f : U −→ R m U ⊂ R n f U f C 1 ∂f ∂x i i =1, ··· ,n U Df : U −→ L(R n , R m ) g :[a, b] −→ R g (a, b) g(b) − g(a)=g  (c)(b − a), c a<c<b. n>1,m=1 f : U → R U ⊂ R n f U [x, x + h]={x + th, t ∈ [0, 1]}⊂U f(x + h) − f (x)=Df(x + θh)h, 0 <θ<1. g(t)=f (x + th) m>1 f : R → R 2 ,f(x)=(x 2 ,x 3 ) f(1) −f(0) = Df(c)(1 −0) ⇔ (1, 1) − (0, 0) = (2c, 2c 2 ) f(x, y)=(e x cos y, e y sin y) f : U → R m U ⊂ R n [x, x + h] ⊂ U f(x + h) − f(x)≤ sup t∈[0,1] Df(x + th)h. T T  =sup h=1 Th Th≤T h. g(t)=f(x + th) g  (t)=Df(x + th)h. g(1) −g(0) =  1 0 g  (t)dt =  1 0 Df(x + th)hdt,  1 0 (φ 1 (t), ··· ,φ m (t))dt =(  1 0 φ 1 , ··· ,  1 0 φ m )  f : U → R m U Df(x)=0, ∀x ∈ U f ≡ f : U → R n U ⊂ R n C 1 K U L>0 f f(x) − f(y)≤Lx − y, ∀x, y ∈ K. 0 <L<1 f : K → K f K f : U −→ R m U ⊂ R n Df : U → L(R n , R m ), L(R n , R m ) R n → R m m × n R mn Df a ∈ U a R n −→ L(R n , R m ) ≡ R mn L(R n ,L(R n , R m )),L(R n ,L(R n ,L(R n , R m ))), ··· ∂f ∂x i U ∂ ∂x j  ∂f ∂x i  (a) f (i, j) a D j D i f(a) ∂ 2 f ∂x j ∂x i (a). ∂ k f ∂x i k ···∂x i 1 (a). f k U f C k U f ≤ k U f(x, y)=yx 2 cos y 2 ∂ 2 f ∂y∂x = ∂ 2 f ∂x∂y = ∂ 2 f ∂x∂y = ∂ 2 f ∂y∂x f(x, y)=xy x 2 − y 2 x 2 + y 2 (x, y) =(0, 0) f(0, 0) = 0 ∂ 2 f ∂y∂x (0, 0) = 1 ∂ 2 f ∂x∂y (0, 0) = −1 f 2 x f C 2 ∂ 2 f ∂x i ∂x j (x)= ∂ 2 f ∂x j ∂x i (x), ∀i, j f ∈ C k ≤ k 2 S h,k = f(x + h, y + k) − f(x + h, y) −f(x, y + k)+f(x, y) g k (u)=f(u, y + k) − f (u, y) S h,k = g k (x + h) − g k (x)=g  k (c)h, c ∈ (x, x + h) =( ∂f ∂x (c, y + k) − ∂f ∂x (c, y))h = ∂ 2 f ∂y∂x (c, d)hk, d ∈ (y,y + k). S h,k g h (v)=f(x + h, v) −f(x, v) S h,k = ∂ 2 f ∂x∂y (c  ,d  )kh, c  ∈ (x, x + h),d  ∈ (y, y + k). h, k → 0 S h,k  g :(a, b) → R ∈ C k x, x + h ∈ (a, b) θ ∈ (0, 1) g(x+h)=g(x)+ 1 1! g  (x)h+ 1 2! g  (x)h 2 +···+ 1 (k −1)! g (k−1) (x)h k−1 + 1 k! g k (x+θh)h k . f : R n → R g(t)=f (x + th),t∈ [0, 1]. ∇ =(D 1 , ··· ,D n )=( ∂ ∂x 1 , ··· , ∂ ∂x n ). h =(h 1 , ··· ,h n ) h∇ = h 1 ∂ ∂x 1 + ···+ h n ∂ ∂x n , (h∇) k = n  i 1 ,···,i k =1 h i 1 ···h i k ∂ k ∂x i 1 ···∂x i k . f (h∇)f = h 1 ∂f ∂x 1 + ···+ h n ∂f ∂x n . (h∇) k f =  i 1 ,···,i k ∂ k f ∂x i 1 ··· ∂ k f ∂x i k h i 1 ···h i k k h 1 , ··· ,h n . f : U → R C k U ⊂ R n [x, x + h] ⊂ U θ ∈ (0, 1) f(x + h)=f(x)+h∇f(x)+···+ 1 (k −1)! (h∇) k−1 f(x)+ 1 k! (h∇) k f(x + θh) g(t)=f(x + th) g (k) (t)=(h∇ ) k f(x + th).  C k f x k x T k x (h)=f(x)+h∇f (x)+···+ 1 k! (h∇) k f(x), R k (x, h)= 1 k!  (h∇) k f(x + θh) −(h∇) k f(x)  , 0 <θ<1. |f(x + h) − T k x (h)| = |R k (x, h)| = o(h k ) f ∈ C k f ∈ C ∞ f x 0 Tf(x)= ∞  k=0 1 k! ((x − x 0 )∇) k f(x 0 ). Tf(x) f(x)= ∞  k=0 e −k cos k 2 x Tf(x) Tf(x)=f(x) f(x)=e − 1 x 2 . f f f x 0 f |f (k) (x)|≤M k , ∀x ∈ (a, b), ∀k ∈ N (a, b) f : U → R,U⊂ R n f a ∈ U f(a) ≥ f(x) x a f a ∈ U f(a) ≤ f(x) x a f a f f a f Df(a)=0 max, min f f U f a ∈ U Df(a)=0 i g i (t)=f(a + te i ) t =0 g  i (0) = ∂f ∂x i (a)=0. Df(a)=0  f(x)=x 3 f(x, y)=x 2 − y 2 f f  f f  (a) > 0 f a f  (a) < 0 f x 2 + y 2 , −x 2 − y 2 ,x 2 − y 2 ,x 2 , −x 2 , 0. (0, 0) (0, 0) f C 2 f a Hf(a) R n  h → Hf(a)(h)=(h∇) 2 f(a)= n  i,j=1 ∂ 2 f(a) ∂x i ∂x j h i h j ∈ R. f C 2 Df(a)=0 Hf(a) Hf(a)(h) > 0, ∀h ∈ R n \ 0 f a Hf(a) Hf(a)(h) < 0, ∀h ∈ R n \ 0 f a Hf(a) f a f(a + h)=f(a)+Df(a)h + 1 2 Hf(a)(h)+o(h 2 ) Df(a)=0 Hf(a) > 0 m = min h=1 Hf(a)(h) > 0 Hf(a)(h) ≥ mh 2 , ∀h ∈ R n f a  H(h)= n  i,j=1 a ij h i h j ,h∈ R n . D k =det(a ij ) 1≤i,j≤k H D 1 > 0,D 2 > 0, ··· ,D n > 0. H D 1 < 0,D 2 > 0, ··· , (−1) n D n > 0. f(x, y)=x 3 + y 3 − 3xy f ∂f ∂x =3x 2 − 3y =0, ∂f ∂y =3y 2 − 3x =0. (0, 0) (1, 1) f Hf =     ∂ 2 f ∂x 2 ∂ 2 f ∂x∂y ∂ 2 f ∂y∂x ∂ 2 f ∂y 2     =  6x −3 −36y  (0, 0) D 2 = −9 < 0 Hf(0, 0) (0, 0) (1, 1) D 1 =6> 0,D 2 =27> 0 Hf(1, 1) > 0 f (1, 1) f : U → R m f f a f a Df(a) Df(a) f a Df(a) f a f(a) Df(a) f a f(a) Ax = y, A ∈ Mat(n, n). det A =0 A x = A −1 y. x 1 , ··· ,x n      f 1 (x 1 , ··· ,x n )=y 1 ··· f n (x 1 , ··· ,x n )=y n , y 1 , ··· ,y n f : R → R f f  (a) =0 f −1 a f −1 C 1 f : U −→ R n U ⊂ R n f C k (k ≥ 1) a ∈ U det Jf(a) =0 V a W f(a) f : V −→ W f −1 : W −→ V f −1 C k Df −1 (y)=(Df(x)) −1 ,y= f(x),x∈ V. Df(a) −1 a = f (a)=0 Df(0) = I n R n x y y = f(x) 0 y ∈ R n g y (x)=y + x −f(x) g y 0 x g y (x)=x f(x)=y x y g(x)=x − f(x) Dg(0) = 0 g ∈ C 1 r>0 g(x) − g(x  )≤ 1 2 x − x  ,x,x  ∈ B(0,r). y≤r/2,g y : B(0,r) −→ B(0,r) g y (x) − g y (x  )≤ 1 2 x − x  . x ∈ B(0,r) g y f f −1 : B(0,r/2) −→ B(0,r). f −1 y, y  ∈ B(0,r/2) x = f −1 (y),x  = f −1 (y  ) ∈ B(0,r) g x − x  ≤f(x) − f(x  ) + g(x) − g(x  )≤f(x) − f(x  ) + 1 2 x − x  . f −1 (y) − f −1 (y  )≤2y −y   f −1 r>0 f −1 ∈ C k det f ∈ C k det Df(a) =0 r>0 (Df(x)) −1 , ∀x ∈ B(0,r). y = f(x),y  = f(x  ),x,x  ∈ B(0,r) f −1 (y) − f −1 (y  ) − (Df(x)) −1 (y −y  ) = x −x  − (Df(x)) −1 (Df(x)(x − x  )+ +o(x − x  ) = (Df(x)) −1 (ox − x  ) = o(y −y  ) . Df −1 (y)=(Df(x)) −1 , y = f(x). Jf −1 (y)= 1 det Jf(x) (A ij (x)) n×n A ij (x) Jf(x)= f x Jf −1 C k−1 f −1 C k  f : U → V C k C k f f, f −1 C k f a f a f(a) u(x, y)=e x cos y, v(x, y)=e x sin y x, y u, v det J(u, v)=      e x cos y −e x sin y e x sin ye x cos y      = e 2x =0. f : R 2 → R 2 ,f(x, y)=(u(x, y),v(x, y)) (x, y) det Jf(x, y) =0, ∀(x, y) ∈ R 2 f : R → R f(x)=x 3 f −1 (y)= 3 √ y f  (0) = 0 f : U → R m ,U⊂ R n f C 1 a ∈ U n<m Df(a) rankDf(a)=n g f(a) 0 R m g ◦f(x 1 , ··· ,x n )=(x 1 , ··· ,x n , 0, ··· , 0) n>m Df(a) rankDf(a)=m h 0 a R n f ◦ h(x 1 , ··· ,x n )=(x 1 , ··· ,x m ) Df(a) Jf(a) n Φ:U × R m−n → R m , Φ(x, y n+1 , ··· ,y m )=f(x)+(0, ··· , 0,y n+1 , ··· ,y m ) Φ ∈ C 1 JΦ(a, 0) Φ (a, 0) f (x)=Φ(x, 0) g =Φ −1 Df(a) Jf(a) m Ψ:U → R m × R n−m , Ψ(x)=(f(x) − f(a),x m+1 − a m+1 , ··· ,x n − a n ). Ψ ∈ C 1 JΨ(a) Ψ a Ψ f(x)=pr ◦ (Ψ(x) − f(a)) pr m h =(Ψ− f(a)) −1  F (x, y)=0 y x y = g(x) g F (x, y)=x 2 + y 2 − 1=0 y = g(x) x = a ∈ (−1, 1) ∂F ∂y =0 F 0x y x a = ±1      a 11 x 1 + ··· +a 1n x n +b 11 y 1 + ··· +b 1m y m =0 ··· ··· ··· ··· ··· ··· a m1 x 1 + ··· +a mn x n +b m1 y 1 + ··· +b mn y m =0 A =(a ij ) m×n ,B=(b ij ) m×m det B =0 y x y = −B −1 Ax      F 1 (x, y)=0 ··· F m (x, y)=0 x ∈ R n ,y∈ R m y x F : A → R m ,A⊂ R n × R m (a, b) ∈ A F C k (k ≥ 1) F (a, b)=0 F y D(F 1 , ··· ,F m ) D(y 1 , ··· ,y m ) (a, b) = det ∂F ∂y (a, b)=           ∂F 1 ∂y 1 (a, b) ··· ∂F 1 ∂y m (a, b) ··· ··· ··· ∂F m ∂y 1 (a, b) ··· ∂F m ∂y m (a, b)           =0. U ⊂ R n a V ⊂ R m b g : U → V C k F (x, y)=0, (x, y) ∈ U × V ⇐⇒ y = g(x),x∈ U, y ∈ V. Dg(x)=−  ∂F ∂y  −1 ∂F ∂x (x, g(x)),x∈ U. f(x, y)=(x, F (x, y)) f −1 (x, z)=(x, G(x, z)) (x, z) (a, 0) g(x)=G(x, 0)  F (x, g(x)) = 0,x∈ U Dg ∂F i ∂x j + m  k=1 ∂F i ∂y k ∂g k ∂x j =0,i=1, ··· ,m; j =1, ··· ,k, Dg m =1 ∂F ∂y (a, b) =0 y = g(x 1 , ··· ,x n ) (a, b) y = g(x) F (x 1 , ··· ,x n ,y)=0,x ∈ U dF = ∂F ∂x 1 dx 1 + ···+ ∂F ∂x n dx n + ∂F ∂y dy =0. ∂F ∂y dg = −  ∂F ∂x 1 dx 1 + ···+ ∂F ∂x n dx n  , ∂g ∂x j = − ∂F/∂x j ∂F/∂y ,j=1, ··· ,n.  xu + yv 2 =0 xv 3 + y 2 u 6 =0 . y)=x 2 − y 2 f f  f f  (a) > 0 f a f  (a) < 0 f x 2 + y 2 , −x 2 − y 2 ,x 2 − y 2 ,x 2 , −x 2 , 0. (0, 0) (0, 0) f C 2 f a Hf(a) R n  h → Hf(a)(h)=(h∇) 2 f(a)= n  i,j=1 ∂ 2 f(a) ∂x i ∂x j h i h j ∈. 3xy f ∂f ∂x =3x 2 − 3y =0, ∂f ∂y =3y 2 − 3x =0. (0, 0) (1, 1) f Hf =     ∂ 2 f ∂x 2 ∂ 2 f ∂x∂y ∂ 2 f ∂y∂x ∂ 2 f ∂y 2     =  6x −3 −36y  (0, 0) D 2 = −9 < 0 Hf(0, 0) (0, 0) (1, 1) D 1 =6& gt;. j) a D j D i f(a) ∂ 2 f ∂x j ∂x i (a). ∂ k f ∂x i k ···∂x i 1 (a). f k U f C k U f ≤ k U f(x, y)=yx 2 cos y 2 ∂ 2 f ∂y∂x = ∂ 2 f ∂x∂y = ∂ 2 f ∂x∂y = ∂ 2 f ∂y∂x f(x, y)=xy x 2 − y 2 x 2 + y 2 (x, y) =(0,