a P (A) f k(x) = A k g k " g k (x) = ( 1) k x ą k. $ & g k (x) = x k (0, 1) f k, f, g : [0, 8) Ñ R f k (x) ď g(x) k P N x P [0, 8) g(x)dx g(x)dx ă 8

Transcription

1 M M, d A Ď M f k : A Ñ R A a P A f kx = A k xña k P N ta k u 8 k=1 f kx = f k x. xña kñ8 kñ8 xña M, d N, ρ A Ď M f k : A Ñ N tf k u 8 k=1 f : A Ñ N A f A 8ř " x ď k, g k x = 1 k x ą k. & g k x = % g k x = 1 k x ď k, 1 x x ą k. 1 k? kx R k g k x = x k, 1 f k, f, g : [, 8 Ñ R ą f k g [, R] f k x ď gx k P N x P [, R ą tf k u 8 k=1 f [, R] kñ8 gxdx RÑ8 ż R f k xdx = gxdx ă 8 fxdx g k k=1 ż R ż R f k xdx = f k xdx. kñ8 RÑ8 RÑ8 kñ8

2 " 1 k 1 ď x ă k, f k x f k x = tf k u 8 k=1 kñ8 f k : [, 8 Ñ R f k x = f k xdx = x 1 + kx4 kñ8 f fxdx f k xdx gx gx = x x P [ 1, 1 ] g 1 fx = 8ÿ k=1 g4 k 1 x 4 k 1. M f R f M, d K Ď M U = f P C K; R ˇˇ a ă fx ă b x P K C K; R, } }8 a, b P R F = f P C K; R ˇˇ a ď fx ď b x P K C K; R, } }8 a, b P R A Ď M B = f P C b A; R ˇˇ fx ą x P A Cb A; R, } } 8 δ : C [, 1]; R Ñ R δf = f δ M, d V, } } A Ď M B Ď C b A; V B C,α [, 1]; R! C,α fx fy [, 1]; R f P C [, 1]; R ˇ ă 8, x,yp[,1] x y α α P, 1] f P C,α [, 1]; R fx fy }f} C,α = fx +. xp[,1] x,yp[,1] x y α x y

3 C,α [, 1]; R, } } C,α B = f P C [, 1]; R ˇˇ }f}c,α ă 1 B C [, 1]; R, } } 8 tf k u 8 k=1 R ď f k x ď 1 x k P N 8 f kj j=1 f f 8 f kj j=1 R f f : [a, b] Ñ R f 1, f : [a, b] Ñ R fa ă = fc ă fb f 1 x x P [a, b] Φ : [a, b] Ñ R Φx = x fx f 1 x. Φx Φy ď k x x, y P [a, b] k P [, 1 b a f x ą x P [a, b] a ď ra ă c Φ : [ra, b] Ñ [ra, b] x P [ra, b] x k+1 = Φx k P N Y tu tx k u 8 k=1 c c f M, d K Ď M Φ : K Ñ K d Φx, Φy ă dx, y x, y P K x y Φ K

4 f [, 1] f = [, 1] p = fxx n dx n P N Y tu. p k+1 x = p k x + x p k x tp k u 8 k=1 x [ 1, k P N Y tu. x p k+1 x = [ x p k x ][ 1 x + p ] kx ď p k x ď p k+1 x ď x x ď 1 x ď 1 x p k x ď x 1 x k ă k + 1 g : [, 1] Ñ R [, 1] g f : [, 1] Ñ R ε ą g }f g} 8 ă ε 挑戰自我之期中考不考題 p n f [, 1] f p n N p N + 1 x,, x N px = Nÿ k= π k x px k π k x k, π k x = x x x x 1 x x N /x x k = ś 1ďjďN j k x x j 挑戰自我之期中考不考題 [, 1] hx = nÿ a j e bjx, j=1 a j b j P R C [, 1]; R

5 X, } } X Y, } } Y BX, Y, } } BX,Y BX, Y, } } BX,Y Y, } }Y P, 1 Ď C b, 1; R, 1 d dx : P, 1 Ñ C b, 1 d dx : P, 1, } } 8 Ñ Cb, 1, } } 8 }p 1 } 8 = 8. }p} 8 =1 X, } } X Y, } } Y U Ď X f : U Ď X Ñ Y f a P U L P BX, ε ą, D δ ą Q }fx fa Lx a} Y ď ε}x a} X x P Da, δ. f : n Ñ n fl = L 1 f n f n f L P n DfL fl + h fl h P BR n, R n L + h 1 L 1 DfL I : C [, 1]; R Ñ R If = fx dx. I f P C [, 1]; R DIf If + h If h P C [, 1]; R C [, 1]; R R C I δi DI

6 U = R z x, P R ˇˇ x ě f : U Ñ R & fx, y = % 1 x a x + y y ą, π y =, π 1 x a y ă, x + y 1, xy & a x, y,, fx, y = x + y % x, y =,. U Ď R n f : U Ñ R Bf,, Bf U M ą Bx 1 Bx n f U ˇ Bf x ˇ ď x P U j = 1,, n. Bx j 共筆 & xy x + y, fx, y = x + y % x + y =. U Ď R n f : U Ñ R a P U D u fa D u fa = nÿ j=1 u = u 1,, u n Bf Bf au j = a,, Bf a u Bx j Bx 1 Bx n & x 3 y x, y,, fx, y = x 4 + y % x, y =,.

7 ux, y, z = xe y, vx, y, z = yz x fu, v = u + v u hx, y, z = f ux, y, z, vx, y, z r, θ, φ R 3 x = r θ φ, y = r θ φ, z = r φ. x, y, z ÞÑ r, θ, φ r, θ, φ ÞÑ x, y, z fx, y, z R 3 f f : R Ñ R x P R ď f 1 x ď fx gx = e x fx f f fx, y, z = x + 1 yz a =, π, 1 u = 1,, v =, 1, w =,, 1 Dfau D favu D 3 fawvu f : A Ď R n Ñ R m g : B Ď R m Ñ R l fa Ď B x P A u, v P R n D g fx u, v = D gfx Dfx u, Dfx v + Dgfx D fx u, v. p : R n Ñ R m px = Lx + c L P BR n, R m f : A Ď R m Ñ R s k D k f px u 1,, u k = D k f px Dpx u 1,, Dpx u k.

8 fx, y R f C 1 R B f BxBy B f ByBx B f BxBy = B f ByBx f : R n Ñ R m Df BR n, R m Dfx 1 u = Dfx u x 1, x P R n u P R n f Df U Ď R n f : U Ñ R C k D j fx = j = 1,, k 1 D k fx u, u,, u ă u P R n u f x D δ ą fx ď x P Dx, δ. fx, y = x 3 + x 4xy + y f Qx, y, h, k D fx, y h, k, h, k f A = x, y ˇˇ ď x ď 1, ď y ď 1, x + y ď 1. f : R Ñ R & x + y x y 4x6 y x, y,, fx, y = x 4 + y % x, y =,. f, x, y P R 4x 4 y ď x 4 + y. ď θ ď π 8 ă t ă 8 g θ t = ft θ, t θ. g θ t = f,,

9 , f U Ď R n f : U Ñ R n C 1 Dfx x P U fw W Ď U D Ď R n f : D Ñ R n C 1 f Df D [ Dfx ] x P D f : BD Ñ R n f : D Ñ R n E = x P D ˇˇ D y P D, y x Q fx = fy E D E E = H E = D E D E = H f D f : R Ñ R C 1 a, b P R fa, b = f y a, b U Ď R a V Ď R b x P U y P V fx, y = y = yx ya = b fx, yx = x P U F x, y, z = z = fx, y x = gy, z y = hz, x f x g y h z = 1 F x, y = y = fx f F F x 1, x, y = y = fx 1, x f x1 x 1 f x1 x f x x F A Ď R n f : A Ñ R

10 P A m ď fx ď M x P A mνa ď Lf, P ď Uf, P ď MνA Lf, P 1 ď Uf, P P 1 P A A A A A a, b P R a ă b [a, b] R A Ď [a, b] R R [a, b]za A Ď R n f : A Ñ R f A A = 8 Ť k=1 D 1 k, 1 k = 8Ť k=1 1 k 1 k, 1 k + 1 k R A fx = 1 x, 1 f : [, 1] Ñ R & fx = % 1 p x = q P Q p, q = 1, p x. f [, 1] fxdx A Ď R n f : A Ñ R f k f 的 k 次方 k P N A, B Ă R f : A Ñ R g : fa Ñ R g f A

11 A Ď R n B Ď A f : A Ñ R f1 B A f B ż ż f1 B xdx = fxdx. A A = [a, b] ˆ [c, d] R f : A Ñ R! x P [a, b] ˇ ż d R 1 c fx, ydy sż d fx, ydy c B! y P [c, d] ˇ ż b a fx, ydx sż b f : [, 1] ˆ [, 1] Ñ R & x = x y, fx, y = 1 % p x, y P Q x = q p, q = 1. p f, y : [, 1] Ñ R y P [, 1] fx, : [, 1] Ñ R x R Q fx, ydy s fx, ydy x = q p ż f [, 1] ˆ [, 1] [,1]ˆ[,1] a fx, ydx fx, yda f : [, 1] ˆ [, 1] Ñ R & 1 x, y = k fx, y =, l ă k, l ă n n P N, n n %. f fx, ydydx = fx, ydxdy ż e x 1 x + ydy dx