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

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1 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 m m j1 x j1f j m j1 x j2f j. m j1 x jnf j f 1 f 2. x 1n x 2n x mn f m f(x, y) [φ 1, φ 2,, φ n ] [f 1, f 2,..., f m ] x 11 x 12 x 1n x 21 x 22 x 2n x m1 x m2 x mn

2 [ φ dφ [φ 1, φ 2,, φ n ], s [ 1 df [f 1, f 2,, f m ], x 1 φ,, s 2,, x 2 ] φ s n ] x m ξ 1 ξ 1 ξ s 1 s 2 1 s n ξ 2 ξ 2 ξ dφ df s 1 s 2 2 s n ξ m s 2 ξ m s 1 df dφ ξ m s n 1 2 φ(s) f ξ(s) f(ξ(s)) dφ ds df dx 1 1 [ ] [ ] [ ] dφ df dξ ds dx ds φ(s, t) f(ξ(s, t)) [ φ s, φ ] [ ] [ df ξ t dx s, ξ ] t φ(s) f(ξ(s), η(s)) [ ] [ dφ ds x, y dξ ds ] dξ ds dη ds

3 10 3 φ(s, t) f(ξ(s, t), η(s, t)) [ φ s, φ ] [ t x, ] ξ s y η s ξ t η t 7.5.2

4 f(x) x ξ(s) 2 φ(s) φ(s) f ξ(s) dφ df (s) ds dx (ξ(s))dξ ds (s) dg/ds dξ/ds s df/dx x ξ(s) s x x s s x dξ/ds(s) dξ/ds 0 s ξ(s) s σ(x) dξ ds (s) 1 dσ dx (ξ(s)) dξ/ds(s) dσ/dx(x) x ξ(s) s σ(x) σ(x) x s dξ (s) 0 ds ξ(s) x ξ(s) dξ (s) ds dφ df dx (ξ(s)) (s) ds dξ ds (s) 7. ξ(s) tan s ( π/2 < s < π/2) df dφ (tan s) dx ds (s) cos2 s

5 10 5 dφ ds d(f(tan s)) (s) (s) df tan s (tan s)d ds dx ds 1/ tan s cos 2 s df dx (tan s) 1 cos 2 s 1 f(x, y) x ξ(s, t) y η(s, t) φ φ(s, t) f(ξ(s, t), η(s, t)) [ φs (s, t), φ t (s, t) ] [ f x (ξ(s, t), η(s, t)), f y (ξ(s, t), η(s, t)) ] [ ξ s (s, t) ξ t (s, t) η s (s, t) η t (s, t) s t x y s t 2 2 (s, t)? 1 x ξ(s, t), y η(s, t) x y σ τ x ξ(σ(x, y), τ(x, y)) y η(σ(x, y), τ(x, y)) ] s σ(ξ(s, t), η(s, t)) t τ(ξ(s, t), η(s, t)) (2) x ξ(s, t), y η(s, t) σ τ (2) s t s (s, t) (s, t) 1 s t [ ] [ ] [ ξ s ξ t 1, 0 σx, σ y η s η t [ ] [ ] [ ξ s ξ t 0, 1 τx, τ y η s η t t (s, t) (s, t) 0 t s ] ]

6 10 6 [ ] [ ] [ ] 1 0 σ x σ y ξ s ξ t 0 1 τ x [ ] 1 [ ] ξ s ξ t τ y σ x η s σ y η t η s η t τ x τ y x ξ(s, t), y η(s, t) (x, y) (s, t) [ fx (ξ(s, t), η(s, t)), f y (ξ(s, t), η(s, t)) ] [ φ s (s, t), φ t (s, t) ] [ ξ s ξ t η s η t ] 1 (3). x ξ(s, t), y η(s, t) s σ(x, y), t τ(x, y) σ τ (x, y) (s, t) ξ η 8. x r cos θ y r sin θ 1 (3) f(x, y) φ(r, θ) φ(r, θ) f(r cos θ, r sin θ) 1 r > 0, 0 θ < 2π

7 10 7 f x (x, y) f y (x, y) (r cos θ, r sin θ) x (r cos θ, r sin θ) y φ(r, θ) (3) ξ(r, θ) r cos θ η(r, θ) r sin θ [ ξ r η r ξ θ η θ ] [ cos θ sin θ ] r sin θ r cos θ s r t θ [ ] [ ] [ ] 1 x, φ y r, φ cos θ r sin θ θ sin θ r cos θ 2 2 [ ] 1 [ cos θ r sin θ 1 r cos θ r sin θ sin θ r cos θ r sin θ cos θ ] x φ r cos θ 1 φ r θ sin θ y φ r sin θ + 1 φ r θ cos θ f x f y (x, y) x r cos θ, y r sin θ f φ f f φ f φ x r cos θ 1 r θ sin θ y r sin θ + 1 r θ cos θ

8 10 8 x y r θ f C 2 2 f x (r cos θ, r sin θ) + 2 f (r cos θ, r sin θ) 2 y2 φ R 2 1 R 2 x 2 + y 2 1 x 2 + y 2 1 (x, y) x 2 + y 2 1 f(x, y) x 2 + y 2 1 x 2 + y f(x) f(x) f (x) 0 1 2

9 x 2 + y 2 1 f(x, y) x 2 + y g(x, y) g(x, y) 0 z f(x, y) g(x, y) 0 z f(x, y) f(x, y) g(x, y) f(x, y, z) g(x, y, z) f(x, y) (a, b) g(x, y) 0 g(a, b) 0 (a, b) r (x, y) g(x, y) 0 f(x, y) f(a, b) 3 (x, y) (a, b) (x, y, z) (a, b, c) g(x, y) 0 f(x, y)

10 10 10? f(x, y) 2x 2 + y 2, g(x, y) x 2 + y f(x, y) g(x, y) 0 g(x, y) x cos t, y sin t g(x, y) 0 f(x, y) x y t φ(t) f(cos t, sin t) 2 cos 2 t + sin 2 t cos 2 t + 1 φ(t) 1 φ (t) 2 cos t sin t 0 t n π n 1, 2, 3, φ (t) 2 sin 2 t 2 cos 2 t φ ( 2m 2 π ) ( ) 2m < 0, φ π 2 > 0 2 n 2 x 2 y 2

11 10 11 (x, y) (±1, 0) (0, ±1) 1 (±1, 0) (0, ±1) 1 y f(0, 1) 1 f( 1, 0) 2 x f(1, 0) 2 f(0, 1) 1 1: 37. f(x, y) x ( x + 3y ) g(x, y) x 2 + y x cos t, y sin t? g(x, y) g(x, y) (?) h(x, y) x 4 + 2x 2 + y 2 3? h(x, y) 0 (x, y)? h(x, y) 0 y y ± 3 2x 2 x 4 ±

12 10 12 h(x, y) y 3 2x 2 x 4 y 3 2x 2 x 4 x h(x, y) 0 3 2x 2 x 4 4 (1 + x 2 ) (1 + x 2 ) x 2 1 x 1 f(x, y) y x 1 1 x 1 f(x, y) y ψ(x) f (x, ± ) 3 2x 2 x 4 3 x 4 x 0 3 x ±1 2 (x, y) (0, ± 3) (±1, 0) y? y x f(x, y) y x [ 1, 1] 1 h(x, y) x ±1? y

13 10 13 x x y h(x, y) 0 x x ±1 x 2 0 x y 2 3 < y < 3 x ± y 2 3 < y < 3 y ± 3 h(x, y) y ± 3 2x 2 x 4 1 < x < 1 x ± y 2 3 < y < f(x, y) 2x 2 + y 2 g(x, y) x 2 + y g(x, y) 0 y ± 1 x 2 ( x < 1), x ± 1 y 2 ( y < 1) 4

14 f xx (f x ) x f x 8 (r cos θ, r sin θ) (r, θ) f xx x x ( φ r cos θ 1 ) r r φ θ sin θ cos θ 1 ( φ r cos θ 1 ) r θ r φ θ sin θ sin θ ( φ rr cos θ + 1 r φ 2 θ sin θ 1 ) r φ rθ sin θ cos θ 1 ( φ θr cos θ φ r sin θ 1 r r φ θθ sin θ 1 ) r φ θ cos θ sin θ φ rr cos 2 θ 1 r φ rθ2 sin θ cos θ + 1 r 2 φ θθ sin 2 θ + 1 r φ r sin 2 θ + 1 r 2 φ θ2 sin θ cos θ f yy y f y ( φ r sin θ + 1 ) r r φ θ cos θ sin θ + 1 ( φ r sin θ + 1 ) r θ r φ θ cos θ cos θ ( φ rr sin θ 1 r φ 2 θθ cos θ + 1 ) r φ rθ cos θ sin θ + 1 ( φ θr sin θ + φ r cos θ + 1 r r φ θθ cos θ 1 ) r φ θ sin θ cos θ φ rr sin 2 θ + 1 r φ rθ2 sin θ cos θ + 1 r 2 φ θθ cos 2 θ + 1 r φ r cos 2 θ 1 r 2 φ θ2 sin θ cos θ 2 f x + 2 f 2 y 2 φ 2 r φ 2 r 2 θ + 1 φ 2 r r g(x, y) f x (x, y) ψ(r, θ) g(r cos θ, r sin θ) h(x, y) f y (x, y) χ(r, θ) h(r cos θ, r sin θ)

15 10 15 g x g y ψ r ψ θ h x h y χ r χ θ f φ g x (r cos θ, r sin θ) ψ r (r, θ) cos θ 1 r ψ θ(r, θ) sin θ (4) g x (f x ) x f xx f xx (r cos θ, r sin θ) ψ(r, θ) g(r cos θ, r sin θ) f x (r cos θ, r sin θ) ψ(r, θ) φ r (r, θ) cos θ 1 r φ θ(r, θ) sin θ ψ r (r, θ) ψ θ (r, θ) r θ f φ (4) f xx r ( φ r cos θ 1 ) r φ θ sin θ 1 ( φ r cos θ 1 ) r θ r φ θ sin θ (r cos θ, r sin θ) (r, θ) f φ 2 x y 2 2 r r r r 2 θ 2 2 / x / y 2 37 x cos t, y sin t f(x, y) x ( x + 3y ) φ(t) ( φ(t) f(cos t, sin t) cos t cos t + ) 3 sin t cos 2 t + 3 cos t sin t cos 2t + 2 sin 2t + 1 (2t 2 cos π ) ( φ (t) 2 sin 2t π ) 3 φ (t) 0 t 3n + 1 π 6 n Z

16 ( φ (t) 4 cos 2t π ) 3 6m ( ) 6m + 1 φ π 6 ( ) 6m + 4 < 0 φ π > 0 6 π (m Z) 6m + 4 π (m Z) 6 x y ( ) 3 ± 2, 1 2 ± ( 1 2, 3 2 ) f(x, y) 2x 2 + y 2 y 1 x 2 x 1 φ + (x) φ + (x) x x 0 x < 1 y 1 x 2 x 0 y 1 y 1 x 2 x 1 φ (x) φ (x) x x 0 y 1 x 2 x 0 y 1 x 1 y 2 f(x, y) y 1 ψ + (y) ψ + (y) 2 y 2 y 0 x 1 y 2 y 0 x 1

17 10 17 x 1 y 2 ψ (y) ψ (y) 2 y 2 y 0 x 1 y 2 y 0 x 1 (±1, 0) (0, ±1)

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

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 2009 9 6 16 7 1 7.1 1 1 1 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

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