x ∈
X d(x, X)=0
X, Y f(x)=
d(x, X)
d(x, X)+d(x, Y )
.
f f
−1
(1) = Y, f
−1
(0) = X
U, V X ⊂ U, Y ⊂ V
R
n
X, Y R
n
d(X, Y )= inf
x∈X,y∈Y
d(x, y).
K ⊂ R
n
X K  x → d(x, X)
x
0
∈ K, y
0
∈ X d(x
0
,y
0
)=d(K, X)

K
f : R
n
→ R
m
B ⊂ R
n
f(B)
f : R
n
→ R
m
K
f
−1
(K)
f
f : K → R K M = {x : f (x)=max
K
f}
[0, 1] (0, 1)
f : K −→ f(K) K f
−1
K
g(x)=sin
1
x
(0, +∞)
f : A → R
m

A ⊂ R
n
f
∃L>0: f(x) − f(y)≤Lx − y, ∀x, y ∈ A
f f
f : R
n
−→ R
m
G
f
f : C → R C f(x) =0, ∀x ∈ C
f(x) x ∈ C
x
4
+7x
3
− 9=0
x = x
f :[a, b] → [a, b] f
x
0
: f(x
0
)=x
0
f [0, 2π] f(0) = f(2π) c ∈
(0, 2π) f (c)=f(c + π)
f :[a, b] → R f(a)f(b) < 0
f(x)=0

√
2 <
1
10
x
2
− 2=0 [0, 2]
α ∈ R f(x)=αx, x ∈ R
A : R
2
→ R
2
A =

ab
cd

a, b, c, d > 0 A R
2
+
→ R
2
+
R
+
= {x ∈ R : x>0}
f :[0,
π
2
] → [0,

π
2
]
A

cos ϕ
sin ϕ

= λ(ϕ)

cos f(ϕ)
sin f(ϕ)

f A R
2
+
f
f : R
2
→ R
2
f(x, y)=(ax + by, cx + dy)
a, b, c, d f R
2
f : R
n
→ R
n
f(x)=Ax A =(a
ij

)
n
f :[0,r] → [0,r],f(x)=x
2
r f
f : X → X d(f(x),f(y)) <d(x, y), ∀x, y ∈ X,x = y
f
f :[0, 1] → [0, 1],f(x)=sinx
f : K → K K f
n
= f ◦···◦f
  
n ln
∩
n∈N
f
n
(K)
(f
k
) f (x
k
) x
(f
k
(x
k
)) f(x)
P
k

(x)=1+x + ···+ x
k
k ∈ N f(x)=
1
1 − x
0 <c<1 (P
k
) f [0,c]
f (0, 1)
g
[a, b]
a = a
0
<a
1
< ···<a
n
= b g(x)=A
k
x + B
k
,x ∈ [a
k−1
,a
k
],k=
1, ···,n
A
k
,B

k
g
[a, b]
B
k
(f) f(x)=x
2
x ∈ [0, 1] k
B
k
(f) − f =sup
x∈[0,1]
(|B
k
(f)(x) − x
2
|) <
1
1000
B
k
(f) f (x)=x
3
,x ∈ [0, 1]
(B
k
(f)) f
f(x)=e
x
,x ∈ R

A h(x)=
n

i=0
a
i
e
b
i
x
,n∈ N,a
i
,b
i
∈ R.
f ∈ C[0, 1] A
f [a, b] f
(P
k
) P
0
(x)=0,P
k+1
(x)=P
k
(x)+
1
2
(x − P
k

(x)
2
).
0 ≤
√
x − P
k
(x) ≤
2
√
x
1+k
√
x
0 ≤
√
x − P
k
(x) ≤
2
k
.
(P
k
) [0, 1]  x →
√
x
f(t)=|t| =
√
t

2
,t ∈ [−1, 1]
f ∈ C[0, 1] k =0, 1, ···

1
0
f(x)x
k
dx =0
f ≡ 0 f

1
0
f
2
=0
f :[0, 1] → R (P
k
)
f [0, 1] P
k
P (x) ≤ n
n +1 x
0
, ···,x
n
P (x)=
n

i=0

P (x
i
)π
i
(x),π
i
(x)=

j=i
(x − x
j
)

j=i
(x
i
− x
j
)
)
f : R
n
−→ R
m
∃M f(x)≤M x
2
f
0 Df(0) = 0
f(x) <Mx f
f(x, y)=(xy, y/x)

f(x, y, z)=(x
4
y, xe
z
)
f(x, y, z)=(z
xy
,x
2
, tgxyz)
f(x, y, z)=(e
z
sin x, xyz)
grad f
f(x, y, z)=x sin y/z
f(x, y, z)=e
x
2
+y
2
+z
2
z = x
3
+ y
4
x =1,y =3,z =82
x
2
− y

2
+ xyz =1 (1, 0, 1)
z =

x
2
+2xy −y
2
+1 (1, 1,
√
3)
ax
2
+2bxy + cy
2
+ dx + ey + f =0 (x
0
,y
0
,z
0
)
(2, −1, 2)
S
1
: x
2
+ y
2
+ z

2
=9 S
2
: z = x
2
+ y
2
− 3.
R
3
S
1
: x
2
+ y
2
+ z
2
=3 S
2
: x
3
+ y
3
+ z
3
=3.
S
1
,S

2
(1, 1, 1)
R
3
S
1
: ax
2
+ by
2
+ cz
2
=1 S
2
: xyz =1.
a, b, c S
1
,S
2
c(t)=(3t
2
,e
t
,t+ t
2
) t =1
c(t)=(2cost, 2sint, t) t = π/2
f(x, y, z)=x
2
y sin z (3, 2, 0)

f(x, y)=e
x
2
y (0, 0)
f (a, b) c f

(c) > 0
f(x)=x x f(x)=sinx x
f

(0) > 0 f 0
f [a, b] f

f

(a),f

(b)
γ f

(a) f

(b) g(x)=f(x)−γx
c ∈ (a, b)
f(x)=x
2
sin
1
x
x =0 f (0) = 0 f f


(0, 0)
f(x, y)=
x
y
, y =0;f(x, y)=0, y =0.
f
v ∈ R
n
a
D
v
(a) = lim
t→0
f(a + tv) − f(a)
t
.
f a f a
D
v
f(a)=< grad f(a),v >
f f
f(x, y)=
xy
x
2
+ y
x
2
= −y f (x, y)=0 x

2
= −y
f(x, y)=
x
2
y
x
4
+ y
2
(x, y) =(0, 0) f(0, 0) = 0
f(x, y)=
3

x
3
+ y
3
f(x, y)=
xy

x
2
+ y
2
x, y =0 f (0, 0) = 0.
f(x, y)=
x
2
y

2
x
2
y
2
+(y −x)
2
x, y =0 f (0, 0) = 0.
f(x, y)=|x| + |y|.
f(u, v, w)=u
2
v + v
2
w u = xy, v =sinx, w = e
x
f(u, v)=u
2
+ v sin u u = xe
u
,v = yz sin x
f : R → R F : R
2
→ R F (x, f(x)) ≡ 0
∂F
∂y
=0 f

= −
∂F/∂x
∂F/∂y

y = f(x)
x = r cos ϕ, y = r sin ϕ
f : R
2
→ R F (r, ϕ)=f (x, y)
(D
1
F (r, ϕ))
2
+
1
r
2
(D
2
F (r, ϕ))
2
=(D
1
f(x, y))
2
+(D
2
f(x, y))
2
θ (x, y) (u, v)
x = u cos θ − v sin θ, y = u sin θ + v cos θ
f : R
2
→ R F(u, v)=f(x, y)

(D
1
F (u, v))
2
+(D
2
F (u, v))
2
=(D
1
f(x, y))
2
+(D
2
f(x, y))
2
f
F (x, y)=f (x
2
+ y
2
) x
∂F
∂y
− y
∂F
∂x
=0
F (x, y)=f(xy) x
∂F

∂x
− y
∂F
∂y
=0
F (x, y)=f(ax + by) a
∂F
∂y
− b
∂F
∂x
=0
f,g : R → R C
2
c ∈ R u(x, y)=f (x + cy) − g(x − cy) u
c
2
∂
2
u
∂x
2
=
∂
2
u
∂y
2
v(x, y)=f(3x +2y)+g(x − 2y)
4

∂
2
v
∂x
2
− 4
∂
2
v
∂x∂y
− 3
∂
2
v
∂y
2
=0
f : R
2
−→ R
2
∂f
1
∂x
=
∂f
2
∂y
,
∂f

1
∂y
= −
∂f
2
∂x
.
det Jf(x, y)=0 Df(x, y)=0
f
f : R
n
→ R m f (tx)=t
m
f(x), ∀x ∈ R
n
,t∈
R
+
f
f m ⇔
n

i=1
x
i
∂f
∂x
i
(x)=mf(x), ∀x ∈ R
n

.
f : R
n
→ R C
k
(k>1)
f(x)=f(0) +
n

i=1
g
i
(x)x
i
,g
i
∈ C
k−1
(R
n
).
f : R → R |f

(x)|≤L, ∀x f
|f(x) − f(y)|≤L| x − y|, ∀x, y ∈ R
f : R
n
→ R
n
f :[a, b] → R 0 <m<f


(x) ≤ M ∀x ∈ [a, b]
f(a) < 0 <f(b) f
g(x)=x −
1
M
f(x) [a, b]
x
0
∈ [a, b] x
k+1
= x
k
−
1
M
f(x
k
),k ∈ N (x
k
)
x
∗
f
|x
k+1
− x
∗
|≤
|f(x

0
)|
m

1 −
m
M

k
f : R → R f(a)=b f

(a) =0 δ
|x − a| <δ |f

(x) − f

(a)|≤
1
2
|f

(a)| η =
δ
2
|f

(a)|
|¯y − b| <η
x
0

= a, x
k+1
= x
k
−
f(x
k
) − ¯y
f

(a)
(k ∈ N)
f(x)=¯y, x ∈ [a −δ, a + δ]
f(x, y)=arctgx +arctgy − arctg
x + y
1 − xy
.
f : R
n
→ R
m
Df(x)=A, ∀x A
f f(x)=Ax+
f : U → R U ⊂ R
n
D
1
f(x)=0, ∀x ∈ U f
f(x
1

,x
2
, ···,x
n
)=f(x

1
,x
2
, ···,x
n
), ∀(x
1
, ···,x
n
), (x

1
, ···,x
n
) ∈ U
f(x, y)=xy
x
2
− y
2
x
2
+ y
2

x, y =0 f (0, 0) = 0
∂
2
f
∂x∂y
(0, 0) =
∂
2
f
∂y∂x
(0, 0).
f(x, y)=x
2
+ y
2
(0, 0) (1, 2)
f(x, y)=e
−x
2
−y
2
cos xy (0, 0)
f(x, y)=e
(x−1)
2
cos y (1, 0)
0 f(x)=e
−
1
x

2
x =0 f(0) = 0
f f
C
1
u = f
1
(x, y)
v = f
2
(x, y)
(x
0
,y
0
)
∆=
∂f
1
∂x
∂f
2
∂y
−
∂f
1
∂y
∂f
2
∂x

0 (x
0
,y
0
) x = x(u, v),y = y(u, v)
∂x
∂u
=
1
∆
∂v
∂y
∂x
∂v
= −
1
∆
∂u
∂y
∂y
∂u
= −
1
∆
∂v
∂x
∂y
∂v
=
1

∆
∂u
∂x
f(x, y)=(
x
2
− y
2
x
2
+ y
2
,
xy
x
2
+ y
2
) f
(0, 1)
R
2
 (r, ϕ) → (r cos ϕ, r sin ϕ) ∈ R
2
R
3
 (ρ, ϕ, θ) → (ρ cos ϕ sin θ,ρ sin ϕ sin θ, ρcos θ) ∈ R
3
f : R
2

\{(0, 0)}→R
2
\{(0, 0)},f(x, y)=(x
2
− y
2
, 2xy)
det Df(x, y) =0, ∀(x, y) f R
2
\
{(0, 0)}
f A = {(x, y):x>0} f(A)
Dg(1, 0) g f
f : R
n
→ R
n
f(x)=x
2
x f ∈ C
∞
f
−1
f(x)=
x
2
+ x
2
sin
1

x
x =0 f(0) = 0 f
f

(0) =0 f 0
f : R
n
→ R
n
C
1
f

(x)≤c<1, ∀x g(x)=x + f(x)
g g
y
(x)=y − f(x)
f : R
n+k
→ R
n
C
1
f(a)=0 Df(a) n
c 0 f(x)=c
f : R → R C
1
u = f(x)
v = −y + xf(x)
f


(x
0
) =0 x
0
,y
0
)
x = f
−1
(u),y= −v + uf
−1
(u)
x F (x, y)=y
2
+ y +3x +1=0
y = y(x)
dy
dx
(x
0
,y
0
,z
0
)
z
2
+ xy −a =0,z
2

+ x
2
− y
2
− b =0.
x = f(z),y= g(z)
f

(z),g

(z)
f : R
3
→ R,g: R
2
→ R F (x, y)=f(x, y, g(x, y)).
DF(x, y) f g
F (x, y)=0 x, y D
1
g, D
2
g f

(x
4
+ y
4
)/x = u
sin x +cosy = v
x, y u, v x = π/2,y = π/2

∂x
∂u
(π
3
/4, 1)
x, y, z u, v, w (0, 0, 0)





u(x, y, z)=x + xyz
v(x, y, z)=y + xy
w(x, y, z)=z +2x +3z
2

x
2
− y
2
− u
3
+ v
2
+4 = 0
2xy + y
2
− 2u
2
+3v

4
+8 = 0
u, v x, y x =2,y = −1 u(2, −1) = 2,v(2, −1) =
1 u, v
u, v x, y

xu + yv
2
=0
xv
3
+ y
2
u
6
=0
x =1,y = −1,u =1,v = −1
u = u(x, y),v = v(x, y)
u, v, w x, y, z





3x +2y + z
2
+ u + v
2
=0
4x +3y + z + u

2
+ v + w +2 = 0
x + z + u
2
+ w +2 = 0
x =0,y =0,z =0,u=0,v =0,w = −2
u = u(x, y, z),v= v(x, y, z),w = w(x, y, z)
sin tx +costx = t, |t| <
1
√
2
x = ϕ(t) ϕ
ϕ 0
Q(x, y)=ax
2
+2bxy + cy
2
(a =0)
Q a>0 ac −b
2
> 0
Q a<0 ac − b
2
> 0
Q ac − b
2
< 0
f(x, y)=x
2
+2xy + y

2
+6
f(x, y)=(x
2
+ y
2
)e
−x
2
−y
2
f(x, y)=x
3
− 3xy
2
f(x, y, z)=x
2
+ y
2
+2z
2
+ xyz
f(x, y, z)=xy
2
z
3
(a − x − 2y −3z) x, y, z > 0 a>0
f(x, y, z)=cos2x sin y + z
2
a

1
, ···,a
n
∈ R x
n

i=1
(x − a
i
)
2
min
n x, y
x
x
1
x
2
··· x
m
y y
1
y
2
··· y
m
n p(x)=a
0
+ a
1

x + ···+ a
n
x
n
Q(a
0
, ···,a
n
)=
m

i=1
(p(x
i
) − y
i
)
2
→ min
p(x)=a
0
+ a
1
x

a
1

i
x

2
i
+ a
0

i
x
i
=

i
x
i
y
i
a
1

i
x
i
+ na
0
=

i
y
i
x
y

x
y
f :[0, 1] → R A, B, C

1
0
(f(x) − Ax
2
− Bx − C)
2
dx min
A, B, C















1
5
A +

1
4
B +
1
3
C =

1
0
x
2
f(x)dx
1
4
A +
1
3
B +
1
2
C =

1
0
xf(x)dx
1
3
A +
1
2

B + C =

1
0
f(x)dx
Ax
2
+ Bx + C 2
f n
[a, b]