 2 p : | p k − x|≥δ | p − kx kδ |≥1 |  2 |≤2M  |p−kx|≥kδ r p (x) ≤ 2M k  p=0  p − kx kδ  2 r p (x) ≤ 2M kδ 2 kx(1 −x) ≤ M 2δ 2 k . >0 δ>0 |f(x) − B k (x)|≤|  1 | + |  2 | <+ M 2δ 2 k k ≥ M/2δ 2  sup |x|≤1 |f(x) −B k (x)| < 2.  f(x)=e x R A K ⊂ R n ∀f,g ∈A,α∈ R,f+ g, fg αf ∈A. A ∀x, y ∈ K, x = y, ∃ϕ ∈A: ϕ(x) = ϕ(y). R[x 1 , ··· ,x n ] n R n a 0 + k  p=1 (a p sin px + b p cos px),a p ,b p ∈ R,k ∈ N, R ϕ 1 , ··· ,ϕ s : K → R K k  p 1 +···+p s =0 a p 1 ···p s ϕ p 1 1 (x) ···ϕ p s (x), a p 1 ···p s ∈ R,k ∈ N. K R n A⊂C(K) K K A ∀f ∈ C(K), ∃g k ∈A:(g k ) k∈N f. A = {g : g A} A⊂C(K) (h k ) ⊂ A h h ∈ A A = A A A (h k ) ⊂ A h k (g k,i ) ⊂A h k i →∞  2 (g k = g σ(k),i(k) ) ⊂A h h ∈ A x, y ∈ K, α, β ∈ R h ∈ A h(x)=α, h(y)=β h A ϕ ∈A ϕ(x) = ϕ(y) h(z)= α +(β −α) ϕ(z) −ϕ(x) ϕ(y) −ϕ(x) h h 1 ,h 2 ∈ A max(h 1 ,h 2 ), min(h 1 ,h 2 ) ∈ A max(h 1 ,h 2 )= h 1 + h 2 + |h 1 − h 2 | 2 min(h 1 ,h 2 )= h 1 + h 2 −|h 1 − h 2 | 2 h ∈ A⇒|h|∈A. h M>0 |h(x)| <M,∀x ∈ K (P k ) [−M,M]  t →|t| g k = P k ◦ h (g k ) A |h| f ∈ C(K) ∀>0, ∃g ∈ A : d(f(x),g(x)) <, ∀x ∈ K, i.e. f(x) −<g(x) <f(x)+, ∀x ∈ K. x, y ∈ K h x,y ∈ A : h x,y (x)=f(x),h x,y (y)=f(y). x y ∈ K h x,y (y)=f(y) U y y h x,y (z) <f(z)+, ∀z ∈ U y ∩ K P x = {U y ,y ∈ K} K K U y 1 , ··· ,U y p K h x = min(h x,y 1 , ··· ,h x,y p ) h x ∈ A h x (z) <f(z)+, ∀z ∈ K. x ∈ K h x (x)=f(x) V x x f(z) −<h x (z), ∀z ∈ V x ∩ K. P = {V x ,x ∈ K} K V x 1 , ··· ,V x q K g =max(h x 1 , ··· ,h x q ) g ∈ A f(z) −<g(z),z∈ K. g  R T P k (x)=a k,0 + N k  p=1 (a k,p sin( 2πpx T )+b k,p cos( 2πpx T )). R T>0 C[0,T]  R n n K 1 ⊂ R n 1 K 2 ⊂ R n 2 A 1 A 2 K 1 ,K 2 A 1 A 2 f ∈ C(K 1 × K 2 ) k  i=1 g i (x)h i (y) g i ∈A 1 ,h i ∈A 2 ,k ∈ N K 1 × K 2  [a, b] K R n f ∈ C[0, 1] n k f B k (x 1 , ··· ,x n )=  0≤p 1 ,···,p n ≤k C p 1 k ···C p n k f( p 1 k , ··· , p n k )x p 1 1 ···x p n n (1−x 1 ) k−p 1 ···(1−x n ) k−p n . (B k ) f U R f : U → R a ∈ U f  (a) lim x→a f(x) −f(a) x − a = lim h→0 f(a + h) −f(a) h = f  (a) f(a + h)=f(a)+f  (a)h + o(h) f(x) T (x)=f(a)+f  (a)(x −a) x a U R n f : U → R m a ∈ U A : R n → R m f(a + h) −f(a) −Ah h → 0, h → 0. A f a Df(a) f  (a) f a f(a + h)=f(a)+Df(a)h + o(h), o(h) ϕ(h) lim h→0 ϕ(h) h =0. f a f a T T (x)=f(a)+Df(a)(x −a) f a f a (a, f(a)) f G f = {(x, y) ∈ R n × R m : y = f(x),x∈ U} , T T a = {(x, y) ∈ R n × R m : y = T(x)=f(a)+Df(a)(x −a),x∈ R n }. d((x, f(x)); T a ) ≤ d(f(x),T(x)) = o(x −a) x → a f a Df(a) f a A, B lim h→0 A(h) − B(h) h =0. x ∈ R n \ 0 A(x) − B(x) x = lim t→0 A(tx) −B(tx) tx =0 A(x)=B(x), ∀x ∈ R n A = B f Df(a) lim x→a (f(x) − f(a)) = lim x→a (f(x) − f(a) −Df(a)(x − a)) + lim x→a Df(a)(x − a)=0 f a  0 T DT(a)=T,∀a R → R m h → <A,h> A ∈ R m A ∈ R m f  (a) = lim h→0 f(a + h) − f(a) h . n>1 y h y ∈ R m ,h∈ R n R n −→ R m m ×n R n R m Jf(a) Df(a) j e j ∈ R n Jf(a)e j = j Jf(a). a Df(a)(te j )=f(a + te j ) − f(a)+o(t). j f a D j f(a) ∂f ∂x j (a) D j f(a)= ∂f ∂x j (a) = lim t→0 f(a + te j ) − f(a) t . ∂f ∂x j a =(a 1 , ··· ,a n ) x k = a k k = j x j → f(a 1 , ··· ,x j , ··· ,a n ) a j e ∈ R n \ 0 e f a D e f(a)= ∂f ∂e (a) = lim t→0 f(a + te) −f(a) t . f e a f(x, y)=x y ∂f ∂x (x, y)=yx y−1 , ∂f ∂y (x, y)=x y ln y (x, y > 0). f(x, y)=  |xy| ∂f ∂x (0, 0) = lim t→0 f(t, 0) −f(0, 0) t =0, ∂f ∂y (0, 0) = 0. f(x 1 , ··· ,x n )=(f 1 (x 1 , ··· ,x n ), ··· ,f m (x 1 , ··· ,x n )). f a ∈ U Df(a) f a Jf(a) f a ∂f i ∂x j (a), (i =1, ··· ,m; j = 1, ··· ,m) Jf(a)=       ∂f 1 ∂x 1 (a) ··· ∂f 1 ∂x n (a) ··· ··· ··· ∂f m ∂x 1 (a) ··· ∂f m ∂x n (a)       . Df(a):R n → R m dx =    dx 1 dx n    → dy =    dy 1 dy m    = Jf(a)dx              df 1 = ∂f 1 ∂x 1 (a)dx 1 + ···+ ∂f 1 ∂x n (a)dx n df m = ∂f m ∂x 1 (a)dx 1 + ···+ ∂f m ∂x n (a)dx n f : R 2 −→ R 3 f(x, y)=(x 2 + y 2 ,x+ y,xy) (x, y) ∈ R 2 , Jf(x, y)=    2x 2y 11 yx    . f(x, y)=x 2 + y 2 (x 0 ,y 0 ) T (x, y)=x 2 0 + y 2 0 +2x 0 (x − x 0 )+2y 0 (y −y 0 ) z = x 2 + y 2 R 3 (x 0 ,y 0 ,z 0 ) T z −z 0 =2x 0 (x − x 0 )+2y 0 (y −y 0 ). dz =2x 0 dx +2y 0 dy f a f a f a f a f(x, y)=  |xy| ∂f ∂x (0, 0) = ∂f ∂y (0, 0) = 0 Df(0, 0) f f (0, 0)      f(h, k) −f(0, 0) −  ∂f ∂x (0, 0) ∂f ∂y (0, 0)   h k       √ h 2 + k 2 → 0 , (h, k) → (0, 0).  |hk| √ h 2 + k 2 → 0 (h, k) → (0, 0) f : U → R m U ⊂ R n ∂f ∂x i ,i=1,··· ,n, U f x ∈ U m =1 h =(h 1 , ··· ,h n ) 0 f(x + h) −f(x)= n  j=1 (f(x + v j ) − f(x + v j−1 )), v j =(h 1 , ··· ,h j , 0, ··· , 0). j g j (h j )=f(x + v j ) f(x + v j ) − f(x + v j−1 )= ∂f ∂x j (c j )h j , c j = v j−1 + θ j h j e j , 0 <θ j < 1. x lim h→0 1 h |f(x + h) −f(x) −  j ∂f ∂x j (x)h j | = lim h→0 1 h |  j ( ∂f ∂x j (c j ) − ∂f ∂x j (x))h j | =0, f x  f,g x f + g x D(f + g)(x)=Df(x)+Dg(x) f,g x m =1 fg x D(fg)(x)=Df(x)g(x)+f(x)Dg(x) f,g x g(x) =0 f g x D( f g )(x)= Df(x)g(x) −f(x)Dg(x) g(x) 2 f : U −→ V g : V −→ W U, V, W R n , R m , R p f x g y = f(x) g ◦ f x Dg ◦ f(x)=Dg(f(x))Df(x) f(x + h)=f(x)+Df(x)h + ϕ 1 (h) ϕ 1 (h)=o(h) g(f(x)+k)=g(f (x)) + Dg(f (x))k + ϕ 2 (k) ϕ 2 (k)=o(k) g ◦f(x + h)=g(f(x)+Df(x)h + ϕ 1 (h)    k ) = g(f(x)) + Dg(f(x))Df(x)h + Dg(f(x))ϕ 1 (h)+ϕ 2 (Df(x)h + ϕ 1 (h)) Dg(f(x))ϕ 1 (h)≤Dg(f(x))ϕ 1 (h) = o(h), ϕ 2 (Df(x)h + ϕ 1 (h)) = o(Df(x)h + ϕ 1 (h))=o(h). g ◦f x D(g ◦ f)(x)=D(g(fx))Df(x).  Jh(x)=Jg(f(x))Jf(x) f(x)=(f 1 (x 1 , ··· ,x n ), ··· ,f m (x 1 , ··· ,x n )) g(y)=(g 1 (y 1 , ··· ,y m ), ··· ,g p (y 1 , ··· ,y m )) y = f(x) h(x)=g ◦f(x)=(h 1 (x 1 , ··· ,x n ), ··· ,h p (x 1 , ··· ,x n ))       ∂h 1 ∂x 1 ··· ∂h 1 ∂x n ··· ··· ··· ∂h p ∂x 1 ··· ∂h p ∂x n       =       ∂g 1 ∂y 1 ··· ∂g 1 ∂y m ··· ··· ··· ∂g p ∂y 1 ··· ∂g p ∂y m             ∂f 1 ∂x 1 ··· ∂f 1 ∂x n ··· ··· ··· ∂f m ∂x 1 ··· ∂f m ∂x n       ∂h i ∂x j = ∂g i ∂y 1 ∂f 1 ∂x j + ∂g i ∂y 2 ∂f 2 ∂x j + ··· ∂g i ∂y m ∂f m ∂x j = m  k=1 ∂g i ∂y k ∂f k ∂x j f(x, y) x, y x = r cos ϕ, y = r sin ϕ h(r, ϕ)=f(r cos ϕ, r sin ϕ) ∂h ∂r = ∂f ∂x cos ϕ + ∂f ∂y sin ϕ, ∂h ∂ϕ = ∂f ∂x (−r sin ϕ)+ ∂f ∂y r cos ϕ. f : R n −→ R f x ∇f(x)= f(x)=( ∂f ∂x 1 (x), ··· , ∂f ∂x n (x)). c ∈ R M c = {x ∈ R n : f(x)=c} = f −1 (c) f n =2 γ :(−1, 1) −→ R n γ R n t γ  (t)= dγ(t) dt = lim ∆t→0 γ(t +∆t) −γ(t) ∆t . γ  (t) γ t γ  (t) γ γ(t) γ M c γ(t) ∈ M c , ∀t (f ◦ γ)  (t)=f  (γ(t))γ  (t)=< f(γ(t)),γ  (t) >=0. f(x) M c x M c a =(a 1 , ··· ,a n ) < f(a),x−a>=0 D 1 f(a)(x 1 − a 1 )+···+ D n f(a)(x n − a n )=0. v ∈ R n f(a + tv)=f(a)+ < f(a),v >t+ o(t). < f(a),v > f a v | < f(a),v >|≤ f(a)v = v = λ f(a) ± f(a) f f . R 2 −→ R 3 f(x, y)=(x 2 + y 2 ,x+ y,xy) (x, y) ∈ R 2 , Jf(x, y)=    2x 2y 11 yx    . f(x, y)=x 2 + y 2 (x 0 ,y 0 ) T (x, y)=x 2 0 + y 2 0 +2x 0 (x − x 0 )+2y 0 (y −y 0 ) z = x 2 + y 2 R 3 (x 0 ,y 0 ,z 0 ). T P k (x)=a k,0 + N k  p=1 (a k,p sin( 2 px T )+b k,p cos( 2 px T )). R T>0 C[0,T]  R n n K 1 ⊂ R n 1 K 2 ⊂ R n 2 A 1 A 2 K 1 ,K 2 A 1 A 2 f ∈ C(K 1 × K 2 ) k  i=1 g i (x)h i (y) g i ∈A 1 ,h i ∈A 2 ,k ∈ N K 1 × K 2  [a,.  2 p : | p k − x|≥δ | p − kx kδ |≥1 |  2 |≤2M  |p−kx|≥kδ r p (x) ≤ 2M k  p=0  p − kx kδ  2 r p (x) ≤ 2M kδ 2 kx(1 −x) ≤ M 2 2 k . >0 δ>0 |f(x) − B k (x)|≤|  1 | + |  2 | <+ M 2 2 k k