ(s, t) x 0 ) = f(x 0 ) + df dx (x 0) ( x 0 + ξ s (s 0, t 0 )(s s 0 ) + ξ t (s 0, t 0 )(t t 0 ) x 0 = f(x 0 ) + df dx (x 0) ξ s (s 0, t 0 )(s s 0 ) + df dx (x 0) ξ t (s 0, t 0 )(t t 0 ) )

23 9 23 (f ξ) (s 0, t 0 ) = df s dx (x 0) ξ s (s 0, t 0 ) (f ξ) (s 0, t 0 ) = df t dx (x 0) ξ t (s 0, t 0 ) (9) (s,t) (s 0,t 0 ) f ξ(s, t) p ϖ(s, t) (s s0 ) 2 + (t t 0 ) 2 = (s,t) (s 0,t 0 ) = (s,t) (s 0,t 0 ) f(ξ(s, t)) p(ξ(s, t)) + p(ξ(s, t)) p(ϖ(s, t)) (s s0 ) 2 + (t t 0 ) 2 ( f(ξ(s, t)) p(ξ(s, t)) ξ(s, t) ξ(s 0, t 0 ) ξ(s, t) ξ(s 0, t 0 ) (s s0 ) 2 + (t t 0 ) 2 ) p(ξ(s, t)) p(ϖ(s, t)) + (s s0 ) 2 + (t t 0 ) ξ(s, t) (s 0, t 0 ) p(x) f(x) x 0 1 f(ξ(s, t)) p(ξ(s, t)) (s,t) (s 0,t 0 ) ξ(s, t) ξ(s 0, t 0 ) = x x0 f(x) p(x) x x 0 = ξ(s, t) ξ(s 0, t 0 ) = ξ(s, t) ϖ(s, t) + ξ s (s 0, t 0 )(s s 0 ) + ξ t (s 0, t 0 )(t t 0 ) ϖ(s, t) ξ(s, t) 1 ξ(s, t) ϖ(s, t) (s,t) (s 0,t 0 ) (s s0 ) 2 + (t t 0 ) = 0 2 ξ s (s 0, t 0 ) = a ξ t (s 0, t 0 ) = b s s 0 = h t t 0 = k 8 (factor) (x + 1)(x 1)(x + 2) x + 1 x 1 x + 2

24 9 24 (h, k) (0, 0) ah + bk h2 + k 2 k = 0 h +0 a h 0 a h = 0 k +0 b k 0 b (h, k) (0, 0) a = b = 0 0 h h 2 + k 2, k h 2 + k 2 (0, 0) (h, k) ah + bk h a h2 + k 2 h2 + k + b k a + b 2 h2 + k2 1 f(ξ(s, t)) p(ξ(s, t)) ξ(s, t) ξ(s 0, t 0 ) (s,t) (s 0,t 0 ) ξ(s, t) ξ(s 0, t 0 ) (s s0 ) 2 + (t t 0 ) 2 f(ξ(s, t)) p(ξ(s, t)) (s,t) (s 0,t 0 ) ( a + b ) ξ(s, t) ξ(s 0, t 0 ) = 0 ( a + b ) = p(ξ(s, t)) p(ϖ(s, t)) = f(x 0 ) + f (x 0 )(ξ(s, t) x 0 ) f(x 0 ) f (x 0 )(ϖ(s, t) x 0 ) = f (x 0 )(ξ(s, t) ϖ(s, t)) ϖ(s, t) ξ(s, t) (s 0, t 0 ) 1 (s,t) (s 0,t 0 ) p(ξ(s, t)) p(ϖ(s, t)) (s s0 ) 2 + (t t 0 ) = f (x 0 ) 2 (s,t) (s 0,t 0 ) = f (x 0 ) 0 = 0 ξ(s, t) ϖ(s, t) (s s0 ) 2 + (t t 0 ) 2 (9) p ϖ(s, t) f ξ(s, t) (s 0, t 0 ) 1

25 f(x, y) x = ξ(s) y = η(s) φ(s) φ(s) = f(ξ(s), η(s)) f ξ η x 0 = ξ(s 0 ), y 0 = η(s 0 ) f(x, y) (x 0, y 0 ) 1 p(x, y) ξ(s) s 0 1 ϖ(s) η(s) s 0 1 ρ(s) 9 p(x, y) = f(x 0, y 0 ) + x (x 0, y 0 )(x x 0 ) + y (x 0, y 0 )(y y 0 ) (10) ϖ(s) = x 0 + dξ ds (s 0)(s s 0 ) (11) ρ(s) = y 0 + dη ds (s 0)(s s 0 ) (12) p(x, y) x = ϖ(s), y = ρ(s) s 1 ψ(s) = p(ϖ(s), ρ(s)) ψ(s) φ(s) s 0 1 φ(s) ψ(s) s s 0 s s 0 = 0 (13) 0 9 ρ ϖ r q p r

26 9 26 ψ(s) = f(x 0, y 0 ) + dφ ds (s 0)(s s 0 ) φ(s) 1 (10) (11) (12) ψ(s) ψ(s) = p(ϖ(s), ρ(s)) = f(x 0, y 0 ) + ( x (x 0, y 0 ) x 0 + dξ = f(x 0, y 0 ) + ds (s 0)(s s 0 ) x 0 ( y 0 + dη + y (x 0, y 0 ) ( x (x 0, y 0 ) dξ ds (s 0) + y (x 0, y 0 ) dη ds (s 0) dψ ds (s 0) = x (x 0, y 0 ) dξ ds (s 0) + y (x 0, y 0 ) dη ds (s 0) ) ds (s 0)(s s 0 ) y 0 ) (s s 0 ) (13) φ(s) ψ(s) s s 0 f(ξ(s), η(s)) p(ϖ(s), ρ(s)) = s s 0 f(ξ(s), η(s)) p(ξ(s), η(s)) + p(ξ(s), η(s)) p(ϖ(s), ρ(s)) = s s 0 = f(ξ(s), η(s)) p(ξ(s), η(s)) (ξ(s) x0 ) 2 + (η(s) y 0 ) 2 (ξ(s) x0 ) 2 + (η(s) y 0 ) 2 s s 0 + p(ξ(s), η(s)) p(ϖ(s), ρ(s)) s s s s 0 ξ(s) η(s) (ξ(s) x0 ) 2 + (η(s) y 0 ) 2 s s 0 s s 0 = s s0 (ξ(s) x 0 ) 2 + (η(s) y 0 ) 2 ( ξ(s) x 0 = s s 0 s s 0 (dξ = ) 2 ds (s 0) + (s s 0 ) 2 ) ) 2 ( η(s) y 0 + s s 0 s s 0 ( dη ds (s 0) ) 2 ) 2

27 ξ(s) η(s) s s 0 (ξ(s), η(s)) (x 0, y 0 ) p(x, y) f(x, y) (x 0, y 0 ) 1 (x, y) (x 0, y 0 ) f(x, y) p(x, y) (x x0 ) 2 + (y y 0 ) s s 0 f(ξ(s), η(s)) p(ξ(s), η(s)) (ξ(s) x0 ) 2 + (η(s) y 0 ) 2 = 0 s s p(x, y) 1 (10) x (x 0, y 0 ) = a y (x 0, y 0 ) = b p(ξ(s), η(s)) p(ϖ(s), ρ(s)) = f(x 0, y 0 ) + a(ξ(s) x 0 ) + b(η(s) y 0 ) f(x 0, y 0 ) a(ϖ(s) x 0 ) b(ρ(s) y 0 ) = a(ξ(s) ϖ(s)) + b(η(s) ρ(s)) ϖ(s) ξ(s) ρ(s) η(s) s s s 0 p(ξ(s), η(s)) p(ϖ(s), ρ(s)) s s 0 s s 0 ξ(s) ϖ(s) η(s) ρ(s) = a + b s s0 s s 0 s s0 s s 0 = a 0 + b 0 = (13) ψ(s) φ(s) s f(x, y) = 4 x 2 y 2, ξ(s) = e s, η(s) = e s (1) φ(s) = f(ξ(s), η(s)) s = 0 (2) (ξ(s), η(s)) s = 0 ξ(s) s = 0 1 x η(s) s = 0 1 y f(x, y) (1, 1) 1 (2) φ(s) s = 0

28 φ (s 0 ) φ(s) φ(s 0 ) s s 0 s s 0 = s s0 f(ξ(s), η(s)) f(ξ(s 0 ), η(s 0 )) s s 0 (14) (14) ? f(x, y) = xy φ(s) = ξ(s)η(s) φ(s) ξ(s)η(s) ξ(s 0 )η(s 0 ) = ξ(s)η(s) ξ(s 0)η(s) + ξ(s 0 )η(s) ξ(s 0 )η(s 0 ) s s 0 s s 0 = ξ(s) ξ(s 0) η(s) + ξ(s 0 ) η(s) η(s 0) s s 0 s s 0 ξ(s 0 )η(s) + ξ(s 0 )η(s) (14) φ(s) φ(s 0 ) s s 0 f(ξ(s 0 ), η(s)) + f(ξ(s 0 ), η(s)) = f(ξ(s), η(s)) f(ξ(s 0), η(s 0 )) s s 0 = f(ξ(s), η(s)) f(ξ(s 0), η(s)) + f(ξ(s 0 ), h(s)) f(ξ(s 0 ), η(s 0 )) s s 0 = f(ξ(s), η(s)) f(ξ(s 0), η(s)) ξ(s) ξ(s 0 ) (15) ξ(s) ξ(s 0 ) s s 0 + f(ξ(s 0), η(s)) f(ξ(s 0 ), η(s 0 )) η(s) η(s 0 ) η(s) η(s 0 ) s s 0 (16) (15)+(16) s s 0 ξ η s 0 1 (15) 2 2 (16) 2 ξ(s) ξ(s 0 ) s s 0 s s 0 = dξ ds (s 0) η(s) η(s 0 ) s s0 s s 0 = dη ds (s 0)

29 9 29 η s 0 η s 0 s s0 η(s) = η(s 0 ) f (x 0, y 0 ) y (16) 1 y = η(s) s s 0 f(ξ(s 0 ), η(s)) f(ξ(s 0 ), η(s 0 )) s s 0 η(s) η(s 0 ) = y y0 f(x 0, y) f(x 0, y 0 ) y y 0 = y (x 0, y 0 ) (15) 1 (15) 1 f(ξ(s), η(s 0 )) f(ξ(s 0 ), η(s 0 )) η s s 0 (16) 1 s s 0 f x (x 0, y 0 ) η s 0 s s s 0? s s 0 η(s) y 0 η(s) η(s 0 ) = y 0? f(ξ(s), η(s)) f(ξ(s 0 ), η(s)) ξ(s) ξ(s 0 ) f(ξ(s), η(s 0)) f(ξ(s 0 ), η(s 0 )) ξ(s) ξ(s 0 ) s s 0 f x (ξ(s 0 ), η(s 0 )) f(x, y) f(x, y) = y(3x 2 y 2 ) x 2 + y 2 (x, y) (0, 0) 0 (x, y) = (0, 0) ξ(s) = η(s) = s, s 0 = 0 (ξ(s 0 ), η(s 0 )) = (0, 0) (0, 0) x f(x, 0) f(0, 0) 0 0 (0, 0) = = = 0 x x 0 x x 0 x f x (0, 0) = 0 η s s f(ξ(s), η(0)) f(ξ(0), η(0)) f(s, 0) f(0, 0) = s 0 ξ(s) ξ(0) s 0 s = s 0 0 = 0 f x (0, 0) η(s) 1 1 f(ξ(s), η(s)) f(ξ(0), η(s)) s 0 ξ(s) ξ(0) f x (0, 0) = 0 f(s, s) f(0, s) = s 0 s ( 1 3s 3 s 3 = s 0 s s 2 + s 2 ) s3 = 2 = 2 s 2 s 0

30 9 30 f(x, y) x f C 1. f C 1 f C 1 (15) 1 f x (x 0, y 0 ) (15) 1 f(ξ(s), η(s)) f(ξ(s), η(s 0 )) f(ξ(s 0 ), η(s)) f(ξ(s 0 ), η(s 0 ) f(x, y) y 1 1 g(y) y y 0 g(y) g(y 0 ) y y 0 = g (c) c y y 0 g(y) = g(y 0 ) + g (c)(y y 0 ) c y y 0 g(y) g(y 0 ). f(x, y) y 1 y 1 ψ x (y) := f(a, y) x y 1 ψ x x x f C 1 ψ x (y) y b b y ψ x (y) ψ x (b) y b = dψ x dy (c) c y b ψ f f(x, y) f(x, b) y b = (x, c) y

31 9 31 c b y c b y x x ξ(s) ξ(s 0 ) = x 0 y η(s) b η(s 0 ) = y 0 f(ξ(s), η(s)) f(ξ(s), η(s 0 )) η(s) η(s 0 ) f(ξ(s 0 ), η(s)) f(ξ(s 0 ), η(s 0 )) η(s) η(s 0 ) = (ξ(s), c) y = y (ξ(s 0), c ) c c η(s) η(s 0 ) f(ξ(s), η(s)) = f(ξ(s), η(s 0 )) + y (ξ(s), c)(η(s) η(s 0)) f(ξ(s 0 ), η(s)) = f(ξ(s 0 ), η(s 0 )) + y (ξ(s 0), c )(η(s) η(s 0 )) 2 (15) 1 f(ξ(s), η(s)) f(ξ(s 0 ), η(s)) ξ(s) ξ(s 0 ) { f(ξ(s), η(s0 )) + f y (ξ(s), c)(η(s) η(s 0 )) } { f(ξ(s 0 ), η(s 0 )) + f y (ξ(s 0 ), c )(η(s) η(s 0 )) } = = f(ξ(s), η(s 0)) f(ξ(s 0 ), η(s 0 )) ξ(s) ξ(s 0 ) ξ(s) ξ(s 0 ) + y (ξ(s), c)η(s) η(s 0) ξ(s) ξ(s 0 ) y (ξ(s 0), c ) η(s) η(s 0) ξ(s) ξ(s 0 ) s s 0 ξ η s = s 0 η(s) η(s 0 ) s s 0 ξ(s) ξ(s 0 ) = η(s) η(s 0 ) s s 0 s s 0 s s 0 ξ(s) ξ(s 0 ) ( η(s) η(s 0 ) = s s 0 s s 0 ) ( s s 0 s s 0 ξ(s) ξ(s 0 ) ) = η (s 0 ) ξ (s 0 ) ξ(s) = ξ(s 0 ) = x 0 s s0 η(s) = η(s 0 ) = y 0 c c η(s) η(s 0 ) = y 0 s s0 s s 0 c c η(s 0 ) = y 0 f C 1 f y (x 0, y 0 ) s s 0 y (ξ(s), c) = y (x 0, y 0 ) = s s0 y (ξ(s 0), c ) (17)

32 9 32 (17) s s 0 y (ξ(s), c)η(s) η(s 0) ξ(s) ξ(s 0 ) ( ) ( ) η(s) η(s 0 ) = (ξ(s), c) s s 0 y s s 0 ξ(s) ξ(s 0 ) s s 0 y (ξ(s 0), c ) η(s) η(s 0) ξ(s) ξ(s 0 ) ( ) ( ) = s s 0 y (ξ(s 0), c η(s) η(s 0 ) ) s s 0 ξ(s) ξ(s 0 ) = y (x 0, y 0 ) η (s 0 ) ξ (s 0 ) = y (x 0, y 0 ) η (s 0 ) ξ (s 0 ) s s 0 (17) f(ξ(s), η(s)) f(ξ(s 0 ), η(s)) s s 0 ξ(s) ξ(s 0 ) = s s0 f(ξ(s), η(s 0 )) f(ξ(s 0 ), η(s 0 )) ξ(s) ξ(s 0 ) = x x0 f(x, y 0 ) f(x 0, y 0 ) x x 0 = x (x 0, y 0 ) f(x, y), ξ(s, t), η(s, t) x = ξ(s, t), y = η(s, t) f(x, y) φ(s, t) φ(s, t) = f(ξ(s, t), η(s, t)) x 0 = ξ(s 0, t 0 ), y 0 = η(s 0, t 0 ) f(x, y) (x 0, y 0 ) 1 p(x, y) ξ(s, t) η(s, t) (s 0, t 0 ) 1 ϖ(s, t), ρ(s, t)

33 9 33 p(x, y), ϖ(s, t), ρ(s, t) 1 x (x 0, y 0 ) = a ξ s (s 0, t 0 ) = α 1 η s (s 0, t 0 ) = β 1 y (x 0, y 0 ) = b, ξ t (s 0, t 0 ) = α 2, η t (s 0, t 0 ) = β 2 p(x, y) = f(x 0, y 0 ) + a(x x 0 ) + b(y y 0 ) (18) ϖ(s, t) = x 0 + α 1 (s s 0 ) + α 2 (t t 0 ) (19) ρ(s, t) = y 0 + β 1 (s s 0 ) + β 2 (t t 0 ) (20) p(x, y) x = ϖ(s, t) y = ρ(s, t) s, t 1 ψ(s, t) ψ(s, t) = p(ϖ(s, t), ρ(s, t)) ψ(s, t) φ(s, t) (s 0, t 0 ) 1 φ(s, t) ψ(s, t) (s,t) (s 0,t 0 ) (s s0 ) 2 + (t t 0 ) = 0 (21) 2 ψ(s, t) = f(x 0, y 0 ) + φ s (s 0, t 0 )(s s 0 ) + φ t (s 0, t 0 )(t t 0 ) φ(s, t) 1 (18) (19) (20) ψ(s, t) ψ(s, t) = p(ϖ(s, t), ρ(s, t)) = f(x 0, y 0 ) + a(x 0 + α 1 (s s 0 ) + α 2 (t t 0 ) x 0 ) + b(y 0 + β 1 (s s 0 ) + β 2 (t t 0 ) y 0 ) = f(x 0, y 0 ) + (aα 1 + bβ 1 )(s s 0 ) + (aα 2 + bβ 2 )(t t 0 ) a, b, α 1, α 2, β 1, β 2 φ s (s 0, t 0 ) = x (x 0, y 0 ) ξ s (s 0, t 0 ) + y (x 0, y 0 ) η s (s 0, t 0 ) φ t (s 0, t 0 ) = x (x 0, y 0 ) ξ t (s 0, t 0 ) + y (x 0, y 0 ) η t (s 0, t 0 )

34 9 34 (21). f(x, y), ξ(s, t), η(s, t) r(x, y), σ(s, t), τ(s, t) r(x, y) := f(x, y) p(x, y) σ(s, t) := ξ(s, t) ϖ(s, t) τ(s, t) := η(s, t) ρ(s, t) p(x, y) f(x, y) (x 0, y 0 ) 1 ϖ(s, t) ξ(s, t) ρ(s, t) η(s, t) (s 0, t 0 ) 1 (x,y) (x 0,y 0 ) (s,t) (s 0,t 0 ) (s,t) (s 0,t 0 ) φ(s, t) = f(ξ(s, t), η(s, t)) = p(ξ(s, t), η(s, t)) + r(ξ(s, t), η(s, t)) r(x, y) (x x0 ) 2 + (y y 0 ) = 0 2 σ(s, t) (s s0 ) 2 + (t t 0 ) = 0 2 τ(s, t) (s s0 ) 2 + (t t 0 ) = 0 2 = f(x 0, y 0 ) + a(ξ(s, t) x 0 ) + b(η(s, t) y 0 ) + r(ξ(s, t), η(s, t)) = f(x 0, y 0 ) + a(ϖ(s, t) + σ(s, t) x 0 ) + b(ρ(s, t) + τ(s, t) y 0 ) = f(x 0, y 0 ) + a(α 1 (s s 0 ) + α 2 (t t 0 ) + σ(s, t)) + r(ξ(s, t), η(s, t)) + b(β 1 (s s 0 ) + β 2 (t t 0 ) + τ(s, t)) + r(ξ(s, t), η(s, t)) = f(x 0, y 0 ) + (aα 1 + bβ 1 )(s s 0 ) + (aα 2 + bβ 2 )(t t 0 ) 1 ψ(s, t) : = p(ϖ(s, t), ρ(s, t)) + aσ(s, t) + bτ(s, t) + r(ξ(s, t), η(s, t)) = f(x 0, y 0 ) + (aα 1 + bβ 1 )(s s 0 ) + (aα 2 + bβ 2 )(t t 0 )

35 9 35 φ(s, t) (s s 0 ) 2 + (t t 0 ) 2 φ(s, t) ψ(s, t) (s s0 ) 2 + (t t 0 ) 2 = aσ(s, t) + bτ(s, t) + r(ξ(s, t), η(s, t)) (s s0 ) 2 + (t t 0 ) 2 σ(s, t) = a (s s0 ) 2 + (t t 0 ) + b τ(s, t) 2 (s s0 ) 2 + (t t 0 ) 2 + r(ξ(s, t), η(s, t)) (ξ(s, t) x0 ) 2 + (η(s, t) y 0 ) 2 (ξ(s, t) x0 ) 2 + (η(s, t) y 0 ) 2 (s s0 ) 2 + (t t 0 ) 2 σ(s, t) τ(s, t) 1 (s,t) (s 0,t 0 ) σ(s, t) (s s0 ) 2 + (t t 0 ) 2 = (s,t) (s 0,t 0 ) τ(s, t) (s s0 ) 2 + (t t 0 ) 2 = 0 ξ η ξ(s, t) = x 0 (s,t) (s 0,t 0 ) η(s, t) = y 0 (s,t) (s 0,t 0 ) (s,t) (s 0,t 0 ) r(ξ(s, t), η(s, t)) (ξ(s, t) x0 ) 2 + (η(s, t) y 0 ) 2 = = 0 (x,y) (x 0,y 0 ) (ξ(s, t) x0 ) 2 + (η(s, t) y 0 ) 2 (s s0 ) 2 + (t t 0 ) 2 r(x, y) (x x0 ) 2 + (y y 0 ) 2 (s, t) (s 0, t 0 ) u, v max{ u, v } u 2 + v 2 u + v

36 9 36 (ξ(s, t) x0 ) 2 + (η(s, t) y 0 ) 2 (s s0 ) 2 + (t t 0 ) 2 ξ(s, t) x 0 + η(s, t) y 0 (s s0 ) 2 + (t t 0 ) 2 a 1 s s 0 + a 2 t t 0 + σ(s, t) + b 1 s s 0 + b 2 t t 0 + τ(s, t) (s s0 ) 2 + (t t 0 ) 2 ( a 1 + b 1 ) s s 0 s s 0 + ( a 2 + b 2 ) t t 0 t t 0 σ(s, t) + (s s0 ) 2 + (t t 0 ) + τ(s, t) 2 (s s0 ) 2 + (t t 0 ) 2 σ τ 1 (s, t) (s 0, t 0 ) 0 (s, t) (s 0, t 0 ) a 1 + a 2 + b 1 + b 2 (21) ψ(s, t) φ(s, t) (s 0, t 0 ) 1

37 φ(s) f(x) = sin x ξ(s) = e s φ (s) = f (ξ(s)) ξ (s) = cos(e s ) e s = e s cos(e s ) 32 (1) s t s 1 1 φ s (s, t) = cos(s2 t) 2st = 2st cos(s 2 t) s t φ t (s, t) = cos(s2 t) s 2 = s 2 cos(s 2 t) (2) φ(s, t) = f ξ(s, t), f(x) = sin x, ξ(s, t) = s 2 t φ df ξ (s, t) = (ξ(s, t)) s dx s (s, t) = cos(s2 t) 2st = 2st cos(s 2 t) φ df ξ (s, t) = (ξ(s, t)) t dx t (s, t) = cos(s2 t) s 2 = s 2 cos(s 2 t) 1 2 (1) 33 (1) φ (s) = (e s ) sin s + e s (sin s) = e s sin s + e s cos s = e s (sin s + cos s) (2) φ(s) = f(ξ(s), η(s)), f(x, y) = xy, ξ(s) = e s, η(s) = sin s dφ ds (s) = x (ξ(s), η(s))dξ ds (s) + (ξ(s), η(s))dη y ds (s) = η(s)(e s ) + ξ(s)(sin s) = (sin s) e s + e s cos s = e s (sin s + cos s) (1)

38 (1) t s 1 1 φ s (s, t) = (( cos(st) ) t ) ( cos(s 2 + t 2 ) ) + ( sin(st) ) (( sin(s 2 + t 2 ) ) (2s) ) = t ( cos(st) )( cos(s 2 + t 2 ) ) 2s ( sin(st) )( sin(s 2 + t 2 ) ) s t 1 φ t (s, t) = (( cos(st) ) s ) ( cos(s 2 + t 2 ) ) + ( sin(st) ) (( sin(s 2 + t 2 ) ) (2t) ) = s ( cos(st) )( cos(s 2 + t 2 ) ) 2t ( sin(st) )( sin(s 2 + t 2 ) ) (2) s φ s (s, t) = x (ξ(s, t), η(s, t)) ξ s (s, t) + y (ξ(s, t), η(s, t)) η (s, t) s = ( cos(st) )( cos(s 2 + t 2 ) ) t + ( sin(st) )( sin(s 2 + t 2 ) ) (2s) = t ( cos(st) )( cos(s 2 + t 2 ) ) 2s ( sin(st) )( sin(s 2 + t 2 ) ) t φ t (s, t) = x (ξ(s, t), η(s, t)) ξ t (s, t) + y (ξ(s, t), η(s, t)) η (s, t) t = ( cos(st) )( cos(s 2 + t 2 ) ) s + ( sin(st) )( sin(s 2 + t 2 ) ) (2t) = s ( cos(st) )( cos(s 2 + t 2 ) ) 2t ( sin(st) )( sin(s 2 + t 2 ) ) (1) (1) (2) (1) (2) 35 (1) z = f(x, y) 2 (x, y) f(x, y) (x, y)

39 9 39 x2 + y 2 f(x, y) ( ) 2 f(x, y) = 4 x2 + y 2 x 2 + y 2 f xy (ξ(s), η(s)) = (e s, e s ) xy = 1 1 s = 0 (1, 1) φ(s) s = s φ (0) = 0 (1). ξ(s) η(s) η( s) = ξ(s) ξ( s) = η(s) f(x, y) f(y, x) = 4 y 2 x 2 = 4 x 2 y 2 = f(x, y) x y φ( s) = f(ξ( s), η( s)) = f(η(s), ξ(s)) = f(ξ(s), η(s)) = φ(s) φ(s) z = φ(s) z s = 0 s φ (s) = 0. φ(s) φ (0) = 0 (2) ξ(s) = e s ξ (s) = e s s = 0 ξ(s) 1 ϖ(s) ϖ(s) = ξ(0) + ξ (0)(s 0) = 1 + s η(s) = e s 1 ρ(s) η (s) = e s ρ(s) = η(0) + η (0)(s 0) = 1 s f(x, y) (ξ(0), η(0)) = (1, 1) z = f(x, y) (1, 1, f(1, 1)) = (1, 1, 2) 1 1 (x, y, z) 2

40 = 1 2 x 1 y 1 z 2 = x 1 + y 1 + 2z 2 f(x, y) (1, 1) 1 p(x, y) p(x, y) = (x 1) 1 2 (y 1) f(x, y) (1, 1) p(x, y) p(x, y) = f(1, 1) + f x (1, 1)(x 1) + f y (1, 1)(y 1) = 2 1 (x 1) 1 (y 1) 2 2 p(x, y) x = ϖ(s) y = ρ(s) p(ϖ(s), ρ(s)) = (1 + s 1) 1 2 (1 s 1) = (3) φ(s) 2 1 dφ ds (s) = x (ξ(s), η(s))dξ ds (s) + (ξ(s), η(s))dη y ds (s) x (x, y) = x 4 x2 y 2 y y (x, y) = 4 x2 y 2 dξ dη (s) = es (x) = e s ds ds s = 0, x = ξ(0) = 1, y = η(0) = 1 dφ ds (0) = 1 e 0 1 ( e 0 ) =

φ s i = m j=1 f x j ξ j s i (1)? φ i = φ s i f j = f x j x ji = ξ j s i (1) φ 1 φ 2. φ n = m j=1 f jx j1 m j=1 f jx j2. m

φ s i = m j=1 f x j ξ j s i (1)? φ i = φ s i f j = f x j x ji = ξ j s i (1) φ 1 φ 2. φ n = m j=1 f jx j1 m j=1 f jx j2. m 2009 10 6 23 7.5 7.5.1 7.2.5 φ s i m j1 x j ξ j s i (1)? φ i φ s i f j x j x ji ξ j s i (1) φ 1 φ 2. φ n m j1 f jx j1 m j1 f jx j2. m j1 f jx jn x 11 x 21 x m1 x 12 x 22 x m2...... m j1 x j1f j m j1 x

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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 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 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx

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