Rn

Rn

Rn

Rn

Rn

R3

Rn

Rn

R3

Rn

Rn

n ∈ N R

x = (x1, . . . , xn) ∈ Rn

xi ∈ R i = 1, . . . ,n

e1 := (1, . . . , 0), . . . , en := (0, . . . , 1) ∈ Rn

+ : Rn × Rn → Rn,

x+ y = (x1, . . . , xn) + (y1, . . . ,yn) := (x1 + y1, . . . , xn + yn) ∈ Rn,

x = (x1, . . . , xn) y = (y1, . . . ,yn) Rn

: R × Rn → Rn,

αx = α(x1, . . . , xn) := (αx1, . . . ,αxn) ∈ Rn,

α ∈ Rx ∈ Rn

xi + yi ∈ Rαxi ∈ R i = 1, . . . ,n

R

Rn

Rn

Rn

RRn

R

x+ (y+ z) = (x+ y) + z ∀ x, y, z ∈ Rn,

x+ y = y+ x ∀ x, y ∈ Rn,

∃ 0 := (0, . . . , 0) ∈ Rn ∀ x ∈ Rn : 0+ x = x,

∀ x = (x1, . . . , xn) ∈ Rn ∃ − x := (−x1, . . . ,−xn) ∈ Rn : −x+ x = 0,

1x = x ∀ x ∈ Rn,

R

α(βx) = (αβ)x ∀ α,β ∈ R, x ∈ Rn,

α(x+ y) = αx+αy ∀ x, y ∈ Rn, α ∈ R,

R

(α+β)x = αx+βx ∀ α,β ∈ R, x ∈ Rn.

R

e1, . . . , en Rn

n x ∈ Rn

x = (x1, . . . , xn) = x1e1 + · · ·+ xnen.

Rn

Rn R

Rn

x · y :=n∑

i=1

xiyi ∀ x = (x1, . . . , xn), y = (y1, . . . ,yn) ∈ Rn,

· : Rn × Rn → R

x · y = y · x ∀ x, y ∈ Rn,

(αx+βy) · z = α(x · z) +β(y · z) ∀ α,β ∈ R, x, y, z ∈ Rn,

x · x ≥ 0 ∀ x ∈ Rn x · x = 0 ⇔ x = 0.

Rn

R

x ∈ Rn

∥x∥ :=√x · x =

( n∑

i=1

x2i

)1/2∀ x ∈ Rn,

∥ · ∥ : Rn → R

∥x∥ ≥ 0 ∀ x ∈ Rn ∥x∥ = 0 ⇔ x = 0,

∥αx∥ = |α|∥x∥ ∀ α ∈ R, x ∈ Rn,

∥x+ y∥ ≤ ∥x∥+ ∥y∥ ∀ x, y ∈ Rn.

|x · y| ≤ ∥x∥∥y∥ ∀ x, y ∈ Rn.

Rn

x, y ∈ Rn

x, y ∈ Rn

∀ (α,β) ∈ R2 \ {(0, 0)} : αx+βy = 0.

λ ∈ R

0 < ∥λx+ y∥2 = (λx+ y) · (λx+ y) = λ2∥x∥2 + 2λx · y+ ∥y∥2

λ

4(x · y)2 − 4∥x∥2∥y∥2 < 0,

x, y ∈ Rn \ {0}λ ∈ R \ {0} y = λx |λ|∥x∥2 = |λ|∥x∥2

x, y ∈ Rn 0 = 0 ✷

Rn

∥x∥∥y∥ = 0∥x∥∥y∥ > 0 x, y ∃ λ > 0 : y = λx

∥x+ y∥2 = (x+ y) · (x+ y) = ∥x∥2 + 2x · y+ ∥y∥2

≤ ∥x∥2 + 2|x · y|+ ∥y∥2 ≤ ∥x∥2 + 2∥x∥∥y∥+ ∥y∥2 = (∥x∥+ ∥y∥)2,

x · y = ∥x∥∥y∥.

x, y = 0x, y λ = 0 y = λxx · y = λ∥x∥2 > 0 λ > 0 x, y

y = λx λ > 0 ✷

∣∣∥x∥− ∥y∥∣∣ ≤ ∥x− y∥ ∀ x, y ∈ Rn.

Rn

∥x∥∥y∥ = 0 ∥x∥∥y∥ > 0 x, y

∥x∥ = ∥x− y+ y∥ ≤ ∥x− y∥+ ∥y∥ ⇒ ∥x∥− ∥y∥ ≤ ∥x− y∥,∥y∥ = ∥y− x+ x∥ ≤ ∥y− x∥+ ∥x∥ ⇒ ∥y∥− ∥x∥ ≤ ∥y− x∥ = ∥x− y∥

−∥x− y∥ ≤ ∥x∥− ∥y∥ ≤ ∥x− y∥,

✷

X R∥ · ∥ : X → R

Rn

Rn

∞ ,

∥x∥∞ = max{|xi| : i = 1, . . . ,n} ∀ x = (x1, . . . , xn) ∈ Rn

1

∥x∥1 =n∑

i=1

|xi| ∀ x = (x1, . . . , xn) ∈ Rn.

1 2 p = 1p = 2 p

∥x∥p =( n∑

i=1

|xi|p)1/p

∀ x = (x1, . . . , xn) ∈ Rn, p ∈ R, p ≥ 1,

∞ 1

∥ · ∥1, ∥ · ∥2 : X → R Xc,C > 0

c∥x∥1 ≤ ∥x∥2 ≤ C∥x∥1 ∀ x ∈ X.

1 ℓ1 L1

1

C∥x∥2 ≤ ∥x∥1 ≤ 1

c∥x∥2 ∀ x ∈ X.

∞ 1Rn x ∈ Rn

∥x∥∞ ≤ ∥x∥1 ≤ n∥x∥∞,

∥x∥∞ ≤ ∥x∥ ≤√n∥x∥∞,

1√n∥x∥ ≤ ∥x∥1 ≤ n∥x∥.

x = (x1, . . . , xn) ∈ Rn i = 1, . . . ,n|xi| ≤ ∥x∥∞ = max{|xi| : i = 1, . . . ,n}

∥x∥1 =n∑

i=1

|xi| ≤ n∥x∥∞ ∥x∥2 =n∑

i=1

|xi|2 ≤ n∥x∥2∞.

i = 1, . . . ,n

|xi| ≤n∑

i=1

|xi| = ∥x∥1, ∥x∥∞ = max{|xi| : i = 1, . . . ,n} ≤ ∥x∥1

|xi|2 ≤

n∑

i=1

|xi|2 = ∥x∥2, ∥x∥2∞ = max{|xi|

2 : i = 1, . . . ,n} ≤ ∥x∥2.

|xi| ≤ ∥x∥∞ ∀ i = 1, . . . ,n|xi|

2 ≤ ∥x∥2∞ ∀ i = 1, . . . ,n m := max{|xi|2 : i =

1, . . . ,n} ≤ ∥x∥2∞ |xi|2 ≤ m ∀ i = 1, . . . ,n

|xi| ≤√m ∀ i = 1, . . . ,n ∥x∥∞ ≤

√m

∥ · ∥1 ∥ · ∥∥ · ∥∞

∥ · ∥1 ∥ · ∥ ✷

R2 1 ∞

0xy

C = {(x,y) ∈ R2 : ∥(x,y)∥ = 1} = {(x,y) ∈ R2 :√

x2 + y2 = 1},

C1 = {(x,y) ∈ R2 : ∥(x,y)∥1 = 1} = {(x,y) ∈ R2 : |x|+ |y| = 1},

C∞ = {(x,y) ∈ R2 : ∥(x,y)∥∞ = 1} = {(x,y) ∈ R2 : max{|x|, |y|} = 1}.

x

∥(x,y)∥ = 1

y

x

∥(x,y)∥∞ = 1

y

x

∥(x,y)∥1 = 1

y

∞1

Rn

d(x, y) := ∥x− y∥ ∀ x, y ∈ Rn.

d : Rn × Rn → R

d(x, y) = d(y, x) ∀ x, y ∈ Rn,

d(x, y) ≥ 0 ∀ x, y ∈ Rn d(x, y) = 0 ⇔ x = y,

d(x, y) ≤ d(x, z) + d(z, y) ∀ x, y, z ∈ Rn.

Rn

Rn R n ∈ N

Rn

Rn

∥x+ y∥2 = ∥x∥2 + ∥y∥2 ⇔ x · y = 0.

R3

Rn

2∥x∥2 + 2∥y∥2 = ∥x+ y∥2 + ∥x− y∥2.

∥x− y∥ ≤ ∥x∥+ ∥y∥ ∀ x, y ∈ Rn

∞ 1

R3

n Rn n = 1, 2, 3

R3

1

R3

0xyz 0x y z

P R3

1− 1 x R3

x,y, z

e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1),

x = (x,y, z) = xe1 + ye2 + ze3,

1− 1(x,y, z) P x P = (x,y, z) = x

x,y, z P P0x 0y 0z P

0xy Q P 0xy

R3

x0 x

y0

y

(x0,y0)

x

y

(x2,y2)

(x1,y1)

(x1 + x2,y1 + y2)

(x1 − x2,y1 − y2)

(x0,y0) ∈ R2

R2

(x,y, 0) Q0x R Q 0x

(x, 0, 0) x RO = (0, 0, 0) = 0 1

E1 = (1, 0, 0) = e1 0xx

O = (0, 0, 0) = 0 P

Px

(x,y, z)

R3

R2

R2

λ > 1∥x∥

d(x, 0) = ∥x − 0∥ = ∥x∥ x 0x y Rn

d(x, y) = ∥x− y∥Rn

⟨x⟩ := {λx : λ ∈ R}, x ∈ Rn \ {0},

x ∈ Rn

0 ∈ Rn x

R3

x

y

(x,y)

λ(x,y), λ > 1

−(x,y)

−λ(x,y), λ > 1

R2

λ > 1

Rn

⟨x, y⟩ := {λx+ µy : λ,µ ∈ R}, x, y ∈ Rn ,

x, y ∈ Rn

0, x, y

⟨ei⟩ = {xiei : xi ∈ R}, i = 1, . . . ,n,

0xi

⟨ei, ej⟩ = {xiei + xjej : xi, xj ∈ R}, i, j = 1, . . . ,n, i = j,

0xixj⟨v⟩ v ∈ R3 \ {0}

a ∈ R3 R3

a+ ⟨v⟩ = a+ {λv : λ ∈ R} := {a+ λv : λ ∈ R} ⊂ R3,

a v⟨v, w⟩ v, w ∈ R3

a ∈ R3 R3

a+ ⟨v, w⟩ = a+ {λv+ µw : λ,µ ∈ R} = {a+ λv+ µw : λ,µ ∈ R} ⊂ R3,

av, w

R3

y

x

z

a

v

R3 a ∈ R3

v ∈ R3 \ {0}

y

x

z

a

v

w

R3 a ∈ R3

v, w ∈ R3

R3

x1

x2

x

y

ϑ

ϑ ∈ [0,π] 0x1x2

ϑ ∈ [0,π] x, y ∈ Rn \ {0}

cos ϑ =x · y

∥x∥∥y∥ , x, y ∈ Rn \ {0},

cos : [0,π] → [−1, 1] 1− 1ϑ ∈ [0,π]

cosϕ sinϕ x ya ∈ R2 ∥a∥ = 1

(0, 0) 0x ϕ ∈ R(1, 0)

a ϕ > 0ϕ < 0 2kπ ≤ |ϕ| < 2(k+ 1)π k ∈ N

ka

A(ϕ) : R2 → R2

R2 ϕ ∈ R

a =

(cosϕsinϕ

)= A(ϕ)

(10

), ϕ ∈ R.

(0, 0)

R3

x

y

1

a

ϕ

sinϕ

a ′

−ϕ cosϕ

− sinϕ

a⊥

ϕ

•

cosϕ

− sinϕ

b

ϑ

a = (cosϕ, sinϕ) a ′ = (cosϕ,− sinϕ) a⊥ = (− sinϕ, cosϕ)b = (cos(ϕ+ ϑ), sin(ϕ+ ϑ))

(0, 1) ϕ

a⊥ =

(− sinϕcosϕ

)= A(ϕ)

(01

), ϕ ∈ R,

a a⊥ · a = 0(1, 0) (0, 1)

A(ϕ)a, b ∈ R2

A(ϕ)(λa) = λA(ϕ)a A(ϕ)(a+ b) = A(ϕ)a+A(ϕ)b

R3

A(ϕ)

A(ϕ) =

(cosϕ − sinϕsinϕ cosϕ

), ϕ ∈ R.

2π

A(2kπ) = I =

(1 00 1

), k ∈ Z,

k ∈ N(0, 0)

cos(2kπ) = 1, sin(2kπ) = 0, k ∈ Z.

ϕ ϑ ϕ+ ϑ

A(ϕ+ ϑ) = A(ϑ)A(ϕ) = A(ϕ)A(ϑ), ϕ, ϑ ∈ R.

cos(ϕ+ ϑ) = cosϕ cos ϑ− sinϕ sin ϑ,

sin(ϕ+ ϑ) = sinϕ cos ϑ+ cosϕ sin ϑ, ϕ, ϑ ∈ R.

A(ϕ+ 2kπ) = A(ϕ), ϕ ∈ R, k ∈ Z,

2π

cos(ϕ+ 2kπ) = cosϕ, sin(ϕ+ 2kπ) = sinϕ, ϕ ∈ R, k ∈ Z.

A(ϕ)

detA(ϕ) = cos2ϕ+ sin2ϕ = 1, ϕ ∈ R,

a = (cosϕ, sinϕ)k = 0

I = A(−ϕ)A(ϕ) = A(ϕ)A(−ϕ), A(ϕ)−1 = A(−ϕ), ϕ ∈ R,

R3

ϕ

−ϕ

(1, 0) |ϕ|

a ′ =

(cosϕ

− sinϕ

)= A(−ϕ)

(10

)=

(cos(−ϕ)sin(−ϕ)

), ϕ ∈ R,

ϕ

A(ϕ) (1, 0) ϕ = π2 ,π,

3π2 , π4

cosπ

2= 0, cosπ = −1, cos

3π

2= 0, cos

π

4=

1√2,

sinπ

2= 1, sinπ = 0, sin

3π

2= −1, sin

π

4=

1√2.

ϕ = π2

cosϕ = sin(ϕ+

π

2

), ϕ ∈ R,

a = (cosϕ, sinϕ) b = (cos(ϕ+ ϑ), sin(ϕ+ ϑ)) ,

cos sin

a · b = cosϕ cos(ϕ+ ϑ) + sinϕ sin(ϕ+ ϑ)

= cosϕ cos(−ϕ− ϑ)− sinϕ sin(−ϕ− ϑ)

= cos(ϕ−ϕ− ϑ)

= cos ϑ.

2π

R3

x

B((0, 0, 0), 1)

z

y

x

y

B((0, 0, 0), 1)

z

x

y

∂B((0, 0, 0), 1)

z

n = 3

x

B((0, 0), 1)

y

x

B((0, 0), 1)

y

x

∂B((0, 0), 1)

y

n = 2

r > 0 x Rn

B(x, r) := {y ∈ Rn : ∥y− x∥ < r},

B(x, r) := {y ∈ Rn : ∥y− x∥ ≤ r},

∂B(x, r) := {y ∈ Rn : ∥y− x∥ = r},

n = 2

n = 1 ∥x∥ = |x| x = x(x− r, x+ r) [x− r, x+ r] {x−r}∪ {x+ r}

r > 0

Rn

B(x, r)∪ ∂B(x, r) = B(x, r) B(x, r)∩ ∂B(x, r) = ∅.

π

cosπ

3=

1

2, cos

π

6=

√3

2,

sinπ

3=

√3

2, sin

π

6=

1

2.

sin(2ϕ) = 2 sinϕ cosϕ,

cos(2ϕ) = 2 cos2ϕ− 1, ϕ ∈ R,

sinϕ+ sin ϑ = 2 sinϕ+ ϑ

2cos

ϕ− ϑ

2,

cosϕ+ cos ϑ = 2 cosϕ+ ϑ

2cos

ϕ− ϑ

2,

cos ϑ− cosϕ = 2 sinϕ+ ϑ

2sin

ϕ− ϑ

2, ϕ, ϑ ∈ R.

Rn

Rn

d Rn

Rn

U ⊂ Rn

∀ x ∈ U ∃ ε > 0 : B(x, ε) ⊂ U

Rn \U

∅ ⊂Rn Rn ⊂ Rn

Rn ∅ ∅ Rn

Rn

Rn

∅ = U ! Rn

U ⊂ Rn

U = {(x,y) ∈ R2 : x < 0,y ≥ 0}∪ {(x,y) ∈ R2 : x ≥ 0,y > 0},

R2 \U = {(x,y) ∈ R2 : x < 0,y < 0}∪ {(x,y) ∈ R2 : x ≥ 0,y ≤ 0},

(x, 0) x < 0ε > 0 B((x, 0), ε) U

(x,−ε2 ) ∈ B((x, 0), ε) ∩ (R2 \U) B((x, 0), ε) x ≥ 0

R2 \U (x, ε2 ) ∈ B((x, 0), ε)∩U

Rn

Rn

x ∈ Rn r > 0 y ∈ B(x, r) ∥y− x∥ < rε > 0 ∥y− x∥ = r− ε z ∈ B(y, ε)

∥z− x∥ ≤ ∥z− y∥+ ∥y− x∥ < ε+ r− ε = r

z ∈ B(x, r) B(y, ε) ⊂ B(x, r) y ∈B(x, r) B(x, r)

y ∈ Rn \ B(x, r) ∥y− x∥ > rε > 0 ∥y− x∥ = r+ ε z ∈ B(y, ε)

∥z− x∥ ≥ ∥y− x∥− ∥z− y∥ > r+ ε− ε = r,

B(y, ε) ⊂ Rn \ B(x, r) Rn \ B(x, r)B(x, r) ✷

x

yr ε

y ∈ B(x, r) ∥y− x∥ = r− ε B(y, ε) ⊂ B(x, r)

Rn

Rn

Rn

x ∈ ⋃i∈IUi Ui i ∈ I ∃ i0 ∈ I

x ∈ Ui0 Ui0 ε > 0 B(x, ε) ⊂ Ui0 ⊂ ⋃i∈IUi

x ∈ ⋂ki=1Ui Ui i = 1, . . . ,k k ∈ N

∀ i = 1, . . . , k εi > 0 B(x, εi) ⊂ Ui

ε = min{εi : i = 1, . . . , k} > 0 B(x, ε) ⊂ ⋂ki=1Ui ✷

Rn B(x, 1k ) k ∈ N⋂∞

k=1 B(x,1k ) = {x} Rn

{x} ⊂ Rn

Rn \ {x} y ∈ Rn \ {x}y = x B(y, ∥x− y∥) ⊂ Rn \ {x}

Rn ∅,Rn

Rn

Rn

Rn

Ki Rn i ∈ I Ii = 1, . . . , k k ∈ N

Rn \⋂

i∈IKi =

⋃

i∈I(Rn \Ki) Rn \

k⋃

i=1

Ki =k⋂

i=1

(Rn \Ki).

✷

Rn

B(x, 1− 1k+1 ) k ∈ N x ∈ Rn

⋃∞k=1 B(x, 1−

1k+1 ) = B(x, 1)

Rn

∂B(x, r) = {y ∈ Rn : ∥y− x∥ = r}

= {y ∈ Rn : ∥y− x∥ ≤ r}∩ {y ∈ Rn : ∥y− x∥ ≥ r}

= B(x, r)∩ (Rn \B(x, r)).

Rn

U ⊂ Rn

∃ r > 0 : U ⊂ B(0, r)

Rn x ∈ Rn

r > 0

B(x, r), ∂B(x, r) ⊂ B(x, r) ⊂ {y ∈ Rn : ∥y∥ ≤ r+ ∥x∥} ⊂ B(0, r+ ∥x∥+ ε)

ε > 0Rn

U Rn \U U0x

U

Rn

x

y

(x1,y1)

U

(x2,y2)

(x3,y3)(x0,y0)

(x0,y0) (x3,y3) U (x2,y2)(x1,y1)

U Rn \U0x

URn U

U ⊂ Rn x ∈ Rn

U ∃ ε > 0 : B(x, ε) ⊂ U

U x Rn \U

U xU

UintU U◦ extU bdU ∂U U

U ⊂ Rn

Rn

Rn = intU∪ extU∪ bdU,

U ⊂ R2

∂U = R × {0}, intU = R × (0,∞), extU = R × (−∞, 0),

x ∈ R ε > 0B((x, 0), ε) U R2 \U y = 0

B((x,y), |y|) U y > 0 R2 \U y < 0

U ⊂ Rn

intU ⊂ U

intU

U ⇔ intU = U

U ⊂ V ⊂ Rn ⇒ intU ⊂ intV ,

extU = int (Rn \U) ⊂ Rn \U.

x ∈ B(x, ε) ε > 0x ∈ intU B(x, ε) ⊂ U ε > 0

B(x, ε) Rn y ∈ B(x, ε)ε(y) > 0 B(y, ε(y)) ⊂ B(x, ε) ⊂ U

y ∈ B(x, ε) U B(x, ε) ⊂ intUx ∈ intU intU

⇒⇐

✷

intB(x, r) = B(x, r) = int B(x, r),

extB(x, r) = Rn \ B(x, r) = ext B(x, r),

bdB(x, r) = ∂B(x, r) = bd B(x, r).

∂B(x, r)

B(x, r) ⊂ B(x, r)

B(x, r) ⊂ int B(x, r) extB(x, r) ⊃ Rn \ B(x, r).

intU ⊂ U extU ⊂ Rn \ U B(x, r) = B(x, r) ∪∂B(x, r)

∂B(x, r)int B(x, r) extB(x, r)

y ∈ ∂B(x, r) ε > 0

x+(1+

min{ε, r}

2r

)(y− x) ∈ B(y, ε)∩ (Rn \ B(x, r)),

x+(1−

min{ε, r}

2r

)(y− x) ∈ B(y, ε)∩ B(x, r),

y int B(x, r)y B(x, r) extB(x, r)

y B(x, r)Rn \ B(x, r)

Rn

U ⊂ Rn

U

U ⊂ Rn

U ⊂ Rn Rn

U U U

U :=⋂

K∈KK, K = {K ⊂ Rn : K , K ⊃ U}.

U ⊂ Rn

U ⊂ U

U

U ⊂ K ⊂ Rn K ⇒ U ⊂ K

U ⇔ U = U

x ∈ U K x ∈ U ⊂ KK ∈ K x ∈ ⋂K∈K K

U Rn

K ∈ K ⋂L∈K L ⊂ K

⇒ U ⊂ U U U ⊂ UU = U

⇐✷

URn U ⊂ Rn

U UU

U ⊂ Rn x ∈ Rn

U ∃ ε > 0 : U∩ B(x, ε) = {x}

U ∀ ε > 0 : U∩ B(x, ε) \ {x} = ∅

U x ∈ U x U

U U ′

U U U∪U ′

U ⊂ Rn

x U ⇒ x ∈ U∩ ∂U

x ∈ U ⇒ x U

intU ⊂ U ′

extU ⊂ Rn \U ′

U ′

U U ⊂ Rn

U ⊂ Rn U = U∪U ′

⊃ U ⊂ UU ′ ⊂ U Rn \ U ⊂ Rn \U ′ x ∈ Rn \ U

U ε > 0 B(x, ε) ⊂ Rn \ U ⊂ Rn \UB(x, ε)∩U = ∅ x U

⊂ Rn \ (U ∪U ′) ⊂ Rn \ Ux ∈ Rn \U U ε > 0

B(x, ε) ∩ U = ∅ U ⊂ Rn \ B(x, ε) U ⊂ Rn \ B(x, ε)B(x, ε) ⊂ Rn \ U x ∈ Rn \ U ✷

Rn

U ⊂ Rn ⇔ U ′ ⊂ U

U = U ⇔ U ′ ⊂ U

U = U∪U ′ ⇔ U ′ ⊂ U,

✷

U ⊂ Rn U = U◦ ∪ ∂U

Rn \ U = extU U U ⊂ U

Rn \ U = int (Rn \ U) ⊂ int (Rn \U) = extU.

x ∈ extU ε > 0 B(x, ε) ⊂ Rn \Ux ∈ (Rn \U) ∩ (Rn \U ′) = Rn \ (U ∪U ′) = Rn \ U

✷

B(x, r)B(x, r)

U = Rn−1 × (0,∞) = {x = (x1, . . . , xn) ∈ Rn : xn > 0}U◦, extU,∂U

x = (x1, . . . , xn) ∈ U B(x, xn) ⊂ Uy = (y1, . . . ,yn) ∈ B(x, xn)

|yn − xn| ≤ ∥y− x∥∞ ≤ ∥y− x∥ < xn,

0 < yn < 2xn y ∈ U U

U◦ = U = Rn−1 × (0,∞).

x ∈ Rn−1 × {0} x = (x1, . . . , xn−1, 0) xi ∈ Ri = 1, . . . ,n− 1 ε > 0

x+ε

2en ∈ B(x, ε)∩U x−

ε

2en ∈ B(x, ε)∩ (Rn \U)

en = (0, . . . , 0, 1) ∥± ε2 en∥ = ε

2 < εx± ε

2 en = (x1, . . . , xn−1,±ε2 )ε2 > 0 ε > 0

B(x, ε) Rn \U Ux ∈ Rn−1 × {0} U

Rn−1 × {0} ⊂ ∂U.

V = Rn−1 × (−∞, 0) = {x = (x1, . . . , xn) ∈ Rn : xn < 0}U x ∈ V y ∈ B(x,−xn) |yn− xn| <

−xn 2xn < yn < 0 y ∈ V

Rn−1 × (−∞, 0) ⊂ extU.

(extU)∪∂U = Rn−1× (−∞, 0] (extU)∩∂U =∅

∂U = Rn−1 × {0}, extU = Rn−1 × (−∞, 0).

U ⊂ Rn

intU =⋃

A∈AA, A = {A ⊂ Rn : A , A ⊂ U}.

RN

U ⊂ Rn

x ∈ ∂U ⇔ ∀ ε > 0 : B(x, ε)∩U = ∅ B(x, ε)∩ (Rn \U) = ∅.

U,V ⊂ Rn U ⊂ V ⇒ U ⊂ V

R R2

n = 1, 2, 3

U ⊂ R2

V ⊂ R2 V = {(0, 0)}∪ (B((0, 0), 2) \ B((0, 0), 1))∪ ∂B((0, 0), 3)

Rn

∥ · ∥∥ · ∥∞

Rn

Rn (xν)ν∈N ⊂ Rn (xν) ⊂ Rn

(xν) ⊂ R

| · | R∥ · ∥ Rn

Rn

∥x− y∥ Rn

d(x,y)

Rn

Rn

RN

x

y

(x1,y1)

(x2,y2)

(x3,y3)

(x4,y4)

(x5,y5)

(x6,y6)

R2

N Rn

N ∋ ν 4→ xν ∈ Rn,

Rn

(xν)ν∈N = (x(1)ν , . . . , x

(n)ν )ν∈N ⊂ Rn, (xν) ⊂ Rn, xν ∈ Rn, ν ∈ N.

xν ∈ Rn (xν) ⊂ Rn

(xν) ⊂ Rn x0 ∈ Rn

x0 0

∥xν − x0∥ → 0 ν→ ∞,

xν → x0 ν→ ∞, xν → x0, (x(1)ν , . . . , x

(n)ν ) → (x

(1)0 , . . . , x

(n)0 ).

x0 ∈ Rn (xν) ⊂ Rn

(xν) ⊂ Rn x0 ∈ Rn xν → x0

(xν,yν) ∈ R2 ν ∈ N (xν,yν) =

( 1ν, 0),

(0,

1

ν2

),

( 1ν,1

ν

),

( 1ν,1

ν3

),

(sin(1/

√ν), e−ν

)

(0, 0) ∈ R2

∥(xν,yν)− (0, 0)∥ = ∥(xν,yν)∥ =√

x2ν + y2ν → 0,

RN

x

y

x

y

x

y

x

y

x

y

∥(xν,yν)∥2 = x2ν + y2ν → 0

f(x) = x2 x ≥ 0f−1(y) =

√y y ≥ 0 x = 0 y = 0

(xν,yν)xν → 0 yν → 0

xν → 0yν → 0

0 ≤ x2ν ≤ x2ν + y2ν → 0 ⇒ x2ν → 0 ⇔ |xν| → 0 ⇔ xn → 0

(yν)

(xν,yν) → (0, 0) ⇔ xν → 0 yν → 0,

n ∈ N

Rn n ≥ 2R(xν) ⊂ Rn x0

n xν+1 − xνxν xν+1 n

n

n

RN

R

Rn

0(xν) ⊂ R x0 ∈ R |xν − x0| → 0

| · | R

(xν,yν, zν) =( 1νsinν,

1

νcosν, 1−

1

ν

), ν ∈ N,

(0, 0, 1) ∈ Rn

∥(xν,yν, zν)− (0, 0, 1)∥ = ∥(xν,yν, zν − 1)∥ =1

ν∥(sinν, cosν,−1)∥ =

√2

ν→ 0.

Rn

xν → x0 ⇔ ∥xν − x0∥ → 0 ⇔ ∥(xν − x0)− 0∥ → 0 ⇔ xν − x0 → 0

∥xν − x0∥ ν ∈ N

R

xν → x0 ⇔ ∀ ε > 0 ∃ ν0 ∈ N ∀ ν ∈ N, ν ≥ ν0 : ∥xν − x0∥ < ε.

Rn

R

(xν) ⊂ Rn

limν→∞

xν

RN

xν → x0 xν → y0 x0 = y0 ∥x0 − y0∥ > 0

ε = ∥x0−y0∥2 > 0

∃ ν1 ∈ N ∀ ν ∈ N, ν ≥ ν1 : ∥xν − x0∥ <∥x0 − y0∥

2,

∃ ν2 ∈ N ∀ ν ∈ N, ν ≥ ν2 : ∥xν − y0∥ <∥x0 − y0∥

2,

∀ ν ∈ N, ν ≥ max{ν1,ν2}

∥x0 − y0∥ ≤ ∥x0 − xν∥+ ∥xν − y0∥ <∥x0 − y0∥

2+

∥x0 − y0∥2

= ∥x0 − y0∥,

α ∈ R α < α ✷

(xν) ⊂ Rn

∃ r > 0 : (xν) ⊂ B(0, r)

xν → x0 ε = 1

∃ ν0 ∈ N ∀ ν ∈ N, ν ≥ ν0 : ∥xν − x0∥ < 1

∥xν∥ ≤ ∥xν − x0∥+ ∥x0∥

∃ ν0 ∈ N ∀ ν ∈ N, ν ≥ ν0 : ∥xν∥ < 1+ ∥x0∥.

∀ ν ∈ N : ∥xν∥ ≤ max{∥x1∥, . . . , ∥xν0∥, 1+ ∥x0∥} =: r0

r > r0 ✷

Rn xν → x0 yν →y0 R αν → α βν → β

ανxν +βνyν → αx0 +βy0.

R(αν), (βν) ⊂ R

∥ανxν +βνyν − (αx0 +βy0)∥ ≤ ∥ανxν −αx0∥+ ∥βνyν −βy0∥≤ |αν|∥xν − x0∥+ |αν −α|∥x0∥+ |βν|∥yν − y0∥+ |βν −β|∥y0∥ → 0.

✷

RN

∞ ∥x∥∞ = max{|xi| : i = 1, . . . ,n}

∥x∥∞ ≤ ∥x∥ ≤√n∥x∥∞ ∀ x ∈ Rn.

Rn

R

xν =(x(1)ν , . . . , x

(n)ν

)∈ Rn, ν ∈ N, x0 =

(x(1)0 , . . . , x

(n)0

)∈ Rn.

xν → x0 ⇔ x(i)ν → x

(i)0 ∀ i = 1, . . . ,n.

⇒ ν ∈ N

|x(i)ν − x

(i)0 | ≤ ∥xν − x0∥∞ ≤ ∥xν − x0∥ ∀ i = 1, . . . ,n,

i = 1, . . . ,n

∀ ε > 0 ∃ ν0 ∈ N ∀ ν ∈ N, ν ≥ ν0 : |x(i)ν − x

(i)0 | ≤ ∥xν − x0∥ < ε,

R x(i)ν → x

(i)0 i = 1, . . . ,n

⇐: x(i)ν → x

(i)0 i = 1, . . . ,n i = 1, . . . ,n

∀ ε > 0 ∃ νi ∈ N ∀ ν ∈ N, ν ≥ νi : |x(i)ν − x

(i)0 | < 1√

nε.

i = 1, . . . ,n νi ∈ Ni = 1, . . . ,n

ν0 := max{νi : i = 1, . . . ,n}i = 1, . . . ,n

∀ ε > 0 ∃ ν0 ∈ N ∀ ν ∈ N, ν ≥ ν0 : |x(i)ν − x

(i)0 | < 1√

nε.

ε > 0 ν ≥ ν0∥xν − x0∥∞ < 1√

nε ∥xν − x0∥ < ε

∀ ε > 0 ∃ ν0 ∈ N ∀ ν ∈ N, ν ≥ ν0 : ∥xν − x0∥ < ε,

xν → x0 ✷

i ν0

RN

RRn

(xν) ⊂ Rn

∀ ε > 0 ∃ ν0 ∈ N ∀ ν,µ ∈ N, ν,µ ≥ ν0 : ∥xν − xµ∥ < ε.

(xν) ⊂ Rn

⇒ xn → x0

∀ ε > 0 ∃ ν0 ∈ N ∀ ν ∈ N, ν ≥ ν0 : ∥xν − x0∥ <ε

2

∀ ε > 0 ∃ ν0 ∈ N ∀ ν,µ ∈ N, ν,µ ≥ ν0 :

∥xν − xµ∥ ≤ ∥xν − x0∥+ ∥xµ − x0∥ < ε.

⇐ xν = (x(1)ν , . . . , x

(n)ν ) ∈ Rn ν ∈ N

∞

|x(i)ν − x

(i)µ | ≤ ∥xν − xµ∥∞ ≤ ∥xν − xµ∥ ∀ i = 1, . . . ,n,

R (x(i)ν )ν∈N

i = 1, . . . ,n R(xν) ⊂ Rn

✷

R Rn

(xν) ⊂ Rn

(xkν) ⊂ (xν)

xν = (x(1)ν , . . . , x

(n)ν ) ∈ Rn ν ∈ N

r > 0 ∥xν∥ < r ν ∈ N∞ i = 1, . . . ,n

|x(i)ν | < r ∀ ν ∈ N,

R R

RN

(x(i)ν )ν∈N ⊂ R i = 1, . . . ,n

R

(kν)ν∈N ⊂ N

(x(i)ν ) i = 1, . . . ,n (x

(i)kν

)i = 1, . . . ,n

(xℓν) (xν) (x(1)ℓν

) (x(i)ℓν

)i = 2, . . . ,n

(xℓmν ) (xℓν) (x(2)ℓmν

)

(x(1)ℓmν

) (x(1)ℓν

)

(x(i)ℓmν

) i = 3, . . . ,nn

(xkν) (xν) (x(i)kν

) i = 1, . . . ,n

✷

Rn

U ⊂ Rn

URn U

U ⊂ Rn x ∈ Rn

x ∈ U ′ ⇔ ∃ (xν) ⊂ U \ {x} : xν → x.

x ∈ Rn

U ε > 0 U∩ B(x, ε) \ {x} = ∅ x ∈ U ′

ν ∈ N xν ∈ U \ {x} ∥xν − x∥ < 1ν → 0

(xν) ⊂ U \ {x} xν → xε > 0 ν0 ∈ N ∥xν0 − x∥ < ε U∩B(x, ε) \ {x} = ∅ ✷

UU Rn

U

U ⊂ Rn x ∈ Rn

x ∈ U ⇔ ∃ (xν) ⊂ U : xν → x.

RN

U = U ∪ U ′ x ∈ Uxν = x ν ∈ N (xν) ⊂ U

xν → x ∥xν − x∥ = 0 → 0 x ∈ U ′

(xν) ⊂ U \ {x} ⊂ U xν → x(xν) ⊂ U xν → x ν ∈ N xν = x

x ∈ U xν = x ν ∈ N x ∈ U ′

✷

Rn

Rn

U ⊂ Rn ⇔ ∀ (xν) ⊂ U xν → x ∈ Rn : x ∈ U.

⇒ U ⊂ Rn (xν) ⊂ U xν → x ∈ Rn

x ∈ U = U⇐ x ∈ U (xν) ⊂ U xν → x

x ∈ U U = U U ✷

Rn

Rn

U ⊂ Rn ⇔ ∀ (xν) ⊂ U ∃ (xkν) ⊂ (xν) x ∈ U : xkν → x.

⇒: (xν) ⊂ U U ⊂ Rn

(xν)(xkν) xkν → x ∈ Rn

x ∈ U⇐: U ν ∈ N xν ∈ U∥xν∥ ≥ ν (xkν) (xν) ∥xkν∥ ≥

kν ≥ ν (xkν)U

(xν) ⊂ U xν → x ∈ Rn (xkν)(xν) xkν → x x ∈ U

U ✷

Rn

xν → x ∈ Rn ⇒ ∥xν∥ → ∥x∥ ∈ R

RN

x ∈ Rn {x}

R2 R3 R3

f : U → R, U ⊂ R,

y = f(x) ∈ R x ∈ U

Rn

Rm n,m ∈ Nn = m = 1

U ⊂ Rn n ∈ N f : U → Rm m ∈ N

Rn ⊃ U ∋ x = (x1, . . . , xn) 4→ f(x) =

⎛

⎜⎝f1(x)

fm(x)

⎞

⎟⎠ =

⎛

⎜⎝f1(x1, . . . , xn)

fm(x1, . . . , xn)

⎞

⎟⎠ ∈ Rm,

n n ≥ 2

m ≥ 2 m = 1

, m = n ≥ 2

fj : U → R j = 1, . . . ,mf U ⊂ Rn Rm f

f(U) := {y ∈ Rm : ∃ x ∈ U : f(x) = y} = {f(x) ∈ Rm : x ∈ U} ⊂ Rm

f f

Γf := {(x, f(x)) : x ∈ U} ⊂ Rn+m

f

x ∈ Rn

f : U → Rm, m ≥ 2

f(x) = y ∈ Rm

f : U → R.

nf : U → Rm U ⊂ Rn U Rm

x ∈ U f(x) ∈ Rm

Rn

f, g : U → Rm U ⊂ Rn

f g

f+ g : U → R, (f+ g)(x) := f(x) + g(x) ∀ x ∈ U,

f α ∈ R

αf : U → R, (αf)(x) := αf(x) ∀ x ∈ U,

U f

f h : V → Rk f(U) ⊂ V ⊂ Rm

h ◦ f : U → R, (h ◦ f)(x) := h(f(x)) ∀ x ∈ U.

n,m ∈ N0 : U → Rm 0(x) := 0 ∈ Rm x ∈ U

f,g : U → R U ⊂ Rn

f g

fg : U → R, (fg)(x) := f(x)g(x) ∀ x ∈ U,

g(x) = 0 ∀ x ∈ U f g

f

g: U → R,

(f

g

)(x) :=

f(x)

g(x)∀ x ∈ U,

f : U → R U ⊂ Rn n ≥ 2

n = 1

(x,y) ∈ U ⊂ R2

(x,y, z) ∈ R3

U ⊂ R2

0xy0z

z = f(x,y) ∈ R|z| 0xy z > 0

z < 0 U = [a,b]× [c,d]f : U → R f(x,y) > 0

(x,y) ∈ U

Γf = {(x,y, f(x,y)) ∈ R3 : (x,y) ∈ U}

z

x

y

(x,y, f(x,y))

(x,y)

U

f : U → R U ⊂ R2 R3

R3 U

f

U ⊂ R2

(x,y) ∈ Uz = f(x,y)

(x,y)

n ≥ 2n ≥ 3

n ≥ 3n = 2

n ≥ 2

f : U → R

U ⊂ Rn

Γf = {(x, f(x)) ∈ Rn+1 : x ∈ U},

Rn+1

R

Rm

R3

f : U → Rn U ⊂ Rn n ≥ 2

x ∈ U ⊂ Rn

U Rn f(x) ∈ Rn

x

0x r > 0

U = R × B((0, 0), r) = {(x,y, z) ∈ R3 : y2 + z2 ≤ r2}.

x ∈ U f(x) = (α, 0, 0)α ∈ R

f : U → R3, f(x) = (α, 0, 0), α ∈ R,

α > 00x α < 0

α = 0

U f(x,y, z)(x,y, z) ∈ U

(x,y, z) ∈ U Ux

f : U → R3, f(x,y, z) = c(r2 − y2 − z2, 0, 0), c > 0.

0 = (0, 0, 0) ∈ R3

(x,y, z) ∈ R3 \ {0},

(x,y, z)

f(x,y, z) = −c(x,y, z)

∥(x,y, z)∥3 , (x,y, z) ∈ R3 \ {0}, c > 0

∥f(x,y, z)∥ = c1

∥(x,y, z)∥2 .

f(x) = x, x ∈ Rn,

x ∈ Rn

f(x) = x x0

f : U → Rn n ≥ 2 U ⊂ R

y

x

z

n ≥ 2

I ⊂ RRn

γ : I → Rn

t ∈ R

Rn

γ : I → Rn γ(t) ∈ Rn Rn

t ∈ I

x

y

R2

γ : I → Rn

γ(I) = {γ(t) : t ∈ I} ⊂ Rn

Rn

R2 R3

x

y

cos t

sin t

t

Rn

Γf = {(t, f(t)) : t ∈ I ⊂ R} ⊂ R2

f : I → R I ⊂ RR2

Γf = γ(I), γ : I → R2, γ(t) = (t, f(t)),

γ

R2

(0, 0) 1

C = γ([0, 2π]) = {γ(t) = (cos t, sin t) : t ∈ [0, 2π]} ⊂ R2,

γ : [0, 2π] → R2

R3 R3

γ : R → R3, γ(t) = a+ tv,

a, v ∈ R3 v = 0

yx

z

γ(t) = (t cos(αt), t sin(αt), t) ∈ R3, t ≥ 0, α > 0.

γ([0,∞)) ⊂ R3 γ

R3 f : U → R3 U ⊂ R2

R3

f(x,y) = (x,y, f(x,y)) ∈ R3, (x,y) ∈ U ⊂ R2, f : U → R,

f f

R3

R3

∂B((0, 0, 0), 1) = {(x,y, z) ∈ R3 : x2 + y2 + z2 = 1}

f±(x,y) = ±√

1− x2 − y2, (x,y) ∈ B((0, 0), 1),

R3

f(λ,µ) = a+ λ v+ µ w ∈ R3, (λ,µ) ∈ R2,

a, v, w ∈ R3 v, w ∈ R3

R3

R3

R3

R3

R3

R3

n ≥ 2

f : U → R, U ⊂ Rn,

R

Ux ∈ U f(x) = c ∈ R

(x,y) ∈ Uf

Un = 2

c c

cn ≥ 3

c

f : U → R U ⊂ Rn c ∈ RU f x ∈ U f c

Lf(c) := {x ∈ U : f(x) = c}

c fn = 2 c c f

Lf(c) = {(x,y) ∈ U : f(x,y) = c},

n = 3 c c f

Lf(c) = {(x,y, z) ∈ U : f(x,y, z) = c}.

U f Lf(c) = ∅ c ∈ R \ f(U)n = 2 n = 3

f(x,y) = x2 + y2, (x,y) ∈ R2,

Γf = {(x,y, x2 + y2) ∈ R3 : (x,y) ∈ R2},

c ∈ RR2 (0, 0)

√c > 0 c > 0

Lf(c) = {(x,y) ∈ R2 : x2 + y2 = c} = ∂B((0, 0),

√c)

c > 0

y

x

z

0xy

(0, 0) 0x 0y c = 0

Lf(0) = {(0, 0)},

c < 0

Lf(c) = ∅ c < 0

c ∈ R cLf(c) c ∈ R

f ⋃

c∈R

Lf(c) =⋃

c≥0

Lf(c) = R2,

(x,y) ∈ R2 Lf(∥(x,y)∥2)f

Lf(c)× {c} R3 c ∈ R

Γf =⋃

c∈R

Lf(c)× {c} =⋃

c≥0

Lf(c)× {c}

c > 0

Lf(c)× {c} = {(x,y, c) ∈ R3 : (x,y) ∈ Lf(c)} = {(x,y, c) ∈ R3 : x2 + y2 = c}

(0, 0, c)√c z = c

fLf(c)× {0} z = 0

(0, 0, c)

Lf(c)× {c} = (0, 0, c) + Lf(c)× {0} := {(0, 0, c) + (x,y, 0) ∈ R3 : (x,y) ∈ Lf(c)}.

Lf(c)R2 Lf(c)× {c}

Lf(c) = {(x,y) ∈ R2 : (x,y, z) ∈ Lf(c)× {c}}.

c > 0R2

f(x,y, z) = x2 + y2 + z2, (x,y, z) ∈ R3

Γf = {(x,y, z, x2 + y2 + z2) ∈ R4 : (x,y, z) ∈ R3}

c > 0

Lf(c) = {(x,y, z) ∈ R3 : x2 + y2 + z2 = c} = ∂B((0, 0, 0),

√c)

c > 0

R3 (0, 0, 0)√c > 0

R3

R3 R3

Lf(0) = {(0, 0, 0)} Lf(c) = ∅ c < 0

R2

f : R2 → R, f(x,y) = d ∀ (x,y) ∈ R2,

d ∈ R z = d R3

Γf = {(x,y,d) ∈ R3 : (x,y) ∈ R2} = {(x,y, z) ∈ R3 : z = d},

c R2 c = dc = d

Lf(c) =

{R2 c = d

∅ c = d

R2

U ⊂ Rn

f : U → R, f(x) = d ∀ x = (x1, . . . , xn) ∈ U,

d ∈ R U× {d} xn+1 =d Rn+1

Γf = {(x,d) := (x1, . . . , xn,d) ∈ Rn+1 : x ∈ U} = {(x, xn+1) ∈ Rn+1 : xn+1 = d},

c U c = dc = d

Lf(c) =

{U c = d

∅ c = d

n = 3

U = [α1,β1]× [α2,β2]× [α3,β3] ⊂ R3,

c = d

f(x,y) = h− x2 − y2, (x,y) ∈ B((0, 0),

√h), h > 0

ff z = c c ∈ [0,h]

f

f(x,y) = x2 − y2, (x,y) ∈ R2,

c ∈ Rf

x = α y = β z = γ α,β,γ ∈ R

f : Rn → RLf(c) c ∈ R

f(x,y) = x2 + y2, c = 0, 1, 4, 9,

f(x,y) = exy, c = e−2, e−1, 1, e, e2, e3,

f(x,y) = cos(x+ y), c = −1, 0,1

2,

√2

2, 1,

f(x,y, z) = x+ y+ z, c = −1, 0, 1,

f(x,y, z) = x2 + 2y2 + 3z2, c = 0, 6, 12,

f(x,y, z) = sin(x2 + y2 + z2), c = −1,−1

2, 0,

√2

2, 1.

U ⊂ Rn f : U → R x0 ∈ Rn Uℓ ∈ R f ℓ x x0 f

x0 ℓ f(x) → ℓ, x → x0

∀ (xν) ⊂ U \ {x0} : xν → x0 ⇒ f(xν) → ℓ.

xν → x0 Rn f(xν) → ℓR

U ⊂ Rn f : U → R x0 ∈ Rn Uℓ ∈ R

f(x) → ℓ, x → x0 ⇔ ∀ ε > 0 ∃ δ > 0 ∀ x ∈ U∩B(x0, δ) \ {x0} : |f(x)− ℓ| < ε.

⇒: ∃ ε > 0 ∀ δ > 0 ∃ x ∈ U ∩ B(x0, δ) \ {x0} : |f(x)− ℓ| ≥ ε∀ ν ∈ N ∃ xν ∈ U∩B(x0,

1ν ) \ {x0} : |f(xν)− ℓ| ≥ ε ∃ (xν) ⊂

U \ {x0} xν → x0 f(xν) → ℓ⇐: (xν) ⊂ U \ {x0} xν → x0 ε > 0 ∃ δ > 0 ∀ x ∈ U ∩

B(x0, δ) \ {x0} : |f(x)− ℓ| < ε ∃ ν0 ∈ N ∀ ν ∈ N,ν ≥ ν0 : xν ∈U∩ B(x0, δ) \ {x0} ∀ ν ∈ N,ν ≥ ν0 : |f(xν)− ℓ| < ε ✷

U ⊂ Rn f : U → R x0 ∈ Rn

U f x x0lim

x→x0

f(x)

x x0 f ℓ1 ℓ2|ℓ1 − ℓ2| > 0 i = 1, 2

∃ δi > 0 ∀ x ∈ U∩ B(x0, δi) \ {x0} : |f(x)− ℓi| <|ℓ1 − ℓ2|

2

δ := min{δ1, δ2} > 0

∀ x ∈ U∩ B(x0, δ) \ {x0} : |ℓ1 − ℓ2| ≤ |ℓ1 − f(x)|+ |f(x)− ℓ2| < |ℓ1 − ℓ2|,

✷

limx→x0

f(x) = ℓ⇔ limx→x0

|f(x)− ℓ| = 0.

x0 U δ0 > 0 B(x0, δ0) ⊂U ∀ δ > 0 x ∈ B(x0, δ) ⇔ η := x− x0 ∈ B(0, δ)

limx→x0

f(x) = ℓ⇔ ∀ ε > 0 ∃ δ ∈ (0, δ0) ∀ x ∈ B(x0, δ) \ {x0} : |f(x)− ℓ| < ε

⇔ ∀ ε > 0 ∃ δ ∈ (0, δ0) ∀ η ∈ B(0, δ) \ {0} : |f(x0 + η)− ℓ| < ε

⇔ limη→0

f(x0 + η) = ℓ.

f,g : U → R U ⊂ Rn x0 Ulim

x→x0

f(x) = ℓ ∈ R limx→x0

g(x) = m ∈ R

limx→x0

(f+ g)(x) = ℓ+m

limx→x0

(αf)(x) = α ℓ α ∈ R

limx→x0

(fg)(x) = ℓm

limx→x0

(f

g

)(x) =

ℓ

mm = 0

limx→x0

(h ◦ f)(x) = h(ℓ) h : V → R f(U) ⊂ V ⊂ R ℓ ∈ V.

(xν) ∈ U \ {x0} xν → x0 (f(xν)) ⊂ Vf(xν) → ℓ ∈ V h : V → R ℓ (h ◦ f)(xν) =h(f(xν)) → h(ℓ)

h(y) = αy y ∈ R ✷

f : U → R U ⊂ Rn x0 Ulim

x→x0

f(x) = ℓ ∈ R

limx→x0

|f(x)| = |ℓ|

limx→x0

√|f(x)| =

√|ℓ|

h(y) = |y| y ∈ R h(y) =√

|y| y ∈ R ✷

f(x,y) = x (x,y) ∈ R2

R3

Γf = {(x,y, x) ∈ R3 : (x,y) ∈ R2}

z = x c ∈ R

Lf(c) = {(x,y) ∈ R2 : x = c} = {(c,y) ∈ R2 : y ∈ R}

xy x = c

lim(x,y)→(x0,y0)

f(x,y) = lim(x,y)→(x0,y0)

x = x0,

|f(x,y)− x0| = |x− x0| ≤ ∥(x,y)− (x0,y0)∥

∀ ε > 0 ∃ δ := ε > 0 ∀ (x,y) ∈ B((x0,y0), δ)∀ (x,y) ∈ R2 ∥(x,y)− (x0,y0)∥ < δ |f(x,y)− x0| < ε

f(x,y) = xy (x,y) ∈ R2

Γf = {(x,y, xy) ∈ R3 : (x,y) ∈ R2}

z = xy c ∈ R

Lf(c) = {(x,y) ∈ R2 : xy = c},

xy y =c

xx = (x,y)

x0 = (x0,y0)lim

x→x0

xy = limx→x0

x · limx→x0

y = x0y0

f(x,y) = sin(x2+y2)x2+y2 = sin(∥x∥2)

∥x∥2 = f(x) ∥x∥ > 0 f

x = (x,y) 0 = (0, 0)

f : U → R U ⊂ Rn

x0 ∈ U

∀ (xν) ⊂ U : xν → x0 ⇒ f(xν) → f(x0)

A ⊂ U f : U → R x0 ∈ A

f : U → R U

Af : U → R A ⊂ U

f|A : A → R, f|A(x) := f(x) ∀x ∈ A

f AA

f : U → R x0 ∈ Ux0 U

f x0 ⇔ limx→x0

f(x) = f(x0)

⇔ ∀ ε > 0 ∃ δ > 0 ∀ x ∈ U∩ B(x0, δ) : |f(x)− f(x0)| < ε

f x0 f(x0) f f(x0)x x0 f(x) → f(x0), x → x0

f,g : U → R x0 ∈ U ⊂ Rn

x0

f+ g

αf α ∈ R

fg

f

gg(x0) = 0

h ◦ f h : V → R f(U) ⊂ V ⊂ R f(x0)

x0U x0

✷

f : U → R x0 ∈ U ⊂ Rn

|f| : U → R, |f|(x) := |f(x)| ∀ x ∈ U,√

|f| : U → R,√

|f|(x) :=√

|f(x)| ∀ x ∈ U,

x0

h(y) = |y| y ∈ R h(y) =√

|y| y ∈ R ✷

U ⊂ Rn f : U → R

C(U) := {f : U → R : f }.

f,g ∈ C(U), α ∈ R ⇒ f+ g,αf, fg, |f|,√

|f| ∈ C(U)

f : U → R U ⊂ Rn f(U)f U

max f := max f(U) = max{f(x) ∈ R : x ∈ U},

min f := min f(U) = min{f(x) ∈ R : x ∈ U}

∃ xm, xM ∈ U : min f = f(xm) ≤ f(x) ≤ f(xM) = max f ∀ x ∈ U.

f(U) ⊂ Rm = 1 R

min ff(U) ⊂ R

inf f := inf f(U) = inf{f(x) ∈ R : x ∈ U} ∈ R,

∀ ν ∈ N ∃ (xν) ⊂ U : f(xν) ∈[inf f, inf f+

1

ν

)

f(xν) → inf f. f(U)inf f = min f ∈ f(U) ∃ xm ∈ U : f(xm) = min f. ✷

f : U → R U ⊂ Rn

∀ ε > 0 ∃ δ > 0 ∀ x, y ∈ U, ∥x− y∥ ≤ δ : |f(x)− f(y)| < ε

U ⊂ Rn f : U → R f

m = 1 ✷

f(x,y) = (x2 + y2)exy, g(x,y, z) =sin(ex + ey + ez)

ln(x2 + y2 + z2), h(s, t) = e−s cos(st).

f : R2 → R

f(x,y) =

{xy2

x2+y2 , (x,y) = (0, 0),

0, (x,y) = (0, 0).

f R2 \ {(0, 0)}

f (0, 0) y = ax a ∈ Rg(x) = f(x,ax) x ∈ R x = 0

f (0, 0)

R2

f1(x,y) = x4 + y4 − 4x2y2, f6(x,y) = arcsinx√

x2 + y2,

f2(x,y) = ln(x2 + y2), f7(x,y) = arctanx+ y

1− xy,

f3(x,y) =1

ycos x2, f8(x,y) =

x√x2 + y2

,

f4(x,y) = tanx2

y, f9(x,y) = xy

2,

f5(x,y) = arctany

x, f10(x,y) = arccos

√x

y.

U ⊂ R2 (a,b) ∈ U f : U → R

lim(x,y)→(a,b)

f(x,y) = L ∈ R.

ε > 0

limx→a

f(x,y), 0 < |y− b| < ε limy→b

f(x,y), 0 < |x− a| < ε,

R

limx→a

limy→b

f(x,y) = limy→b

limx→a

f(x,y) = L.

f(x,y) =x− y

x+ y, x+ y = 0.

limx→0

limy→0

f(x,y) = 1, limy→0

limx→0

f(x,y) = −1.

lim(x,y)→(0,0) f(x,y)

f(x,y) =x2y2

x2y2 + (x− y)2, x2y2 + (x− y)2 = 0.

limx→0

limy→0

f(x,y) = limy→0

limx→0

f(x,y) = 0,

lim(x,y)→(0,0) f(x,y)

f(x,y) =

⎧⎨

⎩x sin

1

y, y = 0,

0, y = 0.

lim(x,y)→(0,0) f(x,y) = 0

limx→0

limy→0

f(x,y) = limy→0

limx→0

f(x,y).

f(x,y) =x2 − y2

x2 + y2, (x,y) ∈ R2 \ {(0, 0)}.

f (0, 0) y = mx m ∈ Rlimx→0 f(x,mx) f(0, 0) f

(0, 0)

f : R2 → R

f(x,y) =

{0, y ≤ 0 y ≥ x2

1, 0 < y < x2

f (0, 0) 0

g : R → R g(0) = 0

f(x,g(x)) =

{1, x = 0,

0, x = 0.

f (0, 0)

U ⊂ Rn f : U → R aU f(a) = 0 ε > 0 f(x) x ∈ B(a, ε)

f(a)

⇒⇐ (xν) ⊂ U xν → x0

∃ ν0 ∈ N ∀ ν ∈ N,ν ≥ ν0 : xν = x0 f(xν) = f(x0) → f(x0)(xν)

xν = x0 (yn) ⊂ (xn) ∩ U \ {x0}yν → x0 f(yν) → f(x0) ∀ ε > 0 ∃ ν0 ∈ N ∀ ν ∈ N,ν ≥ ν0 :|f(yν)− f(x0)| < ε yν

xν

(xν) ⊂ Uxν → x0 ε > 0 ∃ δ > 0 ∀ x ∈ U ∩ B(x0, δ) : |f(x) − f(x0)| < ε

∃ ν0 ∈ N ∀ ν ∈ N,ν ≥ ν0 : xν ∈ U ∩ B(x0, δ)∀ ν ∈ N,ν ≥ ν0 : |f(xν)− f(x0)| < ε

f : U → R

f : U → Rm m = 1

RRm m ≥ 2

| · | R∥ · ∥

Rm

n

n,m, k ∈ N

U ⊂ Rn f : U → Rm x0 ∈ Rn Uℓ ∈ Rm f ℓ x x0f x0 ℓ f(x) → ℓ, x → x0

∀ (xν) ⊂ U \ {x0} : xν → x0 ⇒ f(xν) → ℓ.

xν → x0 Rn

f(xν) → ℓ Rm

U ⊂ Rn f : U → Rm x0 ∈ Rn

U ℓ ∈ Rm

f(x) = (f1(x), . . . , fm(x)) → ℓ = (ℓ1, . . . , ℓm) x → x0

⇔ ∀ j = 1, . . . ,m : fj(x) → ℓj x → x0

⇔ ∀ j = 1, . . . ,m : limx→x0

fj(x) = ℓj

⇔ ∀ ε > 0 ∃ δ > 0 ∀ x ∈ U∩ B(x0, δ) \ {x0} : ∥f(x)− ℓ∥ < ε

⇔ ∀ ε > 0 ∃ δ > 0 ∀ x ∈ U∩ B(x0, δ) \ {x0} : f(x) ∈ B(ℓ, ε).

∀ (xν) ⊂ U \ {x0} : xν → x0 ⇒ f(xν) → ℓ

⇔ ∀ (xν) ⊂ U \ {x0} : xν → x0 ⇒ fj(xν) → ℓj ∀ j = 1, . . . ,m

⇔ ∀ ε > 0 ∃ δ > 0 ∀ x ∈ U∩ B(x0, δ) \ {x0} : |fj(x)− ℓj| < ε ∀ j = 1, . . . ,m

⇔ ∀ ε > 0 ∃ δ > 0 ∀ x ∈ U∩ B(x0, δ) \ {x0} : ∥f(x)− ℓ∥ < ε.

✷

U ⊂ Rn f : U → Rm x0 ∈ Rn

U f x x0lim

x→x0

f(x)

limx→x0

f(x) = ℓ⇔ limx→x0

∥f(x)− ℓ∥ = 0,

limx→x0

f(x) = ℓ⇔ limη→0

f(x0 + η) = ℓ.

f : U → Rm U ⊂ Rn

x0 ∈ U

∀ (xν) ⊂ U : xν → x0 ⇒ f(xν) → f(x0)

A ⊂ U f : U → Rm x0 ∈ A

f : U → Rm U

x0 U

f x0 ⇔ limx→x0

f(x) = f(x0)

f x0 ⇔ ∀ ε > 0 ∃ δ > 0 ∀ x ∈ U∩ B(x0, δ) : ∥f(x)− f(x0)∥ < ε

⇔ ∀ ε > 0 ∃ δ > 0 ∀ x ∈ U∩ B(x0, δ) : f(x) ∈ B(f(x0), ε)

⇔ ∀ j = 1, . . . ,m : fj x0 f = (f1, . . . , fm)

f, g : U → Rm U ⊂ Rn x0 Ulim

x→x0

f(x) = ℓ ∈ Rm limx→x0

g(x) = m ∈ Rm

limx→x0

(f+ g)(x) = ℓ+ m

limx→x0

(αf)(x) = α ℓ α ∈ R

limx→x0

(h ◦ f)(x) = h(ℓ) h : V → Rk f(U) ⊂ V ⊂ Rm ℓ ∈ V .

limx→x0

∥f(x)∥ = ∥ℓ∥

limx→x0

√∥f(x)∥ =

√∥ℓ∥

(xν) ∈ U \ {x0} xν → x0 (f(xν)) ⊂ Vf(xν) → ℓ ∈ V h : V → Rk ℓ (h ◦ f)(xν) =h(f(xν)) → h(ℓ)

h1(y) = αy ∈ Rm h2(y) = ∥y∥ ∈ R h3(y) =√∥y∥ ∈ R

y ∈ Rm✷

f, g : U → Rm x0 ∈ U ⊂ Rn

x0

f+ g

αf α ∈ R

h ◦ f h : V → Rk f(U) ⊂ V ⊂ Rm f(x0)

∥f∥ ∥f∥ : U → R ∥f∥(x) := ∥f(x)∥ ∀ x ∈ U√∥f∥

√∥f∥ : U → R

√∥f∥(x) :=

√∥f(x)∥ ∀ x ∈ U

U ⊂ Rn f : U → Rm

C(U;Rm) := {f : U → Rm : f }.

C(U;Rm)

f, g ∈ C(U;Rm), α ∈ R ⇒ f+ g, αf ∈ C(U;Rm).

f : U → Rm U ⊂ Rn f(U)

(yν) ⊂ f(U) (xν) ⊂ U f(xν) = yνU (xkν) ⊂ (xν) xkν → x0 ∈ U

f ykν = f(xkν) → f(x0) ∈f(U) ✷

f : U → Rm U ⊂ Rn

∀ ε > 0 ∃ δ > 0 ∀ x, y ∈ U, ∥x− y∥ < δ : ∥f(x)− f(y)∥ < ε.

U ⊂ Rn f : U → Rm f

f

∃ ε > 0 ∀ δ > 0 ∃ x, y ∈ U, ∥x− y∥ < δ : ∥f(x)− f(y)∥ ≥ ε.

ε > 0 δ = 1ν

∀ ν ∈ N ∃ xν, yν ∈ U, ∥xν − yν∥ <1

ν: ∥f(xν)− f(yν)∥ ≥ ε.

(xν) ⊂ U U (xkν) ⊂(xν) xkν → x0 ∈ Uykν → x0 ∈ U

∥ykν − x0∥ ≤ ∥ykν → xkν∥+ ∥xkν − x0∥ ≤ 1

kν+ ∥xkν − x0∥ → 0.

f

f(xkν) → f(x0), f(ykν) → f(x0)

f(xkν) − f(ykν) → 0 ε > 0ν0 ∈ N ∥f(xkν0 )− f(ykν0 )∥ < ε ✷

f1(x,y) =

(xy

x2 − y2

), f2(x,y) =

⎛

⎝1ex

ey

⎞

⎠ ,

f3(x,y, z) =

(sin(xyz)cos(x+ y)

), f4(x,y) =

(sin(ln x)ln(sin x)

),

f5(x,y, z) =

(x2 + y2 − z2

2− tan x

), f6(u, v) =

⎛

⎝e−u

ev

sin(uv)

⎞

⎠ .

Rn Rm

A : Rn → Rm, A(x) = Dx+ b, D ∈ Rm×n b ∈ Rm

U ⊂ Rn g : U → Rm g(U) ⊂ Rm

f = (f1, . . . , fm) : U → Rm x0 ∈ U f(x0) = 0

fjg : U → R, (fjg)(x) = fj(x)g(x), j = 1, . . . ,m,

f · g : U → R, (f · g)(x) = f(x) · g(x),x0

Rn

Sn−1 := ∂B(0, 1) = {x ∈ Rn : ∥x∥ = 1}

Sn−1

Sn−1

Sn−1 ⊂ B(0, r) ∀ r > 1.

(xν) ⊂Sn−1 xν → x Rn x ∈ Sn−1

x ∈ Sn−1 ⇔ g(x) := ∥x∥ = 1,

g : Rn → R

1 = g(xν) → g(x) = 1,

x ∈ Sn−1

Q : Rn → R, Q(x) = xTA x, A ∈ Rn×n ,

QQ

x = (x1, . . . , xn) x 4→ xii = 1, . . . ,n Q

Q(x) = x · (Ax)

ε δ

f : U → R U ⊂ Rn n ≥ 2 f

x = (x1, . . . , xn) ∈ U ii = 1, . . . ,n f x

i

∂f

∂xi(x) := lim

h→0

f(x+ hei)− f(x)

h∈ R, i = 1, . . . ,n,

ei = (0, . . . , 0, 1, 0, . . . , 0) ∈ Rn ,

δij :=

{1, j = i,

0, j = i,j = 1, . . . ,n,

x ∈ U x

∂f

∂xi(x) ∈ R ∀ i = 1, . . . ,n,

U ix ∈ U i

f i

∂f

∂xi: U → R,

δij

Ux ∈ U

∂f

∂xi: U → R ∀ i = 1, . . . ,n,

U

∂f

∂xi∈ C(U) ∀ i = 1, . . . ,n.

f x i

∂f

∂xi(x) =

∂f(x)

∂xi=

∂

∂xif(x) = ∂xif(x) = ∂if(x) = fxi(x).

x ∈ Ui U x

U(hν) ⊂ R \ {0} x+ hνei ∈ U hν → 0

f : U → R x = (x1, . . . , xn) ∈U ⊂ Rn i xi ∈ R f

ixj j = i x

fi(x) := f(x1, . . . , xi−1, x, xi+1, . . . , xn), x ∈ (xi − ε, xi + ε)

ε > 0

{x1}× · · ·× {xi−1}× (xi − ε, xi + ε)× {xi+1}× · · ·× {xn} ⊂ U,

∂f

∂xi(x)

= limh→0

f(x1, . . . , xi−1, xi + h, xi+1, . . . , xn)− f(x1, . . . , xi−1, xi, xi+1, . . . , xn)

h

= limh→0

fi(xi + h)− fi(xi)

h

= f ′i(xi).

f x ifi xi

C1 f ∈ C1(U)

f(x,y) = ex2+y2

= e∥(x,y)∥2, (x,y) ∈ R2.

f1(x) = ex2+y2

, x ∈ R, f2(y) = ex2+y2

, y ∈ R,

f ′1(x) = 2xex2+y2

, x ∈ R, f ′2(y) = 2yex2+y2

, y ∈ R.

f(x,y) ∈ R2 (x,y) ∈ R2

∂f

∂x(x,y) = 2xex

2+y2 ∂f

∂y(x,y) = 2yex

2+y2

f

∂f

∂x: R2 → R

∂f

∂y: R2 → R,

f

f(x) = ∥x∥, x ∈ Rn.

f Rn Rn \ {0}

∂f

∂xi(x) =

xi∥x∥ ∀ i = 1, . . . ,n, ∀ x ∈ Rn \ {0},

0

limh→0

∥0+ hei∥− ∥0∥h

= limh→0

|h|

h.

f Rn \ {0}Rn

limx→0

∂f

∂xi(x) = lim

x→0

xi∥x∥ .

h : (0,∞) → R

f(x) = h(∥x∥), x ∈ Rn \ {0},

∂f

∂xi(x) = h ′(∥x∥) xi

∥x∥ , ∀ i = 1, . . . ,n, ∀ x ∈ Rn \ {0}.

h(z) = ez2

n = 2 h(z) = zRn

Rn \ {0}0

h R

f : Rn → R n ≥ 2

f(x) =

⎧⎨

⎩

x1 · · · xn∥x∥n x = 0

0 x = 0

f Rn \ {0}

∂f

∂xi(x) =

x1 · · · xi−1xi+1 · · · xn(∥x∥2 −nx2i )

∥x∥n+2∀ i = 1, . . . ,n, ∀ x ∈ Rn \ {0}.

f 0

∂f

∂xi(0) = lim

h→0

f(0+ hei)− f(0)

h= lim

h→0

0

h= 0 ∀ i = 1, . . . ,n.

f Rn

Rn \ {0} Rn

∂f

∂xi

( 1ν

∑

j =i

ej

)=

1√(n− 1)n

ν→ ∞ ν→ ∞ ∀ i = 1, . . . ,n,

limx→0

∂f

∂xi(x)

f 0 0

f( 1νei

)= 0 → 0 ν→ ∞ ∀ i = 1, . . . ,n,

f( 1ν

n∑

i=1

ei

)=

1√nn

→ 1√nn

ν→ ∞,

limx→0

f(x)

f = (f1, . . . , fm) : U → Rm U ⊂ Rn n ≥ 2 f

x = (x1, . . . , xn) ∈ U ii = 1, . . . ,n

x ∈ U

U i

U

U

fj j = 1, . . . ,m f

f = (f1, . . . , fm) : U → Rm U ⊂ Rn n ≥ 2x ∈ U f x

Jf(x) :=∂(f1, . . . , fm)

∂(x1, . . . , xn)(x) :=

⎛

⎜⎜⎜⎜⎜⎝

∂f1∂x1

(x) · · · ∂f1∂xn

(x)

∂fm∂x1

(x) · · · ∂fm∂xn

(x)

⎞

⎟⎟⎟⎟⎟⎠∈ Rm×n,

f xm = 1 f : U → R f x

f x

Jf(x) = grad f(x) =

(∂f

∂x1(x), . . . ,

∂f

∂xn(x)

)∈ Rn.

Jf(x) ∈ Rm×n

f xfj j = 1, . . . ,m x

Jf(x) =

⎛

⎜⎝grad f1(x)

grad fm(x)

⎞

⎟⎠ ∈ Rm×n.

U ⊂ Rn n ≥ 2

f, g : U → Rm ϕ,ψ : U → R ψ(x) = 0

x ∈ U

f+ g : U → Rm, f · g : U → R, ϕf : U → Rm,f

ψ: U → Rm

x

Jf+g(x) = Jf(x) + Jg(x) ∈ Rm×n,

grad (f · g)(x) = f(x)T Jg(x) + g(x)T Jf(x) ∈ Rn,

Jϕf(x) = ϕ(x)Jf(x) + f(x) gradϕ(x) ∈ Rm×n,

J fψ(x) =

ψ(x)Jf(x)− f(x) gradψ(x)

ψ2(x)∈ Rm×n,

f(x) g(x)

⎛

⎜⎜⎜⎜⎜⎝

∂(f1 + g1)

∂x1(x) · · · ∂(f1 + g1)

∂xn(x)

∂(fm + gm)

∂x1(x) · · · ∂(fm + gm)

∂xn(x)

⎞

⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎝

∂f1∂x1

(x) · · · ∂f1∂xn

(x)

∂fm∂x1

(x) · · · ∂fm∂xn

(x)

⎞

⎟⎟⎟⎟⎟⎠+

⎛

⎜⎜⎜⎜⎜⎝

∂g1∂x1

(x) · · · ∂g1∂xn

(x)

∂gm∂x1

(x) · · · ∂gm∂xn

(x)

⎞

⎟⎟⎟⎟⎟⎠.

grad

( m∑

j=1

fjgj

)(x)

=m∑

j=1

grad (fjgj)(x)

=m∑

j=1

(∂(fjgj)

∂x1(x), . . . ,

∂(fjgj)

∂xn(x)

)

=m∑

j=1

(fj(x)

∂gj∂x1

(x) + gj(x)∂fj∂x1

(x), . . . , fj(x)∂gj∂xn

(x) + gj(x)∂fj∂xn

(x)

)

=m∑

j=1

(fj(x) gradgj(x) + gj(x) grad fj(x)

)

=(f1(x) · · · fm(x)

)

⎛

⎜⎜⎜⎜⎜⎝

∂g1∂x1

(x) · · · ∂g1∂xn

(x)

∂gm∂x1

(x) · · · ∂gm∂xn

(x)

⎞

⎟⎟⎟⎟⎟⎠

+(g1(x) · · · gm(x)

)

⎛

⎜⎜⎜⎜⎜⎝

∂f1∂x1

(x) · · · ∂f1∂xn

(x)

∂fm∂x1

(x) · · · ∂fm∂xn

(x)

⎞

⎟⎟⎟⎟⎟⎠.

⎛

⎜⎜⎜⎜⎜⎝

∂(ϕf1)

∂x1(x) · · · ∂(ϕf1)

∂xn(x)

∂(ϕfm)

∂x1(x) · · · ∂(ϕfm)

∂xn(x)

⎞

⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎝

ϕ(x)∂f1∂x1

(x) · · · ϕ(x)∂f1∂xn

(x)

ϕ(x)∂fm∂x1

(x) · · · ϕ(x)∂fm∂xn

(x)

⎞

⎟⎟⎟⎟⎟⎠+

⎛

⎜⎜⎜⎜⎜⎝

f1(x)∂ϕ

∂x1(x) · · · f1(x)

∂ϕ

∂xn(x)

fm(x)∂ϕ

∂x1(x) · · · fm(x)

∂ϕ

∂xn(x)

⎞

⎟⎟⎟⎟⎟⎠

= ϕ(x)Jf(x) +

⎛

⎜⎝f1(x)

fm(x)

⎞

⎟⎠(∂ϕ

∂x1(x) · · · ∂ϕ

∂xn(x)

).

J fψ(x) =

(1

ψ

)(x)Jf(x) + f(x) grad

(1

ψ

)(x),

grad

(1

ψ

)(x) =

(∂

∂x1

(1

ψ

)(x), . . . ,

∂

∂xn

(1

ψ

)(x)

)= −

gradψ(x)

ψ2(x).

✷

f(x,y) = x3 − 2x2y2 + 4xy3 + y4 + 10 (x,y) ∈ R2

f(x,y) = (x2 + y2)exy (x,y) ∈ R2

f(x,y, z) = xyz sin(x+ y+ z) (x,y, z) ∈ R3

f(x,y, z) =xey

z(x,y, z) ∈ R3 z = 0

f(x,y) =

⎧⎨

⎩

xy

x2 + y2, (x,y) ∈ R2,

0, (x,y) = (0, 0)

nR1, . . . ,Rn R

1

R=

1

R1+ · · ·+ 1

Rn.

∂R/∂Rk RRk

P V T

PV = cT c > 0

∂V

∂T

∂T

∂P

∂P

∂V= −1.

P Vm T

(P+

a

V2m

)(Vm − b) = RT a,b,R > 0

∂Vm

∂T

∂T

∂P

∂P

∂Vm= −1.

f : U → Rm U ⊂ Rn

x ∈ U D : Rn → Rm

limη→0

f(x+ η)− f(x)−Dη

∥η∥ = 0,

U x ∈ U

limη→0

f(x+ η)− f(x)−Dη

∥η∥ = 0 ⇔ limy→x

f(y)− f(x)−D(y− x)

∥y− x∥ = 0,

limη→0

∥f(x+ η)− f(x)−Dη∥∥η∥ = 0 ⇔ lim

y→x

∥f(y)− f(x)−D(y− x)∥∥y− x∥ = 0.

D : Rn → Rm

D =

⎛

⎜⎝d11 · · · d1n

dm1 · · · dmn

⎞

⎟⎠ ∈ Rm×n,

{ei : i = 1, . . . ,n} ⊂ Rn, {ej : j = 1, . . . ,m} ⊂ Rm.

Rn Rm

Dη =

⎛

⎜⎝d11 · · · d1n

dm1 . . . dmn

⎞

⎟⎠

⎛

⎜⎝η1

ηn

⎞

⎟⎠ ∈ Rm ∀ η =

⎛

⎜⎝η1

ηn

⎞

⎟⎠ ∈ Rn.

m = 1 f : U → RU ⊂ Rn x ∈ UD = (d1, . . . ,dn) ∈ Rn

limη→0

f(x+ η)− f(x)−D · η∥η∥ = 0.

m = n = 1f : U → R U ⊂ R

x ∈ U D ∈ R

limη→0

f(x+ η)− f(x)−Dη

η= 0.

x ∈ U D ∈ Rf′(x) = D f x

Df = (f1, . . . , fm) : U → Rm

x ∈ U (dj1, . . . ,djn) ∈ Rn j = 1, . . . ,m

limη→0

fj(x+ η)− fj(x)− (dj1, . . . ,djn) · η∥η∥ = 0 ∀ j = 1, . . . ,m,

fj j = 1, . . . ,m fx

f = (f1, . . . , fm) : U → Rm U ⊂ Rn

x ∈ U D ∈ Rm×n

f x

f x

∂fj∂xi

(x) = dji ∀ j = 1, . . . ,m, i = 1, . . . ,n.

limη→0

(f(x+ η)− f(x)

)= limη→0

(∥η∥ f(x+ η)− f(x)−Dη

∥η∥ +Dη

)

= limη→0

∥η∥ limη→0

f(x+ η)− f(x)−Dη

∥η∥ + limη→0

Dη = 0 0+ 0 = 0,

D ∈ Rm×n

limη→0

Dη = 0,

∥Dη∥2 =m∑

j=1

((dj1, . . . ,djn) · η

)2

≤m∑

j=1

∥(dj1, . . . ,djn)∥2∥η∥2

=m∑

j=1

n∑

i=1

d2ji∥η∥2 =: ∥D∥2∥η∥2,

∥D∥ > 0

∀ ε > 0 ∀ δ ∈(0,

ε

∥D∥

]∀ η ∈ B(0, δ) : ∥Dη∥ ≤ ∥D∥∥η∥ < ε

∥D∥ = 0∀ j = 1, . . . ,m

limη→0

fj(x+ η)− fj(x)− (dj1, . . . ,djn) · η∥η∥ = 0,

δ0 > 0 B(x, δ0) ⊂ U

∀ ε > 0 ∃ δ ∈ (0, δ0) ∀ η ∈ B(0, δ) \ {0} :

|fj(x+ η)− fj(x)− (dj1, . . . ,djn) · η|∥η∥ < ε

η = hei h ∈ R i = 1, . . . ,n

∀ ε > 0 ∃ δ ∈ (0, δ0) ∀ h ∈ (−δ, 0)∪ (0, δ) :|fj(x+ hei)− fj(x)− djih|

|h|< ε,

limh→0

fj(x+ hei)− fj(x)− djih

h= 0

∂fj∂xi

(x) = limh→0

fj(x+ hei)− fj(x)

h= dji.

✷

D ∈ Rm×n

f = (f1, . . . , fm) : U → Rm x = (x1, . . . , xn) ∈ U ⊂ Rn

Df(x) = Jf(x) =

⎛

⎜⎜⎜⎜⎜⎝

∂f1∂x1

(x) · · · ∂f1∂xn

(x)

∂fm∂x1

(x) · · · ∂fm∂xn

(x)

⎞

⎟⎟⎟⎟⎟⎠∈ Rm×n,

f x

f x

Df(x) : Rn → Rm,

f : U → Rm U ⊂ Rn

Df : U → Rm×n Df : U → L(Rn,Rm), x 4→ Df(x),

f U L(Rn,Rm)Rn Rm

m = 1 f : U → R x ∈ Uf x f x

Df(x) = grad f(x) =

(∂f

∂x1(x), . . . ,

∂f

∂xn(x)

)∈ Rn.

Df(x) ∈ Rm×n f =(f1, . . . , fm) : U → Rm x ∈ U ⊂ Rn

fj j = 1, . . . ,m x

Df(x) =

⎛

⎜⎝Df1(x)

Dfm(x)

⎞

⎟⎠ =

⎛

⎜⎝grad f1(x)

grad fm(x)

⎞

⎟⎠ = Jf(x) ∈ Rm×n.

U ⊂ Rn f : U → Rm

f

f x ∈ Uf x f x

Jf = Df : U → Rm×n.

f : U → R U ⊂ Rn

∂f∂xi

: U → R i = 1, . . . ,nx ∈ U f x

U δ0 > 0 B(x, δ0) ⊂ Uη = (η1, . . . ,ηn) ∈ B(0, δ0) \ {0}

y(k) := x+k∑

i=1

ηiei ∈ B(x, δ0) ⊂ U, ∀ k = 1, . . . ,n,

∥y(k) − x∥ = ∥(η1, . . . ,ηk, 0, . . . , 0)∥ =

√√√√k∑

i=0

η2i ≤

√√√√n∑

i=1

η2i = ∥η∥ < δ0.

y(k) − y(k−1) = ηkek k = 1, . . . ,n, y(0) := x,

ϑk ∈ [0, 1]

f(y(k)

)− f(y(k−1)) = f

(y(k−1) + ηkek

)− f(y(k−1))

= ηk∂f

∂xk

(y(k−1) + ϑkηkek

).

f(x+ η)− f(x) = f(y(n))− f(y(0)

)=

n∑

k=1

f(y(k))− f(y(k−1))

=n∑

k=1

ηk∂f

∂xk

(y(k−1) + ϑkηkek

)

|f(x+ η)− f(x)− grad f(x) · η| =∣∣∣

n∑

k=1

ηk

( ∂f∂xk

(y(k−1) + ϑkηkek

)−∂f

∂xk(x))∣∣∣

≤ ∥η∥n∑

k=1

∣∣∣∂f

∂xk

(y(k−1) + ϑkηkek

)−∂f

∂xk(x)∣∣∣.

∂f∂xk

x k = 1, . . . ,n

ε > 0 δ ∈ (0, δ0) η ∈ B(0, δ)∣∣∣∣∂f

∂xk(x+ η)−

∂f

∂xk(x)

∣∣∣∣ <ε

n,

η ∈ B(0, δ)

∥y(k−1) + ϑkηkek − x∥ = ∥(η1, . . . ,ηk−1, ϑkηk, 0, . . . , 0)∥ =

√√√√k−1∑

i=1

η2i + ϑ2kη

2k

≤

√√√√k∑

i=1

η2i ≤

√√√√n∑

i=1

η2i = ∥η∥ < δ

m∑

i=n

ai := 0 m < n η ∈ B(0, δ) \ {0}

|f(x+ η)− f(x)− grad f(x) · η|∥η∥ <

n∑

k=1

ε

n= ε.

✷

f = (f1, . . . , fm) : U → Rm

U ⊂ Rn ∂fj∂xi

: U → R

j = 1, . . . ,m i = 1, . . . ,n x ∈ U fx

fj fx

fj xf ✷

f : U → Rm

U ⊂ Rn f

f = (f1, . . . , fm) : U → Rm U ⊂ Rn

U

Df : U → Rm×n,

A : U → Rm×k, U ⊂ Rn A(x) =

⎛

⎜⎝a11(x) · · · a1k(x)

am1(x) · · · amk(x)

⎞

⎟⎠ ,

U x ∈ U x ∈ U

limy→x

∥A(y)−A(x)∥ = 0, ∥A(x)∥2 =m∑

j=1

k∑

i=1

a2ji(x)

A x ∈ U

limy→x

|aji(y)− aji(x)| = 0 ∀ j = 1, . . . ,m, i = 1, . . . , k.

U ⊂ Rn f : U → Rm

f ⇐⇒ f

=⇒ f

=⇒{f

f .

f

ff

f

f f

✷

f : U → R U ⊂ Rn C1

f ∈ C1(U)f : U → Rm U ⊂ Rn C1

f ∈ C1(U;Rm) m ≥ 2

f(x,y) =

{xy

∥(x,y)∥ , (x,y) ∈ R2 \ {(0, 0)},

0, (x,y) = (0, 0)..

f R2 \ {(0, 0)} (0, 0)

|xy|

∥(x,y)∥ ≤ ∥(x,y)∥2∥(x,y)∥ = ∥(x,y)∥ ∀(x,y) ∈ R2 \ {(0, 0)}

lim(x,y)→(0,0)

f(x,y) = 0,

(0, 0)

∂f

∂x(0, 0) = lim

h→0

f(h, 0)− f(0, 0)

h= 0 = lim

h→0

f(0,h)− f(0, 0)

h=∂f

∂y(0, 0).

(0, 0)

lim(x,y)→(0,0)

f(x,y)− f(0, 0)− grad f(0, 0) · (x,y)∥(x,y)∥ = lim

(x,y)→(0,0)

xy

∥(x,y)∥2

f(x,y) =

{∥(x,y)∥2 sin 1

∥(x,y)∥ , ∥(x,y)∥ = 0,

0, ∥(x,y)∥ = 0,(x,y) ∈ R2.

f R2 \ {(0, 0)} (x,y) ∈ R2 \{(0, 0)} f (x,y)

Df(x,y) = grad f(x,y) =

(∂f

∂x(x,y),

∂f

∂y(x,y)

),

h(t) := t2 sin1

t, t > 0 =⇒ h′(t) = 2t sin

1

t− cos

1

t,

∂f

∂x(x,y) = 2x sin

1

∥(x,y)∥ −x

∥(x,y)∥ cos1

∥(x,y)∥ ,

∂f

∂y(x,y) = 2y sin

1

∥(x,y)∥ −y

∥(x,y)∥ cos1

∥(x,y)∥ .

f (0, 0)

∂f

∂x(0, 0) = lim

h→0

f(h, 0)− f(0, 0)

h= lim

h→0h sin

1

h= 0,

∂f

∂y(0, 0) = lim

h→0

f(0,h)− f(0, 0)

h= lim

h→0h sin

1

h= 0,

grad f(0, 0) = (0, 0).

f (0, 0) Df(0, 0) = grad f(0, 0) = (0, 0)

lim(x,y)→(0,0)

f(x,y)− f(0, 0)− grad f(0, 0) · (x,y)∥(x,y)∥

= lim(x,y)→(0,0)

f(x,y)

∥(x,y)∥ = lim(x,y)→(0,0)

∥(x,y)∥ sin 1

∥(x,y)∥ = 0.

f(0, 0)

limx→0

∂f

∂x(x, 0) = lim

x→0

(2x sin

1

|x|−

x

|x|cos

1

|x|

),

limy→0

∂f

∂y(0,y) = lim

y→0

(2y sin

1

|y|−

y

|y|cos

1

|y|

),

∂f

∂x

( 1

2πν, 0)= −1 → −1,

∂f

∂x

( 2

πν, 0)=

4

πν(−1)ν+1 → 0.

U ⊂ Rn

f, g : U → Rm ϕ,ψ : U → R ψ(x) = 0

x ∈ U

f+ g : U → Rm, f · g : U → R, ϕf : U → Rm,f

ψ: U → Rm

x

D(f+ g)(x) = Df(x) +Dg(x) ∈ Rm×n,

D(f · g)(x) = g(x)TDf(x) + f(x)TDg(x) ∈ Rn,

D(ϕf)(x) = ϕ(x)Df(x) + f(x)Dϕ(x) ∈ Rm×n,

D

(f

ψ

)(x) =

ψ(x)Df(x)− f(x)Dψ(x)

ψ2(x)∈ Rm×n,

f(x) g(x)

f g ϕ ψ x

f+ g

f · g ϕf fψ x

limη→0

(f+ g)(x+ η)− (f+ g)(x)−(Df(x) +Dg(x)

)η

∥η∥

= limη→0

f(x+ η)− f(x)−Df(x)η

∥η∥ + limη→0

g(x+ η)− g(x)−Dg(x)η

∥η∥ = 0.

limη→0

(f · g)(x+ η)− (f · g)(x)− grad (f · g)(x) · η∥η∥

=m∑

j=1

limη→0

(fjgj)(x+ η)− (fjgj)(x)−(fj(x) gradgj(x) + gj(x) grad fj(x)

)· η

∥η∥

=m∑

j=1

limη→0

(fj(x+ η)− fj(x)− grad fj(x) · η

∥η∥ gj(x+ η)

+ fj(x)gj(x+ η)− gj(x)− gradgj(x) · η

∥η∥

+(gj(x+ η)− gj(x)

) grad fj(x) · η∥η∥

)= 0.

limη→0

(ϕfj)(x+ η)− (ϕfj)(x)−(ϕ(x) grad fj(x) + fj(x) gradϕ(x)

)· η

∥η∥ = 0

j = 1, . . . ,m

limη→0

1

∥η∥

(1

ψ(x+ η)−

1

ψ(x)+

gradψ(x) · ηψ2(x)

)

= limη→0

((ψ(x+ η)−ψ(x)

)gradψ(x) · η

ψ(x+ η)ψ2(x)∥η∥ −ψ(x+ η)−ψ(x)− gradψ(x) · η

∥η∥ψ(x+ η)ψ(x)

)

= 0.

✷

U ⊂ Rn V ⊂ Rm f : U → Rm f(U) ⊂ V g : V → Rk

f x ∈ U g y := f(x)g ◦ f : U → Rk x

D(g ◦ f)(x) = Dg(f(x))Df(x).

A := Df(x) B := Dg(y) = Dg(f(x))

limη→0

(g ◦ f)(x+ η)− (g ◦ f)(x)−BAη

∥η∥ = 0.

V δ1 > 0 B(y, δ1) ⊂ VU f x ∈ Uδ2 > 0 B(x, δ2) ⊂ U f(B(x, δ2)) ⊂ B(y, δ1) ⊂ V

η ∈ B(0, δ2) \ {0} ⊂ Rn ξ ∈ B(0, δ1) \ {0} ⊂ Rm.

x+ η ∈ B(x, δ2) \ {x} ⊂ U, f(x+ η) ∈ B(y, δ1) ⊂ V , y+ ξ ∈ B(y, δ1) \ {y} ⊂ V .

limη→0

f(x+ η)− f(x)−Aη

∥η∥ = 0,

limξ→0

g(y+ ξ)− g(y)−Bξ

∥ξ∥ = 0,

f(x+ η) = f(x) +Aη+ ϕ(η) limη→0

ϕ(η)

∥η∥ = 0,

g(y+ ξ) = g(y) +Bξ+ ψ(ξ) limξ→0

ψ(ξ)

∥ξ∥ = 0.

(g ◦ f)(x+ η) = g(f(x+ η)) = g(f(x) +Aη+ ϕ(η))

= g(f(x)) +BAη+Bϕ(η) + ψ(Aη+ ϕ(η)).

limη→0

Bϕ(η) + ψ(Aη+ ϕ(η))

∥η∥ = 0

limη→0

(g ◦ f)(x+ η)− (g ◦ f)(x)−BAη

∥η∥ = limη→0

Bϕ(η) + ψ(Aη+ ϕ(η))

∥η∥ = 0.

limη→0

ϕ(η)

∥η∥ = 0 limξ→0

Bξ = 0 = B0

limη→0

Bϕ(η)

∥η∥ = 0.

ε = 1

∃ δ3 ∈ (0, δ2) ∀ η ∈ B(0, δ3) \ {0} : ∥ϕ(η)∥ ≤ ∥η∥.

limξ→0

ψ(ξ)

∥ξ∥ = 0

ψ(ξ) = ∥ξ∥ψ1(ξ) limξ→0

∥ψ1(ξ)∥ = 0,

η ∈ B(0, δ3) \ {0}

∥ψ(Aη+ ϕ(η))∥∥η∥ =

∥Aη+ ϕ(η)∥∥ψ1(Aη+ ϕ(η))∥∥η∥

≤ (∥A∥+ 1)∥ψ1(Aη+ ϕ(η))∥.

limη→0

Aη = 0 = 0 limη→0

ϕ(η) = 0 =: ϕ(0) limξ→0

ψ1(ξ) = 0 =: ψ1(0)

limη→0

∥ψ1(Aη+ ϕ(η))∥ = 0,

limη→0

∥ψ(Aη+ ϕ(η))∥∥η∥ = 0,

✷

γ = (γ1, . . . ,γn) :I → Rn U ⊂ Rn γ(I) ⊂ U f : U → R

f ◦ γ : I → R

(f ◦ γ)′(t) = grad f(γ(t)) · γ′(t)

=∂f

∂x1(γ(t))γ′1(t) + · · ·+ ∂f

∂xn(γ(t))γ′n(t) ∀ t ∈ I.

f(x,y) = x+ y R2

f(x,y, z) =xy

z(x,y, z) ∈ R3 z = 0

f(x,y) =

(x+

√y√

x+ y

)x,y > 0

f(x,y, z) =

(1+ ln x

x√y+

√z

)x,y, z > 0

f : U → Rm U ⊂ Rn x ∈ Uf x ε > 0

L > 0

∥f(x)− f(y)∥ ≤ L∥x− y∥ ∀ y ∈ B(x, ε) ⊂ Rn.

f : Rn → Rα f(tx) = tαf(x) t > 0

x ∈ Rn

Df(x) · x = αf(x) ∀ x ∈ Rn.

f,g : R2 → R

f(x,y) = sin(xy) g(x,y) = ex+y.

Df(x,y), Dg(x,y), D(f+ g)(x,y), D(2f)(x,y), D(fg)(x,y), D(f/g)(x,y).

f(u, v) = ln(u2 + v2), (u, v) ∈ R2 \ {(0, 0)},

g1(x,y) = xy, (x,y) ∈ R2,

g2(x,y) =

√x

y, (x,y) ∈ (0,∞)× (0,∞)

F(x,y) = f(g1(x,y),g2(x,y)

)= ln

(x2y2 +

x

y2

), (x,y) ∈ (0,∞)× (0,∞).

F

U ⊂ Rn x0 ∈ U f,g : U → R f x0g x0 g(x0) = 0 fg x0D(fg)(x0) = f(x0)Dg(x0)

f = (f1, . . . , fm) : U → Rm U ⊂ Rn x ∈ U

Jf(x) =

⎛

⎜⎜⎜⎜⎜⎝

∂f1∂x1

(x) · · · ∂f1∂xn

(x)

∂fm∂x1

(x) · · · ∂fm∂xn

(x)

⎞

⎟⎟⎟⎟⎟⎠∈ Rm×n,

j fj : U → Rf x i

∂fj∂xi

(x) = limh→0

fj(x+ hei)− fj(x)

h∈ R, j = 1, . . . ,m, i = 1, . . . ,n,

Jf(x)

f x

Df(x) : Rn → Rm, Df(x)η = Jf(x)η,

limη→0

f(x+ η)− f(x)−Df(x)η

∥η∥ = 0.

Df(x) = Jf(x) ∈ Rm×n f x

f x

⇐⇒ limη→0

fj(x+ η)− f(x)− grad fj(x) · η∥η∥ = 0 ∀ j = 1, . . . ,m,

grad fj(x) =

(∂fj∂x1

(x), . . . ,∂fj∂xn

(x)

)

fj x

f x ⇔ fj x ∀ j = 1, . . . ,m

f x ⇔ fj x ∀ j = 1, . . . ,m.

f : U → R, (x,y) 4→ f(x,y), (x,y) ∈ U, U ⊂ R2 ,

fΓf = {(x,y, f(x,y)) ∈ R3 : (x,y) ∈ U}

z = f(x,y), (x,y) ∈ U,

R3

(x,y) ∈ U 0xy0z (x,y, z) z = f(x,y) ∈ R

f f

(x,y) (x,y, f(x,y)) ∈ R3

0xyf (x0,y0) ∈ U

∂f

∂x(x0,y0) = lim

h→0

f(x0 + h,y0)− f(x0,y0)

h,

∂f

∂y(x0,y0) = lim

h→0

f(x0,y0 + h)− f(x0,y0)

h

(x0 − ε, x0 + ε) ∋ x 4→ f(x,y0), (y0 − ε,y0 + ε) ∋ y 4→ f(x0,y),

ε > 0

(x0 − ε, x0 + ε)× (y0 − ε,y0 + ε) ⊂ U.

∂f

∂x(x0,y0)

(x0,y0, f(x0,y0))

{(x,y0, f(x,y0)) ∈ R3 : x ∈ R (x,y0) ∈ U},

Γf y = y0

z = f(x,y0), y = y0, x ∈ R (x,y0) ∈ U.

y = y00xz

z = f(x0,y0) + (x− x0)∂f

∂x(x0,y0), y = y0, x ∈ R.

∂f

∂y(x0,y0)

(x0,y0, f(x0,y0))

{(x0,y, f(x0,y)) ∈ R3 : y ∈ R (x0,y) ∈ U},

Γf x = x0

z = f(x0,y), x = x0, y ∈ R (x0,y) ∈ U.

x = x00yz

z = f(x0,y0) + (y− y0)∂f

∂y(x0,y0), x = x0, y ∈ R.

(x0,y0, f(x0,y0)) ∈ Γf

z = f(x0,y0) + (x− x0)∂f

∂x(x0,y0) + (y− y0)

∂f

∂y(x0,y0)

= f(x0,y0) + (x− x0,y− y0) · grad f(x0,y0), (x,y) ∈ R2,

grad f(x0,y0)(x0,y0, f(x0,y0))

Γf f

lim(x,y)→(x0,y0)

f(x,y)− f(x0,y0)− (x− x0,y− y0) · grad f(x0,y0)∥(x− x0,y− y0)∥

= 0,

f (x0,y0) ∈ Ugrad f(x0,y0)

(x0,y0, f(x0,y0)) Γf fDf(x0,y0) f (x0,y0)

grad f(x0,y0) =

(∂f

∂x(x0,y0),

∂f

∂y(x0,y0)

)= Df(x0,y0).

(x0,y0, f(x0,y0)) Γff

⎛

⎜⎝x

y

z

⎞

⎟⎠ =

⎛

⎜⎝x0

y0

f(x0,y0)

⎞

⎟⎠+ (x− x0)

⎛

⎜⎝1

0∂f∂x (x0,y0)

⎞

⎟⎠+ (y− y0)

⎛

⎜⎝0

1∂f∂y (x0,y0)

⎞

⎟⎠ ,

x,y ∈ R,

⎛

⎜⎝1

0∂f∂x (x0,y0)

⎞

⎟⎠×

⎛

⎜⎝0

1∂f∂y (x0,y0)

⎞

⎟⎠ =

∣∣∣∣∣∣∣

e1 e2 e3

1 0 ∂f∂x (x0,y0)

0 1 ∂f∂y (x0,y0)

∣∣∣∣∣∣∣=

⎛

⎜⎝− ∂f∂x (x0,y0)

− ∂f∂y (x0,y0)

1

⎞

⎟⎠ .

(x0,y0, f(x0,y0))Γf f

⎛

⎜⎝− ∂f∂x (x0,y0)

− ∂f∂y (x0,y0)

1

⎞

⎟⎠ ·

⎛

⎜⎝x− x0

y− y0

z− f(x0,y0)

⎞

⎟⎠ = 0.

y = y0(x0,y0, f(x0,y0)) {(x,y0, f(x,y0)) ∈ R3 : x ∈ R (x,y0) ∈ U}

⎛

⎜⎝x

y

z

⎞

⎟⎠ =

⎛

⎜⎝x0

y0

f(x0,y0)

⎞

⎟⎠+ (x− x0)

⎛

⎜⎝1

0∂f∂x (x0,y0)

⎞

⎟⎠ ∈ R3, x ∈ R,

x = x0(x0,y0, f(x0,y0)) {(x0,y, f(x0,y)) ∈ R3 : y ∈

R (x0,y) ∈ U},

⎛

⎜⎝x

y

z

⎞

⎟⎠ =

⎛

⎜⎝x0

y0

f(x0,y0)

⎞

⎟⎠+ (y− y0)

⎛

⎜⎝0

1∂f∂y (x0,y0)

⎞

⎟⎠ ∈ R3, y ∈ R.

y = y0 x = x0

Γf y = y0 x = x0

f(x,y) = x2 + y2 = ∥(x,y)∥2, (x,y) ∈ R2.

Γf f

z = x2 + y2, (x,y) ∈ R2.

f (x0,y0)

∂f

∂x(x0,y0) = 2x0,

∂f

∂y(x0,y0) = 2y0,

grad f(x0,y0) = 2(x0,y0).

f (x0,y0) ∈ R2

g(x) = ∥x∥2, x ∈ Rn,

gradg(x) = 2x

limη→0

∥x+ η∥2 − ∥x∥2 − 2x · η∥η∥ = lim

η→0∥η∥ = 0.

(x0,y0, x20 + y20)

z = x20 + y20 + 2x0(x− x0) + 2y0(y− y0), (x,y) ∈ R2

(−2x0,−2y0, 1).

(x0,y0)

(0, 0) z = 0 grad f(0, 0) = (0, 0) (0, 0, 1)

(±1, 0) z = 1± 2(x∓ 1) grad f(±1, 0) = (±2, 0) (∓2, 0, 1)

(0,±1) z = 1± 2(y∓ 1) grad f(0,±1) = (0,±2) (0,∓2, 1)

y = y0 x = x0

z = x2 + y20, y = y0, x ∈ R, z = x20 + y2, x = x0, y ∈ R,

(x0,y0, x20 + y20)

z = x20 + y20 + 2x0(x− x0), y = y0, x ∈ R,

z = x20 + y20 + 2y0(y− y0), x = x0, y ∈ R,

(x0,y0) = (0, 0)

z = x2, y = 0, x ∈ R z = y2, x = 0, y ∈ R,

(0, 0, 0)

z = 0, y = 0, x ∈ R, z = 0, x = 0, y ∈ R,

(x0,y0) = (±1, 0)

z = x2, y = 0, x ∈ R z = 1+ y2, x = ±1, y ∈ R,

(±1, 0, 1)

z = 1± 2(x∓ 1), y = 0, x ∈ R, z = 1, x = ±1, y ∈ R,

(x0,y0) = (0,±1)

z = x2 + 1, y = ±1, x ∈ R z = y2, x = 0, y ∈ R,

(0,±1, 1)

z = 1, y = ±1, x ∈ R, z = 1± 2(y∓ 1), x = 0, y ∈ R.

f : U → R U ⊂ R2

(x0,y0, f(x0,y0)) (x0,y0) ∈ Uy = y0 x = x0

∂f

∂x(x0,y0),

∂f

∂y(x0,y0),

grad f(x0,y0) =

(∂f

∂x(x0,y0),

∂f

∂y(x0,y0)

)

x 4→ f(x,y0) y 4→f(x0,y) y = y0 x = x0

0xy (0, 1) (1, 0)f (x0,y0)

(x0,y0)(α,β) = (0, 0)

f (x0,y0)

(1, 0)(0, 1)

(x0,y0, f(x0,y0))

f (x0,y0)

grad f(x0,y0)

f(x0,y0)

f : U → R U ⊂ Rn x ∈ U ν ∈ Rn ∥ν∥ = 1

Dνf(x) :=∂f

∂ν(x) := lim

h→0

f(x+ hν)− f(x)

h

νν f x

f x

Dei(x) =∂f

∂ei(x) =

∂f

∂xi(x) ∀ i = 1, . . . ,n.

f : U → R U ⊂ Rn

x = (x1, . . . , xn) ∈ Uν = (ν1, . . . ,νn) ∈ Rn ∥ν∥ = 1

Dνf(x) = grad f(x) · ν.

x ∈ U U ε > 0 B(x, ε) ⊂ Ux+ hν ∈ B(x, ε) ∀ h ∈ (−ε, ε)

ϕ(h) =

⎛

⎜⎝ϕ1(h)

ϕn(h)

⎞

⎟⎠ := x+ hν =

⎛

⎜⎝x1 + hν1

xn + hνn

⎞

⎟⎠ ∈ Rn, h ∈ (−ε, ε),

Dϕ(0) =

⎛

⎜⎝ϕ′

1(0)

ϕ′n(0)

⎞

⎟⎠ =

⎛

⎜⎝ν1

νn

⎞

⎟⎠ = ν,

limh→0

ϕ(h)− ϕ(0)− νh

h= 0.

Dνf(x) = limh→0

f(x+ hν)− f(x)

h= lim

h→0

f(ϕ(h))− f(ϕ(0))

h

= limh→0

(f ◦ ϕ)(h)− (f ◦ ϕ)(0)h

= (f ◦ ϕ)′(0).

f◦ ϕ : Rn → Rm m = n = 1

(f ◦ ϕ)′(0) = D(f ◦ ϕ)(0) = Df(ϕ(0))Dϕ(0) = Df(x)Dϕ(0) = grad f(x) · ν.

✷

f x ∀ i = 1, . . . ,n

Dei(x) = grad f(x) · ei =(∂f

∂x1(x), . . . ,

∂f

∂xi(x), . . . ,

∂f

∂xn(x)

)· (0, . . . , 1, . . . , 0)

=∂f

∂xi(x).

grad f(x) = 0 grad f(x) · ν

ν =grad f(x)

∥ grad f(x)∥ ,

ν

Dνf(x) = ∥ grad f(x)∥.

x ∈ U fgrad f(x) ∥ grad f(x)∥

(x0,y0, f(x0,y0)) (x0,y0)f(x0,y0)

grad f(x0,y0)− grad f(x0,y0)

− grad f(x0,y0)

ϑ ∈ [0,π] grad f(x) ν ∥ν∥ = 1

cos ϑ =grad f(x) · ν∥ grad f(x)∥ ,

Dνf(x) = ∥ grad f(x)∥ cos ϑ.f : U → R U ⊂ R2

(x0,y0) Dνf(x0,y0) ν = (ν1ν2) ∈ R2 ∥ν∥ = 1(x0,y0, f(x0,y0))f

Γf = {(x,y, z) ∈ R3 : z = f(x,y), (x,y) ∈ U}

0xy(xy

)=

(x0y0

)+ h

(ν1ν2

), h ∈ R.

⎛

⎝xyz

⎞

⎠ =

⎛

⎝x0 + hν1y0 + hν2

f(x0 + hν1,y0 + hν2)

⎞

⎠ =: γ(h), h ∈ (−ε, ε),

h = 0

Jγ(0) =

⎛

⎝ν1ν2

ddhf(x0 + hν1,y0 + hν2)|h=0

⎞

⎠ =

⎛

⎝ν1ν2

grad f(x0,y0) · ν

⎞

⎠

limh→0

γ(h)− γ(0)− Jγ(0)h

h= 0

⇐⇒ limh→0

ν1h− ν1h

h= 0, lim

h→0

ν2h− ν2h

h= 0,

limh→0

f(x0 + hν1,y0 + hν2)− f(x0,y0)− grad f(x0,y0) · νhh

= 0.

Dγ(0) = Jγ(0) (x0,y0, f(x0,y0)) =γν(0)⎛

⎝xyz

⎞

⎠ = γν(0) +Dγ(0)h =

⎛

⎝x0y0

f(x0,y0)

⎞

⎠+ h

⎛

⎝ν1ν2

grad f(x0,y0) · ν

⎞

⎠

=

⎛

⎝x0y0

f(x0,y0)

⎞

⎠+ hν1

⎛

⎝10

∂f∂x (x0,y0)

⎞

⎠+ hν2

⎛

⎝01

∂f∂y (x0,y0)

⎞

⎠ , h ∈ R,

f(x0,y0, f(x0,y0))

∂f

∂ν(x0,y0) = grad f(x0,y0) · ν.

f(x,y) = x2+y2 (x0,y0) ∈ R2 \ {(0, 0)}

ν =grad f(x0,y0)

∥ grad f(x0,y0)∥=

(x0,y0)

∥(x0,y0)∥,

(x0,y0) ∈ R2 \ {(0, 0)}

⎛

⎝xyz

⎞

⎠ =

⎛

⎝(1+ h)x0(1+ h)y0

(1+ h)2(x20 + y20)

⎞

⎠ , h ∈ R,

(x0,y0, x20 + y20)

⎛

⎝xyz

⎞

⎠ =

⎛

⎝x0y0

x20 + y20

⎞

⎠+ h

⎛

⎝x0y0

2∥(x0,y0)∥

⎞

⎠ , h ∈ R,

∂f

∂ν(x0,y0) = 2∥(x0,y0)∥.

(x0,y0) (0, 0)(x0,y0) = (0, 0)

Dνf(0, 0) = grad f(0, 0) · ν = (0, 0) · ν = 0 ∀ ν ∈ R2, ∥ν∥ = 1,

z = 0(0, 0) (0, 0, 0)

0

grad f(x) f : U → RU ⊂ Rn x ∈ U f(x) f

Lf(f(x)) = {y ∈ U : f(y) = f(x)},

γ : (−ε, ε) → Rn γ((−ε, ε)) ⊂ Lf(f(x)) γ(0) = x

Dγ(0) · grad f(x) = 0.

f ◦ γ : (−ε, ε) → R, (f ◦ γ)(h) = f(x),

D(f ◦ γ)(0) = (f ◦ γ)′(0) = 0,

D(f ◦ γ)(0) = grad f(x) ·Dγ(0).

c > 0f(x,y) = x2 + y2 (0, 0)

√c R2

Lf(c) = {(x,y) ∈ R2 : x2 + y2 = c}.

(x0,y0) ∈ Lf(c)Lf(c)

γ(h) =√c(cosϕ(h), sinϕ(h)), h ∈ (−ε, ε),

γ(0) =√c(cosϕ(0), sinϕ(0)) = (x0,y0)

Dγ(0) =√c(− sinϕ(0), cosϕ(0))ϕ′(0) = (−y0, x0)ϕ

′(0).

Dγ(0) · grad f(x) = (−y0, x0)ϕ′(0) · 2(x0,y0) = 0.

(x0,y0, f(x0,y0))

ab

f a b

x2 + y2 (1, 1) (1/√2, 1/

√2)

sin(xy) (1, 0) (1/2,√3/2)

x2 + zey (0, 0, 1) (1, 0, 1)

exyz (1, 1, 1) (1, 2,−1)

f : R2 → R

f(x,y) =

⎧⎨

⎩

xy2

x2 + y4, x = 0,

0, x = 0.

f (x,y) ∈ R2

M = {(x, x) ∈ R2 : x = 0}

f(x,y) =

{ex − 1, (x,y) ∈ M,

0, (x,y) ∈ M.

f (x,y) ∈ M

Dνf(0, 0) ν ∈ R2 ∥ν∥ = 1

ν ∈ R2 ∥ν∥ = 1 Dνf(0, 0) = grad f(0, 0) · νA := {(x, x) ∈ R2 : x ∈ R} = M∪ {(0, 0)}

(xν, xν) → (x,y) ∈ R2 x = y B := R2 \A ⊂ R2 \Mf|B f

(0, 0)

∂

∂xf(0, 0) = lim

x→0

f(x, 0)− f(0, 0)

x= lim

x→0

0− 0

x= 0,

∂

∂yf(0, 0) = lim

y→0

f(0,y)− f(0, 0)

y= lim

y→0

0− 0

y= 0.

f (x,y) ∈ M(x,y) ∈ M x = y = 0 f(x, x) = ex − 1 = 0

f(x+ h, x) = 0 ∀ h = 0

h 4→ f(x+ h, x) 0

∂

∂xf(x, x) := lim

h→0

f(x+ h, x)− f(x, x)

h

f xy

ν = ± 1√2(1, 1)

Dνf(0, 0) = limh→0

f(0+ hν)− f(0)

h= lim

h→0

e± h√

2 − 1

h= ± 1√

2.

ν = (ν1,ν2) ∈ R2 \ {± 1√2(1, 1)} ∥ν∥ = 1 ν1 = ν2

Dνf(0, 0) = limh→0

f(0+ hν)− f(0)

h= lim

h→0

f(hν1,hν2)

h= lim

h→0

0

h= 0.

ν = 1√2(1, 1)

1√2= Dνf(0, 0) = grad f(0, 0) · ν = (0, 0) · 1√

2(1, 1) = 0.

f(0, 0)

M AM = A

f : U → R U ⊂ Rn n ≥ 2 k ∈ N f

k+ 1 kk+ 1 f

∂k+1f

∂xik+1· · · ∂xi1

:=∂

∂xik+1

∂kf

∂xik · · · ∂xi1: U → R

∀ i1, . . . , ik+1 = 1, . . . ,n.

k = 1

k + 1 k + 1≤ k+ 1

f : U → R U ⊂ Rn

n ≥ 2f

∂f

∂xi: U → R,

∂2f

∂xj∂xi:=

∂

∂xj

∂f

∂xi: U → R ∀ i, j = 1, . . . ,n

k k ≥ 2

∂kf

∂xik · · · ∂xi1, i1, . . . , ik ∈ {1, . . . ,n}

i1 = . . . = ik = i ∈ {1, . . . ,n}

∂kf

∂xki:=

∂kf

∂xik · · · ∂xi1,

iℓ ℓ = 1, . . . , k

f(x,y) = xy+ (x+ 2y)2, (x,y) ∈ R2,

g(x,y, z) = exy + z cos x, (x,y, z) ∈ R3

k k ∈ N

f

∂f

∂x(x,y) = y+ 2(x+ 2y),

∂f

∂y(x,y) = x+ 4(x+ 2y),

∂2f

∂x2(x,y) = 2,

∂2f

∂y∂x(x,y) = 5,

∂2f

∂x∂y(x,y) = 5,

∂2f

∂y2(x,y) = 8,

Ck+1 f ∈ Ck+1(U)

g x = (x,y, z)

∂g

∂x(x) = yexy − z sin x,

∂g

∂y(x) = xexy,

∂g

∂z(x) = cos x,

∂2g

∂x2(x) = y2exy − z cos x,

∂2g

∂y∂x(x) = xyexy,

∂2g

∂z∂x(x) = − sin x,

∂2g

∂x∂y(x) = xyexy,

∂2g

∂y2(x) = x2exy,

∂2g

∂z∂y(x) = 0,

∂2g

∂x∂z(x) = − sin x,

∂2g

∂y∂z(x) = 0,

∂2g

∂z2(x) = 0.

h : R2 → R

h(x,y) :=

⎧⎨

⎩xy

x2 − y2

x2 + y2, (x,y) = (0, 0),

0, (x,y) = (0, 0),

R2 \ {(0, 0)}

∂h

∂x(x,y) = y

x4 − y4 + 4x2y2

(x2 + y2)2,

∂h

∂y(x,y) = x

x4 − y4 − 4y2x2

(x2 + y2)2.

h (0, 0)

∂h

∂x(0, 0) = lim

x→0

h(x, 0)− h(0, 0)

x= 0,

∂h

∂y(0, 0) = lim

y→0

h(0,y)− h(0, 0)

y= 0.

h

lim(x,y)→(0,0)

∂h

∂x(x,y) = lim

(x,y)→(0,0)

∂h

∂y(x,y) = 0

R2 \ {(0, 0)} h

(0, 0)

∂2h

∂x2(0, 0) = lim

x→0

∂h∂x (x, 0)−

∂h∂x (0, 0)

x= 0,

h

∂2h

∂y∂x(0, 0) = lim

y→0

∂h∂x (0,y)−

∂h∂x (0, 0)

y= lim

y→0

y −y4

(y2)2

y= lim

y→0(−1) = −1,

∂2h

∂x∂y(0, 0) = lim

x→0

∂h∂y (x, 0)−

∂h∂y (0, 0)

x= lim

x→0

x x4

(x2)2

x= lim

x→01 = 1,

∂2h

∂y2(0, 0) = lim

y→0

∂h∂y (0,y)−

∂h∂y (0, 0)

y= 0.

h

(0, 0)

R2 \ {(0, 0)}

∂2h

∂x2(x,y) = 4xy3

−x2 + 3y2

(x2 + y2)3,

∂2h

∂y∂x(x,y) =

x6 − 9x2y4 + 9x4y2 − y6

(x2 + y2)3,

∂2h

∂x∂y(x,y) =

x6 − 9x2y4 + 9x4y2 − y6

(x2 + y2)3,

∂2h

∂y2(x,y) = 4x3y

−3x2 + y2

(x2 + y2)3.

∂2h∂y∂x

∂2h∂x∂y

R2 \ {(0, 0)} (0, 0)

(0, 0)(0, 0)

f : U → R U ⊂ Rn n ≥ 2

∂2f

∂xj∂xi=

∂2f

∂xi∂xj∀ i, j = 1, . . . ,n.

✷

f : U → R U ⊂ Rn n ≥ 2∂2f∂xj∂xi

: U → R i, j ∈ {1, . . . ,n} i = j

x ∈ U ∂2f∂xi∂xj

(x)

∂2f

∂xi∂xj(x) =

∂2f

∂xj∂xi(x).

limh→0

∂f∂xj

(x+ hei)−∂f∂xj

(x)

h=

∂2f

∂xj∂xi(x),

∀ ε > 0 ∃ δ > 0 ∀ h ∈ (−δ, 0)∪ (0, δ) :

∣∣∣∣∣∣

∂f∂xj

(x+ hei)−∂f∂xj

(x)

h−

∂2f

∂xj∂xi(x)

∣∣∣∣∣∣< ε.

∂f

∂xj(x) = lim

k→0

f(x+ kej)− f(x)

k,

∀ ε > 0 ∃ δ > 0 ∀ h ∈ (−δ, 0)∪ (0, δ) :∣∣∣∣ limk→0

f(x+ hei + kej)− f(x+ hei)− f(x+ kej) + f(x)

hk−

∂2f

∂xj∂xi(x)

∣∣∣∣ < ε,

∀ ε > 0 ∃ δ > 0 ∀ h ∈ (−δ, 0)∪ (0, δ) :

∣∣∣∣ limk→0

Φ(k)

hk−

∂2f

∂xj∂xi(x)

∣∣∣∣ < ε,

Φ(k) := f(x+ hei + kej)− f(x+ hei)− f(x+ kej) + f(x).

ε > 0 U ∂2f∂xj∂xi

x

δ > 0 B(x, δ) ⊂ U

∣∣∣∣∂2f

∂xj∂xi(y)−

∂2f

∂xj∂xi(x)

∣∣∣∣ <ε

2∀ y ∈ B(x, δ).

h,k ∈ R \ {0} h2 + k2 < δ2

x+ ϑ1hei + ϑ2kej ∈ B(x, δ) ∀ ϑ1, ϑ2 ∈ [0, 1]

h, k, ϑ1, ϑ2∣∣∣∣∂2f

∂xj∂xi(x+ ϑ1hei + ϑ2kej)−

∂2f

∂xj∂xi(x)

∣∣∣∣ <ε

2

∣∣∣∣ limk→0

∂2f

∂xj∂xi(x+ ϑ1hei + ϑ2kej)−

∂2f

∂xj∂xi(x)

∣∣∣∣ ≤ε

2< ε ∀ h ∈ (−δ, 0)∪ (0, δ)

h,kϑ1, ϑ2 ∈ (0, 1)

Φ(k)

hk=

∂2f

∂xj∂xi(x+ ϑ1hei + ϑ2kej)

limk→0

Φ(k)

hk=

1

h

(∂f

∂xj(x+ hei)−

∂f

∂xj(x)

)

ϕ(ℓ) := f(x+ ℓei + kej)− f(x+ ℓei), ℓ ∈ [−|h|, |h|]

ϕ′(ℓ) =∂f

∂xi(x+ ℓei + kej)−

∂f

∂xi(x+ ℓei), ℓ ∈ [−|h|, |h|].

ϑ1 ∈ (0, 1)

Φ(k) = ϕ(h)−ϕ(0) = hϕ′(ϑ1h) = h

(∂f

∂xi(x+ ϑ1hei + kej)−

∂f

∂xi(x+ ϑ1hei)

).

ψ(m) :=∂f

∂xi(x+ ϑ1hei +mej), m ∈ [−|k|, |k|],

ψ′(m) =∂2f

∂xj∂xi(x+ ϑ1hei +mej), m ∈ [−|k|, |k|],

ϑ2 ∈ (0, 1)

Φ(k) = h (ψ(k)−ψ(0)) = hkψ′(ϑ2k) = hk∂2f

∂xj∂xi(x+ ϑ1hei + ϑ2kej).

✷

f : U → R U ⊂ Rn n ≥ 2 k

∂kf

∂xik · · · ∂xi1=

∂kf

∂xiπ(k) · · · ∂xiπ(1)i1, . . . , ik ∈ {1, . . . ,n} π 1, . . . , k

kn = 4 k = 3

∂3f

∂x4∂x2∂x1=

∂

∂x4

∂2f

∂x2∂x1=

∂

∂x4

∂2f

∂x1∂x2=

∂3f

∂x4∂x1∂x2=

∂2

∂x4∂x1

∂f

∂x2

=∂2

∂x1∂x4

∂f

∂x2=

∂3f

∂x1∂x4∂x2=

∂

∂x1

∂2f

∂x4∂x2=

∂

∂x1

∂2f

∂x2∂x4=

∂3f

∂x1∂x2∂x4.

✷

f(x,y) = ln√

x2 + y2 (x,y) ∈ R2 \ {(0, 0)}

∂2f

∂x2+∂2f

∂y2= 0.

f(x,y, z) =1√

x2 + y2 + z2(x,y, z) ∈ R3 \ {(0, 0)}

∂2f

∂x2+∂2f

∂y2+∂2f

∂z2= 0.

U ⊂ R2 f,g ∈ C2(U)

∂f

∂x=∂g

∂y

∂f

∂y= −

∂g

∂xU.

∂2f

∂x2+∂2f

∂y2= 0

∂2g

∂x2+∂2g

∂y2= 0 U.

f,g : R → Rα ∈ R u(x, t) = f(x−αt) + g(x+αt) (x, t) ∈ R2

∂2u

∂t2= α2

∂2u

∂x2.

f : U → R U ⊂ Rn n ≥ 2f x ∈ U

grad f(x) :=

(∂f

∂x1(x), . . . ,

∂f

∂xn(x)

)∈ Rn

f xf

grad f =

(∂f

∂x1, . . . ,

∂f

∂xn

): U → Rn

f

f grad fU

f U

grad : {f : U → R : f } → {f : U → Rn},

grad f =

(∂f

∂x1, . . . ,

∂f

∂xn

).

grad

∇ =

(∂

∂x1, . . . ,

∂

∂xn

),

∇ : {f : U → R : f } → {f : U → Rn},

∇f = grad f =

(∂f

∂x1, . . . ,

∂f

∂xn

)=

(∂

∂x1, . . . ,

∂

∂xn

)f.

f

∇ =

(∂

∂x1, . . . ,

∂

∂xn

)

f ∇

f = (f1, . . . , fn) : U → Rn U ⊂ Rn n ≥ 2

div f :=n∑

i=1

∂fi∂xi

: U → R

f

f ∇ =

(∂

∂x1, . . . ,

∂

∂xn

)f

div f = ∇ · f =(∂

∂x1, . . . ,

∂

∂xn

)· (f1, . . . , fn) =

n∑

i=1

∂

∂xifi.

div

f

∇ · f ≡ 0,

f = (f1, f2, f3) : U → R3 U ⊂ R3

curl f :=

(∂f3∂x2

−∂f2∂x3

,∂f1∂x3

−∂f3∂x1

,∂f2∂x1

−∂f1∂x2

): U → R3

f

f

rot f = curl f.

x = (x1, x2, x3), y = (y1,y2,y3) ∈ R3

x× y := (x2y3 − x3y2, x3y1 − x1y3, x1y2 − x2y1) =

∣∣∣∣∣∣

e1 e2 e3x1 x2 x3y1 y2 y3

∣∣∣∣∣∣,

f ∇ =

(∂

∂x1,∂

∂x2,∂

∂x3

)f

curl f = ∇× f =

(∂

∂x1,∂

∂x2,∂

∂x3

)× (f1, f2, f3) =

∣∣∣∣∣∣

e1 e2 e3∂∂x1

∂∂x2

∂∂x3

f1 f2 f3

∣∣∣∣∣∣

=

(∂

∂x2f3 −

∂

∂x3f2,

∂

∂x3f1 −

∂

∂x1f3,

∂

∂x1f2 −

∂

∂x2f1

).

∂∂xi

fjcurl

f

∇× f ≡ 0,

f : U → R U ⊂ Rn n ≥ 2

∆f :=n∑

i=1

∂2f

∂x2i: U → R

f

f

∆f = ∇ ·∇f =

(∂

∂x1, . . . ,

∂

∂xn

)·(∂

∂x1, . . . ,

∂

∂xn

)f =

n∑

i=1

∂2

∂x2if,

f

∆f = div grad f.

∆

f∆f = ∇2f

u

u

∆u(x) = 0,

u ∈ C2(U) U ⊂ Rn

n = 2 n = 3 x ∈ U ⊂ Rn

t ∈ R

k > 0

∂

∂tu(t, x) = k∆u(t, x) = k

n∑

i=1

∂2

∂x2iu(t, x1, . . . , xn), (t, x) ∈ (0,∞)× Rn

c > 0

∂2

∂t2u(t, x) = c2∆u(t, x) = c2

n∑

i=1

∂2

∂x2iu(t, x1, . . . , xn), (t, x) ∈ R × Rn

∆ ux

f ∈ C2(U;R3) U ⊂ R3 div curl f = 0

f,g ∈ C2(U) U ⊂ Rn

∆(fg) = g∆f+ 2∇f ·∇g+ f∆g.

u(x) =

{c1

1∥x∥n−2 , n ≥ 3,

c2 ln ∥x∥, n = 2,x ∈ Rn \ {0}

Rn \ {0} c1, c2 ∈ Rc1, c2 u

u(t, x) = (4πkt)−n/2e−∥x∥2/(4kt), (t, x) ∈ (0,∞)× Rn

u

f ∈ C2(R) ϑ ∈ Rn c > 0 ω := ∥ϑ∥c

u(t, x) = f(ϑ · x−ωt), (t, x) ∈ R × Rn

f : Rn → R g = (g1, . . . ,gn) : Rn → Rn

div (fg) = f div g+ g · grad f

∇ = ( ∂∂x1, . . . , ∂

∂xn)

∇ · (fg) = f∇ · g+ g ·∇f

∇ · (fg) =n∑

i=1

∂

∂xi(fgi) =

n∑

i=1

(gi

∂

∂xif+ f

∂

∂xigi

)=

n∑

i=1

gi∂

∂xif+ f

n∑

i=1

∂

∂xigi

= g ·∇f+ f∇ · g.

f,g : Rn → R

∇ · (f∇g− g∇f) = f∆g− g∆f.

∇ · (f∇g− g∇f) = ∇ ·(f∂

∂x1g− g

∂

∂x1f, . . . , f

∂

∂xng− g

∂

∂xnf

)

=n∑

i=1

∂

∂xi

(f∂

∂xig− g

∂

∂xif

)

=n∑

i=1

((∂

∂xif

)∂

∂xig+ f

∂2

∂x2ig−

(∂

∂xig

)∂

∂xif− g

∂2

∂x2if

)

= fn∑

i=1

∂2

∂x2ig− g

n∑

i=1

∂2

∂x2if

= f∆g− g∆f.

f = (f1, f2, f3) : R3 → R3

curl curl f = grad div f−∆f, ∆f := (∆f1,∆f2,∆f3),

RN

∇ = ( ∂∂x1, . . . , ∂

∂xn)

∇×(∇× f

)= ∇

(∇ · f

)−∆f

∇× f =

(∂f3∂x2

−∂f2∂x3

,∂f1∂x3

−∂f3∂x1

,∂f2∂x1

−∂f1∂x2

),

∇×(∇× f

)=

⎛

⎜⎜⎜⎝

∂∂x2

(∂f2∂x1

− ∂f1∂x2

)− ∂∂x3

(∂f1∂x3

− ∂f3∂x1

)

∂∂x3

(∂f3∂x2

− ∂f2∂x3

)− ∂∂x1

(∂f2∂x1

− ∂f1∂x2

)

∂∂x1

(∂f1∂x3

− ∂f3∂x1

)− ∂∂x2

(∂f3∂x2

− ∂f2∂x3

)

⎞

⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎝

∂2f2∂x2∂x1

− ∂2f1∂x2

2− ∂2f1∂x2

3+ ∂2f3∂x3∂x1

∂2f3∂x3∂x2

− ∂2f2∂x2

3− ∂2f2∂x2

1+ ∂2f1∂x1∂x2

∂2f1∂x1∂x3

− ∂2f3∂x2

1− ∂2f3∂x2

2+ ∂2f2∂x2∂x3

⎞

⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎝

∂2f2∂x2∂x1

+ ∂2f1∂x2

1+ ∂2f3∂x3∂x1

∂2f3∂x3∂x2

+ ∂2f2∂x2

2+ ∂2f1∂x1∂x2

∂2f1∂x1∂x3

+ ∂2f3∂x2

3+ ∂2f2∂x2∂x3

⎞

⎟⎟⎟⎠−∆f

=

⎛

⎜⎜⎜⎝

∂∂x1

(∂f2∂x2

+ ∂f1∂x1

+ ∂f3∂x3

)

∂∂x2

(∂f3∂x3

+ ∂f2∂x2

+ ∂f1∂x1

)

∂∂x3

(∂f1∂x1

+ ∂f3∂x3

+ ∂f2∂x2

)

⎞

⎟⎟⎟⎠−∆f

=

⎛

⎜⎝

∂∂x1

(∇ · f

)

∂∂x2

(∇ · f

)

∂∂x3

(∇ · f

)

⎞

⎟⎠−∆f

= ∇(∇ · f

)−∆f.

Rn

f : U → Rm, U ⊂ Rn, n,m ∈ N,

x ∈ U ⊂ Rn f(x) ∈ Rm

n,m

RN

m = 1n ≥ 2

f = (f1, . . . , fm) : U → Rm, m ≥ 2,

x ∈ U f

fi : U → R, i = 1, . . . ,m,

fm = 1,n ≥ 2

n = 1,m ≥ 2

f : I → Rm, I ⊂ R, m ∈ N, m ≥ 2.

I 1n ∈ N

f : I → Rn, I ⊂ R, n ∈ N, n ≥ 2.

I = N f : N → Rn Rn

I ⊂ R

R = (−∞,∞) I ⊂ R

I = [α,β], (α,β], [α,β), (α,β), [α,∞), (α,∞), (−∞,β], (−∞,β), R,

α,β ∈ R α < βt ∈ I ⊂ R

f : I → Rn

α = βI = {α} = {β}

I = ∅ ⊂ R

RN

γ(α)

γ(β)

x

y

γ ′(t)γ(t)

γ : [α,β] → R2

γ(t)

tf

Rn

f(t) ∈ Rn t ∈ I

It1, t2 ∈ I t1 < t2 t ∈ [t1, t2]

IRn

f

n ≥ 2γ

γ = (γ1, . . . ,γn) : I → Rn I ⊂ RRn γ(I)

Rn

RN

C = γ(I) ⊂ Rn

C ⊂ Rn

γ : I → Rn I ⊂ RC t ∈ I C =

γ(I) γγ

γ(I)

γ(I)γ

Rn

c

γ : I → Rn

C = γ(I) ⊂ Rn

γ : I → Rn C = γ(I)C ⊂ Rn

γ : I → Rn γ(I) = C

γ1 : t 4→ (cos t, sin t), t ∈ [0, 2π),

γ2 : t 4→ (cos t, sin t), t ∈ R,

γ3 : t 4→ (cos(vt), sin(vt)), t ∈ [0, 2π/v], v > 0,

R2

C = γ1([0, 2π)) = γ2(R) = γ3([0, 2π/v]) = {(x,y) ∈ R2 : x2 + y2 = 1}.

γ(I) ⊂ Rn

γ(I) γ

RN

γλ : t 4→ a+ tλv ∈ Rn, t ∈ R, a ∈ Rn, ∥v∥ = 1, λ ∈ R \ {0},

Rn n = 3

γλ(R) = {a+ tλv ∈ Rn : t ∈ R}.

Γf = {(t, f(t)) ∈ R2 : t ∈ I}f : I → R I ⊂ R

R2

γf(t) = (t, f(t)) ∈ R2, t ∈ I,

γf(I) = Γf

γ : I → Rn

γ : I → Rn 1− 1 γ

I = [α,β] γ(α), γ(β) ∈ Rn

γ : [α,β] → Rn γ γ(α) = γ(β)γ

γ : [α,β] → Rn

[α,β) γ|[α,β) 1− 1

C =γ(I) ⊂ Rn C ⊂ Rn

γ : I → Rn C = γ(I)C ⊂ Rn

γ : [α,β] → Rn γ([α,β]) = C

C ⊂ R2 R2 \C

C

C ⊂ R2

RN

γ3

(cos 0, sin 0) = (cos(2π), sin(2π)) = (1, 0).

[0, 2πv ) t1, t2 ∈ [0, 2πv ) t1 < t2 cos(vt1) =cos(vt2) vt1 ≤ π ≤ vt2 cos x 1− 1

[0,π] [π, 2π] sin(vt1) ≥ 0 ≥ sin(vt2)vt1 = 0 vt2 = π cos 0 = 1 = −1 = cosπ

γ1γ3

γ2 γ1 [0, 2π]γ2

γ1 γ2

γ0(t) = (cos t, sin t), t ∈ [0, 2π],

γ3 v = 1R2 {(x,y) ∈ R2 : x2 + y2 = 1}

γλ λ = 0t1, t2 ∈ R

γλ(t1) = γλ(t2) ⇔ (t1 − t2)λν = 0 ⇔ |t1 − t2| = 0.

γf t1, t2 ∈ I

γf(t1) = γf(t2) ⇒ t1 = t2.

γ : I → Rn

x ∈ γ(I) ⊂ Rn

t1, t2 ∈ I t1 = t2 t1 = t2γ(t1) = γ(t2) = x x

γ(t) = (t2 − 1, t3 − t), t ∈ R,

(0, 0) = γ(−1) = γ(1),

(t21 − 1, (t21 − 1)t1) = (t22 − 1, (t22 − 1)t2) t1 < t2 ⇒ t1 = −1, t2 = 1.

γ (−∞,−1] [1,∞)[−1, 1]

RN

γ ′(1)γ ′(−1)

x

y

γ(−1) = (0, 0) = γ(1)

γ = (γ1, . . . ,γn) : I → Rn I ⊂ R

γi : I → R i = 1, . . . ,nγ : I → Rn

t ∈ I

γ′(t) := Dγ(t) =

⎛

⎜⎝γ′1(t)

γ′n(t)

⎞

⎟⎠ ∈ Rn,

I = [α,β] α,β ∈ R α < βγi : [α,β] → R i = 1, . . . ,n

γ ′i(α) = lim

t→α+

γi(t)− γi(α)

tγ ′i(β) = lim

t→β−

γi(t)− γi(β)

t.

γγ ′ : I → Rn

C1

C1

RN

t ∈ I

γ : I → Rn Rn

γ ′(t) t ∈ Iγ(t) t

γ(t) γ ′(t) = 0γ ′(t) = 0

γ

C ⊂ Rn

γ : I → Rn

γ(I) = C

γ(t) = (t2, t3), t ∈ R,

γ(R) = {(x,y) ∈ R2 : x ≥ 0, y = ±x3/2} = {(x,y) ∈ R2 : x ≥ 0, y2 = x3}

γ′(t) = (2t, 3t2), t ∈ R,

(0, 0)

γ′(t) = limh→0

γ(t+ h)− γ(t)

h

γ(t)

{γ(t) + s

γ′(t)

∥γ′(t)∥ : s ∈ R}⊂ Rn,

γ(t)

{γ(t) + s

γ(t+ h)− γ(t)

∥γ(t+ h)− γ(t)∥|h|

h: s ∈ R

}⊂ Rn, 0 < |h| ≪ 1 ,

RN

x

y

y2 = x3

γ(t)γ(t+ h)

γ(t+h)−γ(t)h

γ(t) γ(t+ h)

h → 0γ ′(t)

γ(t) ∈ Rn t ∈ I ∥γ′(t)∥

0 < ε≪ 1 ε 1

RN

γ ′1(t1)γ ′

2(t2)

γ1 γ2

ϑ

ϑ γ1 γ2

γi : Ii → Rn i = 1, 2γ1(t1) = γ2(t2) ϑ ∈ [0,π]

γ′1(t1) γ′2(t2)

cos ϑ =γ′1(t1) · γ′2(t2)

∥γ′1(t1)∥∥γ′2(t2)∥γi γ1(t1) = γ2(t2)

γ′(t) = (2t, 3t2 − 1) = 0, ∀ t ∈ R.

(0, 0) = γ(−1) = γ(1)

γ′(−1) = (−2, 2) γ′(1) = (2, 2)

γ : [α,β] → Rn α < β Rn P([α,β])P = {t0, t1, . . . , tν} α =: t0 < t1 < . . . < tν := β

RN

γ(β)

γ(tk−1)

γ(t2)

γ(t1)

γ(α)

γ[α,β]

ν ∈ N [α,β] γR γ

L(γ) := sup

{ν∑

k=1

∥γ(tk)− γ(tk−1)∥ : P = {t0, t1, . . . , tν} ∈ P([α,β])

}∈ R.

γ(tk) = γ(tk−1)P γ

γ : [α,β] → Rn

γ : [α,β] → Rn

γg : [0, 1] → R2

RN

γg(t) = (t,g(t))

g(t) =

{t cos(π/t), t ∈ (0, 1],

0, t = 0.

P = {0, 12ν ,

12ν−1 , . . . ,

13 ,

12 , 1}

∥γg(0)− γg( 12ν )∥+

2ν∑

k=2

∥γg( 1k )− γg(1

k−1 )∥

= 12ν∥(1, 1)∥+

2ν∑

k=2

1k(k−1)∥(1, 2k− 1)∥ >

2ν∑

k=2

1k → ∞ ν→ ∞

γ : [α,β] → Rn

L(γ) =

∫β

α

∥∥γ′(t)∥∥ dt.

γ : [α,β] → Rn

∀ ε > 0 ∃ δ > 0 ∀ t, τ ∈ [α,β], |t− τ| ∈ (0, δ) :

∥∥∥∥γ(t)− γ(τ)

t− τ− γ′(t)

∥∥∥∥ < ε.

γ : [α,β] → Rn

γi : [α,β] → R i = 1, . . . ,n γ γ′i : [α,β] → R[α,β]

∀ ε > 0 ∃ δ > 0 ∀ t, s ∈ [α,β], |t− s| < δ ∀ i = 1, . . . ,n :∣∣γ′i(s)− γ

′i(t)

∣∣ < ε√n,

t, τ ∈ [α,β] 0 <|t− τ| < δ ϑi ∈ (0, 1) i = 1, . . . ,n

γi(t)− γi(τ)

t− τ= γ′i (t+ ϑi(τ− t)) ∀ i = 1, . . . ,n,

|t− (t+ ϑi(τ− t))| = |ϑi||τ− t| < δ

∀ ε > 0 ∃ δ > 0 ∀ t, τ ∈ [α,β], |t− τ| ∈ (0, δ) :∥∥∥∥γ(t)− γ(τ)

t− τ− γ′(t)

∥∥∥∥2

=n∑

i=1

∣∣∣∣γi(t)− γi(τ)

t− τ− γ′i(t)

∣∣∣∣2

=n∑

i=1

∣∣γ′i (t+ ϑi(τ− t))− γ′i(t)∣∣2 ≤

n∑

i=1

ε2

n= ε2.

✷

RN

γ = (γ1, . . . ,γn) : [α,β] → Rn

∫β

αγ(t)dt :=

( ∫β

αγ1(t)dt, . . . ,

∫β

αγn(t)dt

)∈ Rn

∥∥∥∫β

αγ(t)dt

∥∥∥ ≤∫β

α∥γ(t)∥ dt.

γγi i = 1, . . . ,n ∥γ∥ :

[α,β] → R ∥γ∥(t) := ∥γ(t)∥

S(i)ν :=

ν∑

k=1

γi

(α+ k

β−α

ν

) 1ν, ν ∈ N, i = 1, . . . ,n,

sν :=ν∑

k=1

∥∥∥γ(α+ k

β−α

ν

)∥∥∥1

ν, ν ∈ N,

limν→∞

S(i)ν =

∫β

αγi(t)dt ∈ R, i = 1, . . . ,n, lim

ν→∞sν =

∫β

α∥γ(t)∥ dt ∈ R,

Sν := (S(1)ν , . . . , S

(n)ν ) =

ν∑

k=1

γ

(α+ k

β−α

ν

)1

ν, ν ∈ N

limν→∞

Sν =

∫β

αγ(t)dt,

limν→∞

∥Sν∥ =∥∥∥∫β

αγ(t)dt

∥∥∥.

∥Sν∥ =∥∥∥ν∑

k=1

γ(α+ k

β−α

ν

) 1ν

∥∥∥ ≤ν∑

k=1

∥∥∥γ(α+ k

β−α

ν

)∥∥∥1

ν= sν ∀ ν ∈ N,

∥∥∥∫β

αγ(t)dt

∥∥∥ = limν→∞

∥Sν∥ ≤ limν→∞

sν =

∫β

α∥γ(t)∥dt.

✷

RN

γ : [α,β] → Rn

γ′ : [α,β] → Rn ∥γ′∥ : [α,β] → Rε >

0 δ1 > 0 P = {t0, t1, . . . , tν} ∈ P([α,β])∥P∥ := max{tk − tk−1 : k = 1, . . . ,ν} < δ1

∣∣∣∣∣

∫β

α

∥∥γ′(t)∥∥ dt−

ν∑

k=1

∥∥γ′(tk)∥∥ (tk − tk−1)

∣∣∣∣∣ <ε

2.

δ ∈ (0, δ1]P ∥P∥ < δ

∥∥∥∥γ(tk)− γ(tk−1)

tk − tk−1− γ′(tk)

∥∥∥∥ <ε

2(β−α)∀ k = 1, . . . ,ν

∣∣∣∣∣

∫β

α

∥∥γ′(t)∥∥ dt−

ν∑

k=1

∥γ(tk)− γ(tk−1)∥∣∣∣∣∣

≤∣∣∣∣∣

∫β

α

∥∥γ′(t)∥∥ dt−

ν∑

k=1

∥∥γ′(tk)∥∥ (tk − tk−1)

∣∣∣∣∣

+ν∑

k=1

∥∥γ(tk)− γ(tk−1)− γ′(tk)(tk − tk−1)

∥∥ <ε

2+

ν∑

k=1

ε(tk − tk−1)

2(β−α)= ε,

{ν∑

k=1

∥γ(tk)− γ(tk−1)∥ : P = {t0, t1, . . . , tν} ∈ P([α,β])

}⊂ R

∫βα ∥γ′(t)∥ dt ∈ R

∫β

α

∥∥γ′(t)∥∥ dt =

ν∑

k=1

∫tk

tk−1

∥∥γ′(t)∥∥ dt

≥ν∑

k=1

∥∥∥∥∥

∫tk

tk−1

γ′(t)dt

∥∥∥∥∥ =ν∑

k=1

∥γ(tk)− γ(tk−1)∥ .

✷

RN

x

y

π 2π

t

γ : [0,ϕ] → R2, γ(t) = (cos t, sin t), ϕ > 0,

L(γ) =

∫ϕ

0∥γ ′(t)∥dt =

∫ϕ

0∥(− sin t, cos t)∥dt =

∫ϕ

0dt = ϕ

ϕ = 2π

γ : R → R2, γ(t) = (t− sin t, 1− cos t),

0xyt = 0

0x

0x

L(γ|[0,2π]) =

∫2π

0∥(1− cos t, sin t)∥dt =

∫2π

0

√2− 2 cos t dt

= 2

∫2π

0sin(t/2)dt = 4

∫π

0sin τdτ = 8.

γ : [α,β] → Rn ϕ : [A,B] → [α,β]A,B ∈ R A < B 1− 1

ζ := γ ◦ϕ : [A,B] → Rn

RN

γϕ

ϕγ

ϕ : [A,B] → [α,β] ϕ−1 : [α,β] → [A,B]ϕ C1

ζ =γ ◦ϕ γγ([α,β]) = ζ([A,B])

x ∈ γ([α,β]) t ∈ [α,β] x = γ(t) τ ∈ [A,B]t = ϕ(τ) x = γ(ϕ(τ)) = ζ(τ) ∈ ζ([A,B]) x ∈ ζ([A,B])

τ ∈ [A,B] x = ζ(τ) = γ(ϕ(τ)) ϕ(τ) ∈ [α,β]x ∈ γ([α,β])

ϕ : [A,B] → [α,β] C1

ϕ′(τ) > 0 ∀ τ ∈ [A,B]

ϕ′(τ) < 0 ∀ τ ∈ [A,B]

C1 ϕϕ′(τ) = 0 ∀ τ ∈ [A,B]

(ϕ−1 ◦ϕ

)′(τ) =

(ϕ−1

)′(ϕ(τ))ϕ′(τ) = 1 ∀ τ ∈ [A,B].

ϕ ′ : [A,B] → [α,β][A,B]

γ : [α,β] → Rn α,β ∈ R α < β

[A,B] A,B ∈ R A < B C1

ϕ(τ) =αB−βA

B−A+β−α

B−Aτ, τ ∈ [A,B],

γ

ψ(T) =βB−αA

B−A−β−α

B−AT , T ∈ [A,B],

ϕ ∈ C([A,B])ϕ ∈ C1([A,B]) ϕ(x) =

x3 x ∈ [−1, 1] ϕ ∈ C1([A,B]) ϕ ′(τ) = 0 ∀ τ ∈ [A,B] C1

ϕ : [A,B] → ϕ([A,B])

RN

ζ = γ ◦ϕ : [A,B] → Rn

ζ(A) = γ(α), ζ(B) = γ(β), ζ ′(τ) =β−α

B−Aγ ′(ϕ(τ)),

η = γ ◦ψ : [A,B] → Rn

η(A) = γ(β), η(B) = γ(α), η ′(T) = −β−α

B−Aγ ′(ψ(T)).

t0 = ϕ(τ0) = ψ(T0) ∈[α,β]

x0 = γ(t0) = ζ(τ0) = η(T0) ∈ C = γ([α,β]) = ζ([A,B]) = η([A,B])

γ ζη

γ : [α,β] → Rn

γγ

γ(t)t ∈ [α,β] t1 < t2

γ(t1) γ(t2)t

Rγ

C ⊂ Rn

C

C ⊂ R2

CC

CC

C

RN

γ−(β)

γ−(α)

x

y

(γ−) ′(τ)

γ−(τ)

CC

π2

(0 −11 0

)

C

γ : [α,β] → Rn

γ−(τ) = γ(α+β− τ), τ ∈ [α,β],

γ

R2

γ : [0, 2π] → R2, γ(t) = (cos t, sin t),

γ ′(t) = (− sin t, cos t) ∥γ ′(t)∥ = 1 ∀ t ∈ [0, 2π],

(0 −11 0

)(− sin tcos t

)=

(− cos t− sin t

),

RN

γ(t)γ C1

ϕ(τ) = ντ, τ ∈ 1

ν[0, 2π] =

{[0, 2πν ], ν > 0,

[2πν , 0], ν < 0,ν = 0,

ζ(τ) = γ(ϕ(τ)) = (cos(ντ), sin(ντ)), τ ∈ 1

ν[0, 2π].

ϕ γ ν > 0ν < 0

t τγ ζ

γ(0) = (1, 0) = ζ(0) = η(2π|ν|

),

γ(π2

)= (0, 1) = ζ

( π2ν

)= η

( 3π

2|ν|

),

γ(π) = (−1, 0) = ζ(πν

)= η

( π|ν|

),

γ(3π

2

)= (0,−1) = ζ

(3π2ν

)= η

( π

2|ν|

),

γ(2π) = (1, 0) = ζ(2πν

)= η(0).

t τ ζγ ν > 0 ν < 0

2πν ≤ τ ≤ 0

τ ν < 0

ψ(τ) = ντ+ 2π = −|ν|τ+ 2π, τ ∈[0,

2π

|ν|

], ν < 0,

γ

η(τ) = γ(ψ(τ)) = (cos(−|ν|τ+ 2π), sin(−|ν|τ+ 2π))

= (cos(|ν|τ),− sin(|ν|τ)), τ ∈[0,

2π

|ν|

].

ηγ

ν = −1 γ

η(τ) = γ−(τ) = (cos τ,− sin τ), τ ∈ [0, 2π],

RN

C1

γ : [α,β] → Rn ϕ :[A,B] → [α,β] C1

ζ = γ ◦ϕ : [A,B] → Rn

ζ′(τ) = γ′(ϕ(τ))ϕ′(τ) ∀ τ ∈ [A,B],

ζ(τ) =γ(ϕ(τ)) ϕ′(τ) ∈ R \ {0}

γ ζ = γ ◦ ϕ

ϕ

ϕ

γi : [αi,βi] → Rn i = 1, 2ϑ γ1(t1) = γ2(t2) ϕi : [Ai,Bi] → [αi,βi]

C1

ϑ′

ζi = γi ◦ϕi : [Ai,Bi] → Rn ζ1(τ1) = ζ2(τ2) τi = ϕ−1i (ti)

ϑ′ = ϑ ϕi

ϑ′ = π − ϑ ϕi

γ : [α,β] → Rn

ϕ : [A,B] → [α,β] C1

ζ = γ ◦ϕ : [A,B] → Rn

L(ζ) =

∫B

A∥ζ′(τ)∥dτ =

∫B

A∥γ′(ϕ(τ))∥|ϕ′(τ)|dτ =

∫β

α∥γ′(t)∥dt = L(γ),

ϕϕ

∫B

A∥γ′(ϕ(τ))∥|ϕ′(τ)|dτ = −

∫B

A∥γ′(ϕ(τ))∥ϕ′(τ)dτ

= −

∫α

β∥γ′(t)∥dt =

∫β

α∥γ′(t)∥dt.

RN

C ⊂ Rn

Rn

C ⊂ Rn γ : [α,β] → Rn

C1 γ([α,β]) = C C γ

L(C) := L(γ) =

∫β

α∥γ ′(t)∥dt.

C1

γ : [α,β] → Rn

s(t) =

∫t

α∥γ′(τ)∥dτ, t ∈ [α,β],

γ

s−1 sγ : [α,β] → Rn C1

γ

s(t) =

∫t

α

∥∥γ′(τ)∥∥ dτ, t ∈ [α,β].

γ ∥γ′(τ)∥ ∈ R τ ∈ [α,β]

s′(t) = ∥γ′(t)∥, t ∈ [α,β].

γ s′(t) > 0 ∀ t ∈ [α,β]

s : [α,β] → [0,L(γ)]

1− 1

s−1 : [0,L(γ)] → [α,β]

(s−1)′(τ) =1

s′((s−1)(τ)), τ ∈ [0,L(γ)].

s−1 C1

✷

RN

γ : [α,β] → Rn

s : [α,β] → [0,L(γ)]

ζ := γ ◦ (s−1) : [0,L(γ)] → Rn

γ ζ

γ : [α,β] → Rn

ζ = γ ◦ (s−1) : [0,L(γ)] → Rn

∥ζ′(τ)∥ = 1 ∀ τ ∈ [0,L(γ)],

γ : [α,β] → Rn

s(t) =

∫t

α∥γ′(t)∥dτ = t−α ∀ t ∈ [α,β].

ζ′(τ) = (γ ◦ (s−1))′(τ) = γ′(s−1(τ))(s−1)′(τ) =γ′(s−1(τ))

∥γ′(s−1(τ))∥ ∀ τ ∈ [0,L(γ)].

✷

Rn

Rn

RN

γ : [α,β] → Rn

s ∈ [α,β]

k(s) =∥∥∥ limh→0

γ ′(s+ h)− γ ′(s)

h

∥∥∥,

γ : [α,β] → Rn

k(s) := ∥γ′′(s)∥, s ∈ [α,β],

γ s

γ : [α,β] → Rn

t ∈ [α,β]

k(t) =∥∥∥ limh→0

T(t+ h)− T(t)

s(t+ h)− s(t)

∥∥∥,

T(t) =γ ′(t)

∥γ ′(t)∥γ(t) γ

t s

k(t) = k(s(t)) ∀ t ∈ [α,β],

k ζ = γ ◦ s−1 : [0,L(γ)] → Rn

γk(t) = k(t) ∀ t ∈ [α,β] k

γ(t) = a+ tb t ∈ R a, b ∈ Rn b = 0γ′(t) = b s(t) = (t− t0)∥b∥

t0 ∈ R

ζ(s) = γ

(s

∥b∥ + t0

)= a+

(s

∥b∥ + t0

)b = a+ t0b+ s

b

∥b∥ ∀ s ∈ R

γ[α,β] α ∈ R β = α+ L(γ)

RN

ζ′(s) =b

∥b∥ ∀ s ∈ R

k(s) = ∥ζ′′(s)∥ = 0 ∀ s ∈ R.

γ(t) = (r cos t, r sin t) t ∈ [0, 2π] r > 0γ′(t) = (−r sin t, r cos t) s(t) =

tr t ∈ [0, 2π]

ζ(s) = γ(sr

)=(r cos

(sr

), r sin

(sr

))∀ s ∈ [0, 2πr]

ζ′(s) =(−r sin

(sr

), r cos

(sr

)) 1

r=(− sin

(sr

), cos

(sr

))∀ s ∈ [0, 2πr]

k(s) = ∥ζ′′(s)∥ =∥∥∥(− cos

(sr

),− sin

(sr

))∥∥∥1

r=

1

r∀ s ∈ [0, 2πr].

fα(x) = α12x

2 x ∈ [−1, 1] α > 0γα([−1, 1])

γα(x) = (x,α12x

2), x ∈ [−1, 1].

γα

sα(x) =

∫x

−1∥γ ′α(t)∥dt =

∫x

−1

√1+α2t2 dt =

1

α

∫αx

−α

√1+ τ2 dτ

=1

α

1

2

(τ√

1+ τ2 + arsinh τ)∣∣∣αx

τ=−α

=1

α

1

2

(αx√

1+α2x2 + arsinh(αx) +α√

1+α2 + arsinhα)

=1

2

(x√

1+α2x2 +1

αarsinh(αx) +

√1+α2 +

1

αarsinhα

).

γαs = sα(x)

RN

t

k(t) =∥T ′α(t)∥

∥γ ′α(t)∥

=

∥∥γ ′′α(t)∥γ ′

α(t)∥2 − γ ′α(t)(γ

′′α(t) · γ ′

α(t))∥∥

∥γ ′α(t)∥4

, t ∈ [−1, 1].

γ ′α(t) = (1,αt), γ ′′

α(t) = (0,α),

k(t) =α

(1+α2t2)3/2 , t ∈ [−1, 1],

t = 0α

Rn

R2

γi : [αi,βi] → Rn i = 1, . . . , k

γi(βi) = γi+1(αi+1) i = 1, . . . , k− 1.

γ : [α,β] → Rn

α = α1, β = α1 +k∑

i=1

(βi −αi),

γ(τ) := γi

(τ+αi −α1 −

i−1∑

j=1

(βj −αj)

)

∀ τ ∈[α1 +

i−1∑

j=1

(βj −αj),α1 +i∑

j=1

(βj −αj)

], i = 1, . . . , k,

RN

γi i = 1, . . . , k

γ = γ1 ⊕ · · ·⊕ γk,

C = γ([α,β]),Ci = γi([αi,βi]) ⊂ Rn i = 1, . . . , k

C = C1 ⊕ · · ·⊕Ck.

γγi i = 1, . . . , k

γ

γ : [α,β] → Rn P = {t0, t1, . . . , tk}[α,β] α = t0 < t1 < . . . < tk = β

γ = γ1 ⊕ · · ·⊕ γk γi = γ|[ti−1,ti], i = 1, . . . ,k.

γ

γ = γ1 ⊕ · · · ⊕ γk : [α,β] → Rn

L(γ) :=k∑

i=1

L(γi) =k∑

i=1

∫βi

αi

∥γ ′i(t)∥dt

γ

C1 γ

C1

γγi i = 1, . . . , k

L(C) =k∑

i=1

L(Ci).

C1

C1

RN

ai ∈ Rn i = 0, . . . , k ai−1 = ai i = 1, . . . ,k

γi(t) = ai−1 + t(ai − ai−1), t ∈ [0, 1], i = 1, . . . ,k.

γ = γ1 ⊕ · · ·⊕ γk : [0, k] → Rn C = γ([0, k]) = C1 ⊕ · · ·⊕Ck,

γi Ci = γi([0, 1])k + 1 ai

L(γ) =k∑

i=1

L(γi) =k∑

i=1

L(Ci) =k∑

i=1

∥ai − ai−1∥.

C L(C) L(γ)

∅ = U = (a,b)× (c,d) ⊂ R2

∂U = ([a,b]× {c})∪ ({b}× [c,d])∪ ([a,b]× {d})∪ ({a}× [c,d])

=: C1 ∪C2 ∪C3 ∪C4

Ci =γi([0, 1]) i = 1, . . . , 4

γ1(t) = (a, c) + t((b, c)− (a, c)

)= (a, c) + t(b− a, 0), t ∈ [0, 1],

γ2(t) = (b, c) + t((b,d)− (b, c)

)= (b, c) + t(0,d− c), t ∈ [0, 1],

γ3(t) = (b,d) + t((a,d)− (b,d)

)= (b,d) + t(a− b, 0), t ∈ [0, 1],

γ4(t) = (a,d) + t((a, c)− (a,d)

)= (a,d) + t(c− d, 0), t ∈ [0, 1],

∂U

∂U = C1 ⊕C2 ⊕C3 ⊕C4

L(∂U) = L(C1) + L(C2) + L(C3) + L(C4) = 2(b− a) + 2(d− c).

RN

f : [a,b] → R y =f(x) a ≤ x ≤ b

γ(t) = (t, f(t)) ∈ R2, t ∈ [a,b],

γ

L(γ) =

∫b

a

√1+

(f ′(t)

)2dt.

f : [a,b] → R r = f(ϕ)a ≤ ϕ ≤ b

γ(ϕ) = f(ϕ)(cosϕ, sinϕ), ϕ ∈ [a,b].

γ

L(γ) =

∫b

a

√(f(ϕ)

)2+(f ′(ϕ)

)2dϕ.

y = a coshx

a0 ≤ x ≤ x0 a > 0

r = aϕ 0 ≤ ϕ ≤ 2π a > 0

r = a(1+ cosϕ) 0 ≤ ϕ ≤ 2π a > 0

a > b > 0

γ(t) = (a cos t,b sin t) ∈ R2, t ∈ [0, 2π].

γ C1

E = γ([0, 2π]) =

{(x,y) ∈ R2 :

x2

a2+

y2

b2= 1

}⊂ R2.

γ E

x2

a2+

y2

b2= 1.

∼ ∼

RN

E

L(E) = L(γ) = a

∫2π

0

√1− ε2 cos2 tdt, ε :=

√a2 − b2

a< 1.

ε e := aε

γ(t) = a(cos3 t, sin3 t), t ∈ [0, 2π], a > 0 ,

γ : R → R3, γ(t) = (r cos t, r sin t, ct), r > 0, c ∈ R \ {0}

γ|[0,T ] T > 0

γ γ(t) t ∈ Rγ′(t) = (−r sin t, r cos t, c) t ∈ R

∥γ′(t)∥ =√

r2 + c2.

γ′ : R → R3

γ|[0,T ] T > 0

L(γ|[0,T ]) =

∫T

0∥γ′(t)∥dt =

∫T

0

√r2 + c2 dt =

√r2 + c2 T .

γ : R → R2, γ(t) = (ect cos t, ect sin t), c ∈ R \ {0}.

γ|[a,b] a < b a,b ∈ R

lima→−∞

L(γ|[a,0])

γ (0, 0)

γ′(t) = ect(c cos t− sin t, c sin t+ cos t)

∥γ′(t)∥ = ect√

(c cos t− sin t)2 + (c sin t+ cos t)2 = ect√

c2 + 1,

L(γ|[a,b]) =

∫b

a∥γ′(t)∥dt =

∫b

aect√

c2 + 1dt =√

c2 + 1ecb − eca

c.

RN

lima→−∞

L(γ|[a,0]) =

√c2 + 1

cc > 0,

c < 0 R

limb→∞

L(γ|[0,b]) =

√c2 + 1

−cc < 0.

κ : R → R2 κ(t) = r(cos t, sin t) r > 0 ∥κ(t)∥ = rt ∈ R t1, t2 ∈ R κ(t1) = γ(t2) ∥γ(t2)∥ =

ect2 = r t2 = log rc γ(t2) = κ(t2) ∥γ(t)∥ = r

t = t2 t2t ∈ R γ κ

ϑ ∈ [0,π] γ(t2) = κ(t2)

cos ϑ =γ′(t2) · κ′(t2)

∥γ′(t2)∥∥κ′(t2)∥

=1√

c2 + 1(c cos t2 − sin t2, c sin t2 + cos t2) · (− sin t2, cos t2)

=1√

c2 + 1∈ (0, 1).

|c| → 0|c| → ∞

γ : (α,β) → Rn

T ′(t) · T(t) = 0 ∀ t ∈ (α,β) T ′(t) γT ′(t) = 0

N(t) :=T ′(t)

∥T ′(t)∥

γ t

∥T(t)∥ = 1 ∀ t ∈ (α,β)

0 =d

dt1 =

d

dt∥T(t)∥2 =

d

dt(T(t) · T(t)) = 2T ′(t) · T(t).

T ′(t) =d

dt

(γ′(t)

∥γ′(t)∥

)=γ′′(t)∥γ′(t)∥2 − γ′(t) (γ′′(t) · γ′(t))

∥γ′(t)∥3 .

RN

γ : [α,β] → R3

γ t = s−1(τ) ∈[α,β] s : [α,β] → [0,L(γ)] γ

k(s(t)) =∥γ′′(t)× γ′(t)∥

∥γ′(t)∥3 .

γ τ ∈ [0,L(γ)]

k(τ) = ∥ζ′′(τ)∥ =

∥∥∥∥d

dτ

(T(s−1(τ))

)∥∥∥∥ =∥T ′(t)∥∥γ′(t)∥

=∥γ′′(t)∥γ′(t)∥2 − γ′(t) (γ′′(t) · γ′(t)) ∥

∥γ′(t)∥4

=

√∥γ′′(t)∥2∥γ′(t)∥2 − (γ′′(t) · γ′(t))2 ∥

∥γ′(t)∥3 ,

=∥γ′′(t)× γ′(t)∥

∥γ′(t)∥3 ,

∥αx−βy∥2 = (αx−βy) · (αx−βy) = α2∥x∥2−αβ(x · y), (αx−βy) · y = 0,

∥x× y∥2 = ∥x∥2∥y∥2 − (x · y)2.

γ : R → R3 γ(t) = (r cos t, r sin t, ct) r > 0 c ∈ R \ {0}

γ′(t) = (−r sin t, r cos t, c)

∥γ′(t)∥ =√

r2 + c2

γ′′(t) = (−r cos t,−r sin t, 0)

∥γ′′(t)∥ = r.

γ′′(t) · γ′(t) = 0,

∥γ′′(t)∥∥γ′(t)∥2 =

r

r2 + c2.

γ[α,β] →Rn

k

k(t) =∥∥∥T ′(t)

s ′(t)

∥∥∥ ∀ t ∈ [α,β]

k ζ = γ ◦ s−1 : [0,L(γ)] → Rn

k(τ) =∥∥∥d

dτ(T(s−1(τ)))

∥∥∥ =∥∥∥T ′(s−1(τ))

s ′(s−1(τ))

∥∥∥ ∀ τ ∈ [0,L(γ)]

τ = s(t)

α = (α1, . . . ,αn) ∈ Nn0 , n ∈ N, N0 := N ∪ {0},

|α| := α1 + . . .+αn

αα! := α1! . . .αn!

αi! := αi · (αi − 1) · . . . · 2 · 1 αi ∈ N 0! := 1

xα := xα11 . . . xαn

n , x = (x1, . . . , xn) ∈ Rn

x0i := 1 |α|f x

Dαf(x) :=∂|α|

∂xα11 . . .∂xαn

nf(x).

U ⊂ Rn f : U → R kx ∈ U η ∈ Rn {x+ tη : t ∈ [0, 1]} ⊂ U g : [0, 1] → Rg(t) := f(x+ tη) k

g(k)(t) =∑

|α|=k

k!

α!Dαf(x+ tη) ηα ∀ t ∈ [0, 1].

k

g(k)(t) =n∑

i1,...,ik=1

∂kf

∂xik . . .∂xi1(x+ tη)ηi1 . . .ηik ∀ t ∈ [0, 1].

k = 1 γ(t) := x+ tη Dγ(t) = γ′(t) = η

g′(t) = Dg(t) = Df(γ(t))Dγ(t) = ∇f(x+ tη) · η =n∑

i=1

∂f

∂xi(x+ tη)ηi.

ν = 2, . . . ,k

g(ν−1)(t) =n∑

i1,...,iν−1=1

∂ν−1f

∂xiν−1. . .∂xi1

(x+ tη)ηi1 . . .ηiν−1.

g(ν)(t) =d

dtg(ν−1)(t)

=n∑

i1,...,iν−1=1

d

dt

(∂ν−1f

∂xiν−1. . .∂xi1

(x+ tη)

)ηi1 . . .ηiν−1

=n∑

i1,...,iν−1=1

(∇ ∂ν−1f

∂xiν−1. . .∂xi1

(x+ tη) · η)ηi1 . . .ηiν−1

=n∑

i1,...,iν−1=1

n∑

iν=1

∂

∂xiν

∂ν−1f

∂xiν−1. . .∂xi1

(x+ tη)ηiν ηi1 . . .ηiν−1

=n∑

i1,...,iν=1

∂νf

∂xiν . . .∂xi1(x+ tη)ηi1 . . .ηiν .

s(i1, . . . , ik) :=∂kf

∂xik . . .∂xi1(x+ tη)ηi1 . . .ηik

k (i1, . . . , ik) ∈ {1, . . . ,n}k

αν ∈ {0, . . . , k} ν ∈ {1, . . . ,n}

α := (α1, . . . ,αn) ∈ Nn0 |α| = α1 + . . .+αn = k.

α

k!

α!=

k!

α1! . . .αn!k (i1, . . . , ik) ∈ {1, . . . ,n}k

(i1, . . . , ik) α

s(i1, . . . , ik) =∂kf

∂xα11 . . .∂xαn

n(x+ tη)ηα1

1 . . .ηαnn = Dαf(x+ tη) ηα

g(k)(t) =n∑

i1,...,ik=1

s(i1, . . . , ik) =∑

(i1,...,ik)∈{1,...,n}k

s(i1, . . . , ik)

=∑

α∈{α∈Nn0 :|α|=k}

k!

α!Dαf(x+ tη) ηα =

∑

|α|=k

k!

α!Dαf(x+ tη) ηα.

✷

U ⊂ Rn x ∈ U η ∈ Rn {x+ tη : t ∈ [0, 1]} ⊂ Uf : U → R k+ 1 ϑ ∈ [0, 1]

f(x+ η) =∑

|α|≤k

Dαf(x)

α!ηα +

∑

|α|=k+1

Dαf(x+ ϑη)

α!ηα.

g : [0, 1] → R g(t) := f(x+ tη)k+ 1

ϑ ∈ [0, 1]

g(1) =k∑

m=0

g(m)(0)

m!+

g(k+1)(ϑ)

(k+ 1)!.

g(m)(0)

m!=

∑

|α|=m

Dαf(x)

α!ηα, m = 1, . . . , k,

g(k+1)(ϑ)

(k+ 1)!=

∑

|α|=k+1

Dαf(x+ ϑη)

α!ηα

✷

f : U → R U ⊂ Rn kx ∈ U

f(x+ η) =∑

|α|≤k

Dαf(x)

α!ηα + o(∥η∥k) η→ 0,

limη→0

1

∥η∥k

⎛

⎝f(x+ η)−∑

|α|≤k

Dαf(x)

α!ηα

⎞

⎠ = 0.

U ⊂ Rn x ∈ U δ > 0B(x, δ) ⊂ U η ∈

Rn ∥η∥ < δ ϑ ∈ [0, 1]

f(x+ η) =∑

|α|≤k−1

Dαf(x)

α!ηα +

∑

|α|=k

Dαf(x+ ϑη)

α!ηα

=∑

|α|≤k

Dαf(x)

α!ηα +

∑

|α|=k

Dαf(x+ ϑη)−Dαf(x)

α!ηα

α ∈ Nn0 |α| = k Dαf |ηα| ≤ ∥η∥|α|

✷

f : U → R U ⊂ Rn kx ∈ U

f(x+ η) =k∑

m=0

Pm(η) + o(∥η∥k) η→ 0,

Pm(η) :=∑

|α|=m

Dαf(x)

α!ηα, η ∈ Rn, m = 0, . . . ,k

m ∈ {0, . . . , k} η = (η1, . . . ,ηn) ∈ Rn

Pm(λη) = λmPm(η) ∀ λ ∈ R ∀ η ∈ Rn.

Tk,f,x(x+ η) :=k∑

m=0

Pm(η), η ∈ Rn

k f xPm m = 0, 1, 2

m = 0α ∈ Nn

0 |α| = 0 ⇒ α = (0, . . . , 0) = 0 ⇒

P0(η) =∑

|α|=0

Dαf(x)

α!ηα =

D0f(x)

0!η0 = f(x) ∀ η ∈ Rn.

m = 1α ∈ Nn

0 |α| = 1 ⇒ α = (0, . . . , 0, 1, 0, . . . , 0) = ei i = 1, . . . ,n ⇒

P1(η) =∑

|α|=1

Dαf(x)

α!ηα =

n∑

i=1

Deif(x)

ei!ηei

=n∑

i=1

∂f

∂xi(x)ηi = ∇f(x) · η ∀ η ∈ Rn.

m = 2α ∈ Nn

0 |α| = 2 ⇒ α = ei + ej i, j = 1, . . . ,n i ≤ j ⇒

P2(η) =∑

|α|=2

Dαf(x)

α!ηα =

n∑

i,j=1i≤j

Dei+ejf(x)

(ei + ej)!ηei+ej

=n∑

i=1

D2eif(x)

(2ei)!η2ei +

n∑

i,j=1i<j

Dei+ejf(x)

(ei + ej)!ηei+ej

=1

2

n∑

i=1

∂2f

∂x2i(x)η2i +

n∑

i,j=1i<j

∂2f

∂xi∂xj(x)ηiηj

=1

2

n∑

i=1

∂2f

∂x2i(x)η2i +

1

2

n∑

i,j=1i =j

∂2f

∂xi∂xj(x)ηiηj

=1

2

n∑

i,j=1

∂2f

∂xi∂xj(x)ηiηj =:

1

2ηTHf(x) η ∀ η ∈ Rn,

Hf(x)

f : U → R U ⊂ Rn

f x ∈ U

Hf(x) :=

⎛

⎜⎜⎜⎜⎜⎜⎝

∂2f

∂x21(x) · · · ∂2f

∂xn∂x1(x)

∂2f

∂x1∂xn(x) · · · ∂2f

∂x2n(x)

⎞

⎟⎟⎟⎟⎟⎟⎠∈ Rn×n

f x

f : U → R U ⊂ Rn

Hf(x) x ∈ U

k = 0, 1, 2

f : U → R U ⊂ Rn x ∈ U

f

f(x+ η) = f(x) + o(1) η→ 0,

f

f(x+ η) = f(x) + grad f(x) · η+ o(∥η∥) η→ 0,

f

f(x+ η) = f(x) + grad f(x) · η+ 1

2ηTHf(x) η+ o(∥η∥2) η→ 0.

f(x,y) = e(x−1)2 cosy, (x,y) ∈ R2,

(x0,y0) = (1, 0)

fR2

grad f(x,y) =

(∂

∂xf(x,y),

∂

∂yf(x,y)

)= e(x−1)2 (2(x− 1) cosy,− siny)

Hf(x,y) =

(∂2

∂x2 f(x,y)∂2

∂y∂xf(x,y)∂2

∂x∂yf(x,y)∂2

∂y2 f(x,y)

)

= e(x−1)2((4(x− 1)2 + 2) cosy −2(x− 1) cosy

−2(x− 1) cosy − cosy

).

(x,y)(x,y) 4→ x (x,y) 4→ y

∂2

∂x∂yf(x,y) =

∂2

∂y∂xf(x,y).

f : R2 → R

f(x,y) = T2,f,(1,0)(x,y) + o((x− 1)2 + y2

)(x,y) → (1, 0).

f (1, 0)

T2,f,(1,0)(x,y) = f(1, 0) + grad f(1, 0)

(x− 1y

)+

1

2((x− 1),y)Hf(1, 0)

(x− 1y

)

= 1+ (0, 0)

(x− 1y

)+

1

2((x− 1),y)

(2 00 −1

)(x− 1y

)

= 1+1

2((x− 1),y)

(2(x− 1)

−y

)

= 1+ (x− 1)2 −1

2y2

= 2− 2x+ x2 −1

2y2.

o(·)

lim(x,y)→(1,0)

2e(x−1)2 cosy− 4+ 4x− 2x2 + y2

(x− 1)2 + y2= 0.

2(x0,y0) = (1, 1)

fi : (0,∞)× (0,∞) → R, i = 1, 2, f1(x,y) =x− y

x+ yf2(x,y) = xy.

f(x,y) = ex siny (x0,y0) ∈ R2

ϑ : R2 → [0, 1]

sin(x+ y) = x+ y−1

2(x2 + 2xy+ y2) sin

(ϑ(x,y)(x+ y)

)∀ (x,y) ∈ R2.

U ⊂ Rn f : U → R x ∈ U

∃ ε > 0 ∀ y ∈ B(x, ε)∩U : f(x) ≤ f(y),

∃ ε > 0 ∀ y ∈ B(x, ε)∩U : f(x) ≥ f(y),

∃ ε > 0 ∀ y ∈ B(x, ε)∩U \ {x} : f(x) < f(y),

∃ ε > 0 ∀ y ∈ B(x, ε)∩U \ {x} : f(x) > f(y),

f x

∀ y ∈ U : f(x) ≤ f(y),

∀ y ∈ U : f(x) ≥ f(y),

∀ y ∈ U \ {x} : f(x) < f(y),

∀ y ∈ U \ {y} : f(x) > f(y),

f x

f : U → R x ∈ Ux f

f(x) ff(x) x ∈ U

U ⊂ Rn f : U → Rf

U ⊂ Rn f : U → Rx ∈ U

grad f(x) = 0.

ε > 0 B(x, ε) ⊂ U

gi : (−ε, ε) → R, gi(t) := f(x+ tei), i = 1, . . . ,n,

t = 0

g′i(0) =∂f

∂xi(x).

∂f

∂xi(x) = 0 ∀ i = 1, . . . ,n.

✷

U ⊂ Rn f : U → Rx ∈ U x grad f(x) = 0

U ⊂ Rn ff : U → R U ⊂ Rn

f UintU U

f : U → Rx ∈ U ⊂ Rn fx

f

f x ∈ ∂Ux f

x

f : U → R, f(x) = c, c ∈ R , U ⊂ Rn,

U

grad f(x) = 0 ∀ x ∈ U.

fi : Rn → R, fi(x1, . . . , xn) = xi, i = 1, . . . ,n,

grad fi(x) = ei = 0 ∀ x = (x1, . . . , xn) ∈ Rn.

f : R2 → R, f(x,y) = x2 − y2,

grad f(x,y) = 2(x,−y) ∀ (x,y) ∈ R2

(0, 0)

∀ ε > 0 : f(ε, 0) > 0 > f(0, ε).

f : Rn → R, f(x) = ∥x∥,x = 0

grad f(x) =x

∥x∥ = 0 ∀ x ∈ Rn \ {0},

x = 0

f(0) = 0

f : U → R, U = [0, 1]× [0, 1], f(x,y) = x+ y,

f(1, 1) = 2 f(0, 0) = 0(0, 1)× (0, 1) U

grad (x+ y) = (1, 1) ∀ (x,y) ∈ R2,

∂U = A∪ B U

A =([0, 1]× {0}

)∪({1}× [0, 1]

), B =

({0}× [0, 1]

)∪([0, 1]× {1}),

(0, 0) (1, 1) A Bf

f(0, 0) = 0 ≤ f(x, 0) = x ≤ f(1, 0) = 1 ≤ f(1,y) = 1+ y ≤ f(1, 1) = 2,

f(0, 0) = 0 ≤ f(0,y) = y ≤ f(0, 1) = 1 ≤ f(x, 1) = x+ 1 ≤ f(1, 1) = 2

x,y ∈ [0, 1]

A ∈ Rn×n

ηTA η ≥ 0 ∀ η ∈ Rn,

ηTA η ≤ 0 ∀ η ∈ Rn,

ηTA η > 0 ∀ η ∈ Rn \ {0},

ηTA η < 0 ∀ η ∈ Rn \ {0},

A

∃ η1, η2 ∈ Rn : ηT1A η1 < 0, ηT2A η2 > 0.

A ∈ Rn×n n

⇔ ≥ 0

⇔ ≤ 0

⇔ > 0

⇔ < 0

⇔ A < 0 > 0

✷

A =

⎛

⎜⎝a11 . . . a1n

an1 . . . ann

⎞

⎟⎠ ∈ Rn×n

∆k :=

∣∣∣∣∣∣∣

a11 . . . a1k

ak1 . . . akk

∣∣∣∣∣∣∣, k = 1, . . . ,n, ∆1 = a11

⇔ ∆k > 0 ∀ k = 1, . . . ,n

⇔ (−1)k∆k > 0 ∀ k = 1, . . . ,n

✷

2× 2

2× 2

A =

(a bb c

)∈ R2×2

⇔ a > 0 |A| > 0

⇔ a < 0 |A| > 0,

⇔ |A| < 0

a = 0

(x,y)

(0 bb c

)(xy

)= y(2bx+ cy)

|A| = −b2 ≤ 0 b = 0 Ac A |A| < 0

|A| < 0 b = 0

y(2bx+ cy) = ±2b (x,y) = (±1−c

2b, 1),

Aa = 0

(x,y)

(a bb c

)(xy

)= x(ax+ by) + y(bx+ cy) =

1

a

((ax+ by)2 + |A|y2

).

A |A| < 0 |A| < 0

(ax+ by)2 + |A|y2 =

{a2 > 0 (x,y) = (1, 0),

|A| < 0 (x,y) = (−ba , 1),

A a ≷ 0 |A| > 0 AA

a ≷ 0 |A| > 0 |A| < 0|A| = 0 ✷

U ⊂ Rn f : U → R x ∈ Ugrad f(x) = 0

Hf(x) ⇒ f x

Hf(x) ⇒ f x

Hf(x) ⇒ f x

A := Hf(x) Uδ0 > 0

ϕ : B(0, δ0) → R B(x, δ0) ⊂ U

f(x+ η) = f(x) +1

2ηTAη+ϕ(η) ϕ(η) = o(∥η∥2) η→ 0,

ϕ

∀ ε > 0 ∃ δ ∈ (0, δ0) ∀ η ∈ B(0, δ) : |ϕ(η)| ≤ ε∥η∥2.

A S := {η ∈ Rn : ∥η∥ = 1}S

η 4→ ηTAη, η ∈ Rn,

SηTAη > 0 η ∈ S

α := min{ηTAη : η ∈ S} > 0.

ηTAη ≥ α∥η∥2 ∀ η ∈ Rn.

η ∈ S η = 0 η ∈ Rn \ {0}η = λη∗ λ := ∥η∥ η∗ := η

∥η∥ ∥η∗∥ = 1

ηTAη = λ2ηT∗Aη∗ ≥ λ2α∥η∗∥2 = α∥η∥2.

ε = α4 δ ∈ (0, δ0)

|ϕ(η)| ≤ α

4∥η∥2 ∀ η ∈ B(0, δ).

f(x+ η) ≥ f(x) +α

2∥η∥2 − |ϕ(η)| ≥ f(x) +

α

4∥η∥2 ∀ η ∈ B(0, δ),

f(x+ η) > f(x) ∀ η ∈ B(0, δ) \ {0},

f xA = Hf(x) −A = H−f(x)

−fx f x

A

∀ δ ∈ (0, δ0) ∃ y1, y2 ∈ B(x, δ) : f(y1) > f(x) > f(y2),

x ff(x)

A η1 ∈ Rn \ {0} α := ηT1Aη1 > 0

f(x+ tη1) = f(x) +α

2t2 +ϕ(tη1) ∀ t ∈

(−δ0

∥η1∥,δ0

∥η1∥

),

ϕ(η) = o(∥η∥2) η→ 0 ε = α4∥η1∥2

> 0 δ1 ∈ (0, δ0∥η1∥

)

|ϕ(tη1)| ≤α

4t2 ∀ t ∈ (−δ1, δ1),

f(x+ tη1) ≥ f(x) +α

4t2 > f(x) ∀ t ∈ (−δ1, 0)∪ (0, δ1).

η2 ∈ Rn \ {0} β := ηT2Aη2 < 0

f(x+ sη2) = f(x) +β

2s2 +ϕ(sη2) ∀ s ∈

(−δ0

∥η2∥,δ0

∥η2∥

).

ε = −β4∥η2∥2

> 0 δ2 ∈ (0, δ0∥η2∥

)

|ϕ(sη2)| ≤−β

4s2 ∀ s ∈ (−δ2, δ2),

f(x+ sη2) ≤ f(x) +β

4s2 < f(x) ∀ s ∈ (−δ2, 0)∪ (0, δ2).

δ ∈ (0, min{δ1∥η1∥, δ2∥η2∥}) B(x, δ) ⊂ B(x, δ0) ⊂ U

∀ y1 := x+ tη1, y2 := x+ sη2 ∈ B(x, δ) \ {x} : f(y1) > f(x) > f(y2).

✷

n = 2

2× 2

U ⊂ R2 f : U → R(x,y) ∈ U

∂f(x,y)

∂x=∂f(x,y)

∂y= 0

∆ :=∂2f(x,y)

∂x2∂2f(x,y)

∂y2−

(∂2f(x,y)

∂x∂y

)2

.

∂2f(x,y)

∂x2> 0 ∆ > 0 ⇒ f (x,y)

∂2f(x,y)

∂x2< 0 ∆ > 0 ⇒ f (x,y)

∆ < 0 ⇒ f (x,y)

f : R2 → R, f(x,y) = c+ x2 + y2, c ∈ R

f

grad f(x,y) = 2(x,y) ∀ (x,y) ∈ R2.

f (0, 0)

Hf(x,y) = 2

(1 00 1

)∀ (x,y) ∈ R2,

(0, 0)

g : R2 → R, g(x,y) = c− x2 − y2, c ∈ R

g

gradg(x,y) = −2(x,y) ∀ (x,y) ∈ R2.

g (0, 0)

Hg(x,y) = −2

(1 00 1

)∀ (x,y) ∈ R2,

(0, 0)

h : R2 → R, h(x,y) = c+ x2 − y2, c ∈ R

h

gradh(x,y) = 2(x,−y) ∀ (x,y) ∈ R2.

f (0, 0)

Hh(x,y) = 2

(1 00 −1

)∀ (x,y) ∈ R2,

(0, 0) h

h(x,y) = x2 − y2

(0, 0, 0)

ff(0, 0) = c

f(0, 0) = c < f(x,y) = c+ ∥(x,y)∥2 ∀ (x,y) ∈ R2 \ {(0, 0)} ⇔ ∥(x,y)∥ > 0.

g g(0, 0) = c

g(0, 0) = c > g(x,y) = c− ∥(x,y)∥2 ∀ (x,y) ∈ R2 \ {(0, 0)} ⇔ ∥(x,y)∥ > 0.

h (0, 0)

h(ε, 0) = c+ ε2 > h(0, 0) = c > h(0, ε) = c− ε2 ∀ ε > 0,

(0, 0)h h

h(0, 0)(0, 0,h(0, 0)) h

h(x, 0) x ∈ R x = 0 h(0,y) y ∈ Ry = 0 0x 0y(0, 0) h

h

(0, 0,h(0, 0))R3 z = c

c = 0

h (0, 0)

fi : R2 → R i = 1, . . . , 4

f1(x,y) = x2 + y4, f2(x,y) = x2, f3(x,y) = x2 + y3, f4(x,y) = x2 − y4.

(0, 0)

grad f1(x,y) = (2x, 4y3), grad f2(x,y) = (2x, 0),

grad f3(x,y) = (2x, 3y2), grad f4(x,y) = (2x,−4y3),

(0, 0)

Hfi(0, 0) =

(2 00 0

), i = 1, . . . , 4.

(0, 0)

f1

f2

f3

(0, 0, 0)

f40x 0y

f(x,y) = sin x siny sin(x+ y), 0 ≤ x,y, x+ y ≤ π.

f R2

intU U

U = {(x,y) ∈ R2 : 0 ≤ x ≤ π, 0 ≤ y ≤ π− x},

intU = {(x,y) ∈ R2 : 0 < x < π, 0 < y < π− x},

f R2

grad f(x,y) =

(siny sin(2x+ y)sin x sin(2y+ x)

),

sin(α+β) = sinα cosβ+ cosα sinβ, α,β ∈ R.

intU sin x, siny ∈ (0, 1]

sin(2x+ y) = sin(2y+ x) = 0,

2x+ y, 2y+ x ∈ (0, 2π)

2x+ y = 2y+ x = π

intU

(x,y) =(π3,π

3

).

f R2

Hf(x,y) =

(2 siny cos(2x+ y) sin(2(x+ y))

sin(2(x+ y)) 2 sin x cos(2y+ x)

)

cosπ = −1 sin π3 = − sin 4π3 =

√32

Hf

(π3,π

3

)=

(−2 sin π3 sin 4π

3sin 4π

3 −2 sin π3

)=

(−√3 −

√32

−√32 −

√3

)

intUf

f(π3,π

3

)=

3√3

8.

U

f(x,y) = 0 ∀ (x,y) ∈ ∂U ⊂ {(x,y) ∈ R2 : x = 0 ∨ y = 0 ∨ x+ y = π}.

f U intUf(U) ⊂ [0, 1] f(intU) ⊂ (0, 1) U f

f Uf 0 ∂U

U

f(x,y) = sin x+ siny+ sin(x+ y), 0 ≤ x,y ≤ π

2.

f R2

grad f(x,y) =

(cos x+ cos(x+ y)cosy+ cos(x+ y)

),

Hf(x,y) =

(− sin x− sin(x+ y) − sin(x+ y)

− sin(x+ y) − siny− sin(x+ y)

).

(0, π2 ) × (0, π2 ) [0, π2 ] × [0, π2 ]

cos x = − cos(x+ y) = cosy

1− 1(0, π2 )

(x, x) ∈(0,π

2

)×(0,π

2

)cos x = − cos(2x) = 1− 2 cos2 x,

cos(α+β) = cosα cosβ− sinα sinβ, α,β ∈ R

sinα ∈ [0, 1] ∀ α ∈ [0,π] sinα ∈ (0, 1) ∀ α ∈ (0,π)

cos2 α+ sin2 α = 1, α ∈ R.

2z2 + z− 1 = (2z− 1)(z+ 1) = 0

z = cos x ∈ (0, 1) x ∈ (0, π2 ) cos π3 = 12

f

(x,y) =(π3,π

3

)f(π3,π

3

)= 2 sin

π

3+ sin

2π

3=

3√3

2,

Hf

(π3,π

3

)=

(− sin π3 − sin 2π

3 − sin 2π3

− sin 2π3 − sin π3 − sin 2π

3

)=

(−√3 −

√32

−√32 −

√3

),

f

f f

ϕ1(x) = f(x, 0) = 2 sin x, 0 ≤ x ≤ π

2,

ϕ2(y) = f(π2,y)= 1+ siny+ cosy, 0 ≤ y ≤ π

2,

ϕ3(x) = f(x,π

2

)= 1+ sin x+ cos x, 0 ≤ x ≤ π

2,

ϕ4(y) = f(0,y) = 2 siny, 0 ≤ y ≤ π

2,

sinα = cos(α−

π

2

), α ∈ R.

ϕ1 ϕ4 [0, π2 ]

f(0, 0) = 0, f(π2, 0)= f(0,π

2

)= 2,

ϕ3 ϕ2 (0, π2 ) x = y = π4

ϕ3

(π4

)= ϕ2

(π4

)= f(π4,π

2

)= f(π2,π

4

)= 1+

√2,

2π cossin 0, π2 ,π tan = sin

cos = 1cot

ϕ3 ϕ2

ϕ ′3(x) = cos x− sin x ϕ ′′

3 (x) = − sin x− cos x.

ϕ3,ϕ2

f ϕ3(x) = f(x, π2 )x = π

4 [0, π2 ]× {π2 } ∂([0, π2 ]× [0, π2 ])f(π4 ,y) y ∈ [0, π2 ] y = π

2f

∂

∂yf(π4,π

2

)= cos

π

2+ cos

(π4+π

2

)= −

1√2< 0,

f(π4,y)> f(π4,π

2

)y ∈

(π2− ε,

π

2

)ε > 0

f (π4 ,π2 ) f(x,y) = f(y, x)

(π2 ,π4 )

f(0, 0) (π2 , 0) (π2 , 0) (π2 ,

π4 )

(π2 ,π4 ) (π2 ,

π2 ) f(π2 ,

π2 ) = 2 f

(0, 0) (0, π2 ) (π4 ,π2 ) (π2 ,

π2 )

f(π3 ,

π3 )

(π2 ,π2 )

fk = 1 f R2

(π2 ,π2 ) η ∈ R2

ϑ ∈ [0, 1]

f(x+ η) = f(x) + grad f(x+ ϑη) · η.

grad f(π2,π

2

)= (−1,−1),

ff ε > 0

f(x,y) = f(π2 ,π2 ) + grad f

((π2 ,

π2 ) + ϑ(x−

π2 ,y− π

2 ))· (x− π

2 ,y− π2 ) > f(π2 ,

π2 )

(x,y) ∈ (π2 − ε, π2 ]× (π2 − ε, π2 ] \ {(π2 ,π2 )} ϑ ∈ [0, 1]

f(π2 ,π2 ) = 2

ff

f(0, 0) = 0

f(π2 ,π2 ) = 2

f(π3 ,π3 ) =

3√3

2

f(x,y) = x3 − y3, (x,y) ∈ R2

f(x,y) = x3 + y3 − 3xy, (x,y) ∈ R2

f(x,y) = x2 + y2 − 2xy+ 1, (x,y) ∈ R2

f(x,y) = x2 + xy+ y2 + x+ y+ 1, (x,y) ∈ R2

f(x,y) = x3y2(1− x− y), (x,y) ∈ R2

f(x,y) =1

y−

1

x− 4x+ y, (x,y) ∈ R2 x = 0 y = 0

f(x,y) = (x2 + 2y2)e−(x2+y2), (x,y) ∈ R2

f(x,y) = (y− x2)(y− 2x2), (x,y) ∈ R2

f(x,y) = sin x siny, (x,y) ∈ R2

k ∈ N x1, . . . , xk ∈ Rn

x ∈ Rn

f(x) =k∑

i=1

∥x− xi∥2, x ∈ Rn,

ξ =1

k

k∑

i=1

xi.

90

E(x, t) x t

E(x, t) = x2(a− x)t2e−t, 0 ≤ x ≤ a, 0 ≤ t.

x t

U ⊂ Rn V ⊂ Rm F = (F1, . . . Fm) : U × V → Rm

(x0, y0) ∈ U× V F(x0, y0) = 0

∂F

∂y(x0, y0) :=

⎛

⎜⎜⎝

∂F1∂y1

(x0, y0) . . . ∂F1∂ym

(x0, y0)

∂Fm∂y1

(x0, y0) . . . ∂Fm∂ym

(x0, y0)

⎞

⎟⎟⎠ ∈ Rm×m

∃ δ, ε > 0 ∀ x ∈ B(x0, δ) ⊂ U ∃ ! g(x) ∈ B(y0, ε) ⊂ V :

F(x, g(x)) = 0 g : B(x0, δ) → B(y0, ε)

✷

δ1 ∈ (0, δ)

∂F

∂y(x, g(x)) ∈ Rm×m ∀ x ∈ B(x0, δ1)

Dg(x) = −

(∂F

∂y(x, g(x))

)−1∂F

∂x(x, g(x)) ∈ Rm×n ∀ x ∈ B(x0, δ1).

G(x) := (x, g(x)) ∈ Rn+m x ∈ B(x0, δ),(F ◦ G

)(x) = F

(G(x)

)= F (x, g (x)) = 0 ∈ Rm ∀ x ∈ B(x0, δ),

D(F ◦ G

)(x) = DF(G(x))DG(x)

=

(∂F

∂x(G(x))

︸ ︷︷ ︸∈ Rm×n

,∂F

∂y(G(x))

︸ ︷︷ ︸∈ Rm×m

)(I

Dg(x)

)} ∈ Rn×n

} ∈ Rm×n

=∂F

∂x(G(x)) +

∂F

∂y(G(x))Dg(x)

=∂F

∂x(x, g(x)) +

∂F

∂y(x, g(x))Dg(x)

= O ∈ Rm×n ∀ x ∈ B(x0, δ),

O ∈ Rm×n I ∈ Rn×n

∂F

∂x(x, y) :=

⎛

⎜⎜⎝

∂F1∂x1

(x, y) . . . ∂F1∂xn

(x, y)

∂Fm∂x1

(x, y) . . . ∂Fm∂xn

(x, y)

⎞

⎟⎟⎠ ∈ Rm×n.

det∂F

∂y(x, g(x)) , x ∈ B(x0, δ),

∂Fj∂yi

(x, g(x)) , j, i = 1, . . . ,m,

∂F∂y (x0, y0)

det∂F

∂y(x0, y0) = 0,

δ1 ∈ (0, δ)

det∂F

∂y(x, g(x)) = 0 ∀ x ∈ B(x0, δ1),

∂F∂y (x, g(x)) x ∈ B(x0, δ1) ✷

n = m = 1

F : (a,b)× (c,d) → R

(x0,y0) ∈ (a,b)× (c,d) F(x0,y0) = 0∂F

∂y(x0,y0) = 0.

∃ δ1, ε > 0 ∀ x ∈ (x0 − δ1, x0 + δ1) ⊂ (a,b) ∃ ! g(x) ∈ (y0 − ε,y0 + ε) ⊂ (c,d) :

F(x,g(x)) = 0 g : (x0 − δ1, x0 + δ1) → (y0 − ε,y0 + ε)

∂F

∂y(x,g(x)) = 0 g′(x) = −

∂F∂x (x,g(x))∂F∂y (x,g(x))

∀ x ∈ (x0 − δ1, x0 + δ1).

gF(x, y) = 0

y

U × V F U × V xy F(x, y) = 0

c ∈ R

Lf(c) = {(x,y) ∈ U : f(x,y) = c}

f : U → RU ⊂ R2 (x0,y0) ∈ Lf(c)

grad f(x0,y0) =

(∂f

∂x(x0,y0),

∂f

∂y(x0,y0)

)= (0, 0),

F(x,y) := f(x,y)− c, (x,y) ∈ U.

∂F∂y (x0,y0) =

∂f∂y (x0,y0) = 0 I1, I2 ⊂ R

(x0,y0) ∈ I1 × I2 ⊂ U

ϕ : I1 → I2

Lf(c)∩ (I1 × I2) = {(x,y) ∈ I1 × I2 : y = ϕ(x)}, ϕ′(x) = −∂f∂x (x,y)∂f∂y (x,y)

∀ x ∈ I1.

∂F∂x (x0,y0) =

∂f∂x (x0,y0) = 0 J1, J2 ⊂ R

(x0,y0) ∈ J1 × J2 ⊂ U

ψ : J2 → J1

Lf(c)∩ (J1 × J2) = {(x,y) ∈ J1 × J2 : x = ψ(y)}, ψ′(y) = −∂f∂y (x,y)∂f∂x (x,y)

∀ y ∈ J1.

y x x y

Γf(c) = {(x,y, z) ∈ R3 : z = f(x,y), (x,y) ∈ R2}

f(x,y) = x2 + y2 (x,y) ∈ R2

c ∈ R c < 0

Lf(c) = ∅, c < 0

{(0, 0)} c = 0Lf(0) = {(0, 0)}

(0, 0)√c > 0 R2 c > 0

Lf(c) = {(x,y) ∈ R2 : x2 + y2 = c}, c > 0

Γ1 := {(x,y) ∈ R2 : x ∈ (−√c,√c), y =

√c− x2},

Γ2 := {(x,y) ∈ R2 : x ∈ (−√c,√c), y = −

√c− x2},

Γ3 := {(x,y) ∈ R2 : y ∈ (−√c,√c), x =

√c− y2},

Γ4 := {(x,y) ∈ R2 : y ∈ (−√c,√c), x = −

√c− y2},

grad f(x,y) = (2x, 2y) = (0, 0), (x,y) ∈ Lf(c), c > 0.

A :=∂F

∂y(x0, y0) ∈ Rm×m

G : U× V → Rm, G(x, y) := y−A−1F(x, y)

∂G

∂y(x, y) = I−A−1 ∂F

∂y(x, y) ∀ (x, y) ∈ U× V ,

I ∈ Rm×m

∂G

∂y(x0, y0) = O ∈ Rm×m,

O ∈ Rm×m

∂G

∂y,∂F

∂y: U× V → Rm×m F(x, y0) : U → Rm F(x0, y0) = 0

U0 × V0 := B(x0, δ)× B(y0, ε) ⊂ U× V, δ, ε > 0,

∥∥∥∥∂G

∂y(x, y)

∥∥∥∥ ≤ 1

2, ∥F(x, y0)∥ <

ε

4∥A−1∥∂F

∂y(x, y) ∈ Rm×m ∀ (x, y) ∈ U0 × V0.

∥G(x, y)− G(x, η)∥ ≤ 1

2∥y− η∥ ∀ x ∈ U0, y, η ∈ V0,

∥y− η∥− ∥A−1(F(x, y)− F(x, η))∥ ≤ 1

2∥y− η∥ ∀ x ∈ U0, y, η ∈ V0,

∥y− η∥ ≤ 2∥A−1∥∥F(x, y)− F(x, η)∥ ∀ x ∈ U0, y, η ∈ V0,

ε

2∥A−1∥ ≤ ∥F(x, y)− F(x, y0)∥ ∀ x ∈ U0, y ∈ ∂V0,

∥F(x, y)∥ ≥ ∥F(x, y)− F(x, y0)∥− ∥F(x, y0)∥ >ε

4∥A−1∥ > ∥F(x, y0)∥

∀ x ∈ U0, y ∈ ∂V0

x ∈ U0

f : V0 → R, f(y) := ∥F(x, y)∥2,

V0

y ∈ V0 ∂V0

V0

grad f(y) = 2∂F

∂y(x, y)F(x, y) = 0,

F(x, y) = 0,

y ∈ V0

g : U0 → V0 F(x, g(x)) = 0 x ∈ U0

g : U0 → V0 G(x, g(x)) = g(x) x ∈ U0

x1, x2 ∈ U0

g(x1)− g(x2) = G(x1, g(x1))− G(x2, g(x2))

= G(x1, g(x1))− G(x1, g(x2))−A−1(F(x1, g(x2))− F(x2, g(x2))

),

M := max

{∥∥∥∥∂F

∂x(x, y)

∥∥∥∥ : (x, y) ∈ U0 × V0

}

∥F(x1, y)− F(x2, y)∥ ≤ M∥x1 − x2∥, ∀ x1, x2 ∈ U0, y ∈ V0,

∥g(x1)− g(x2)∥ ≤ 1

2∥g(x1)− g(x2)∥+ ∥A−1∥∥x1 − x2∥ ∀ x1, x2 ∈ U0,

∥g(x1)− g(x2)∥ ≤ 2∥A−1∥∥x1 − x2∥ ∀ x1, x2 ∈ U0,

gg : U0 → V0

gg

x ∈ U0 F (x, g(x)) ∈ U× VF(x, g(x)) = 0

F(x+ η, g(x+ η)) = B η+Ch(η) + ϕ(η, h(η)) ∀ η ∈ B(0, δ1),

h(η) := g(x+ η)− g(x), B(x, δ1) ⊂ U0, B :=∂F

∂x(x, g(x)), C :=

∂F

∂y(x, g(x))

ϕ(η, h(η)) = o(∥(η, h(η))∥) (η, h(η)) → 0.

F(x+ η, g(x+ η)) = 0 ∀ η ∈ B(0, δ1),

h(η) = −C−1B η−C−1ϕ(η, h(η)) ∀ η ∈ B(0, δ1),

ϕ(η, h(η)) = o(∥η∥) η→ 0,

ϕ(η, h(η)) = o(∥(η, h(η))∥) = (∥η∥+ ∥h(η)∥) = (∥η∥) η→ 0.

✷

x ∈ R |x| < δ δ > 0y(x)

esin(xy) + x2 − 2y− 1 = 0

y′(x) x y(x)

F(x,y) = esin(xy) + x2 − 2y− 1 (x,y) ∈ R2

F(0, 0) = 0

∂

∂yF(x,y) = esin(xy) cos(xy)x− 2 ⇒ ∂

∂yF(0, 0) = −2 = 0.

δ, ε > 0F(x,y) = 0 x ∈ (−δ, δ) y ∈ (−ε, ε)

(−δ, δ) ∋ x 4→ y(x) ∈ (−ε, ε) F(x,y(x)) = 0

y′(x) = −∂∂xF(x,y(x))∂∂yF(x,y(x))

= −esin(xy(x)) cos(xy(x))y(x) + 2x

esin(xy(x)) cos(xy(x))x− 2, x ∈ (−δ, δ).

y(0) = 0 y′(0) = 0

x ∈ Ry(x) x2 + y2 = 1 y′(x) = − x

y(x)

F(x,y) = x2+y2−1 (x,y) ∈R2 F(0, 1) = 0

grad F(x,y) = (2x, 2y) ⇒ grad F(0, 1) = (0, 2) ⇒ ∂

∂yF(0, 1) = 2 = 0.

δ > 0 ε ∈(0, 1) F(x,y) = 0 x ∈ (−δ, δ)y(x) ∈ (1 − ε, 1 + ε) ⇒ y(x) > 0 x 4→ y(x)

y′(x) = −∂∂xF(x,y(x))∂∂yF(x,y(x))

= −x

y(x), x ∈ (−δ, δ).

y(0) = 1 y′(0) = 0

x,y, z ∈ Rz(x,y)

x4 + 2x cosy+ sin z = 0

F(x,y, z) = x4 + 2x cosy+sin z (x,y, z) ∈ R3 F(0, 0, 0) = 0

∇F(x,y, z) = (4x3 + 2 cosy,−2x siny, cos z) ⇒ ∂

∂zF(0, 0, 0) = 1 = 0.

δ, ε > 0(x,y) ∈ B((0, 0), δ) F(x,y, z) = 0 z(x,y) ∈ (−ε, ε)

(x,y) 4→ z(x,y)

∇z(x,y) = −

(∂∂xF(x,y, z(x,y)),

∂∂yF(x,y, z(x,y))

)

∂∂zF(x,y, z(x,y))

= −

(4x3 + 2 cosy,−2x siny

)

cos(z(x,y)), (x,y) ∈ B((0, 0), δ).

z(0, 0) = 0 ∇z(0, 0) = (−2, 0)

x ∈ R y(x)z(x) {

x2 + y2 − 2z2 = 0,

x2 + 2y2 + z2 = 4

F(x,y, z) =

(F1(x,y, z)F2(x,y, z)

)=

(x2 + y2 − 2z2

x2 + 2y2 + z2 − 4

), (x,y, z) ∈ R3.

y(x) z(x) F(x,y, z) = 0 x ∈ (−δ, δ) δ > 0y(0) z(0) F(0,y, z) = 0

z(0) = 2√5

y(0) = 2√2√5

F

(0,

2√2√5,

2√5

)= 0.

∂F(x,y, z)

∂(y, z)=

(∂F1(x,y,z)

∂y∂F1(x,y,z)

∂z∂F2(x,y,z)

∂y∂F2(x,y,z)

∂z

)= 2

(y −2z2y z

)

yz = 0

(∂F(x,y, z)

∂(y, z)

)−1

=1

10yz

(z 2z

−2y y

),

δ > 0 ε ∈(0, 2√

5

)

x ∈ (−δ, δ) F(x,y, z) = 0

(y(x), z(x)) ∈ B

((2√2√5,

2√5

), ε

)(⇒ y(x), z(x) > 0)

x 4→ (y(x), z(x))

(y′(x)z′(x)

)= −

(∂F(x,y(x), z(x))

∂(y, z)

)−1(∂F1(x,y(x),z(x))

∂x∂F2(x,y(x),z(x))

∂x

)

= −1

10y(x)z(x)

(z(x) 2z(x)

−2y(x) y(x)

)(2x2x

)=

(−3x5y(x)

x5z(x)

), x ∈ (−δ, δ).

u(x,y) v(x,y){x2 + y2 − u2 − v2 = 0,

x2 + 2y2 + 3u2 + 4v2 = 1

F(x,y,u, v) =

(F1(x,y,u, v)F2(x,y,u, v)

)=

(x2 + y2 − u2 − v2

x2 + 2y2 + 3u2 + 4v2 − 1

), (x,y,u, v) ∈ R4.

u(x,y) v(x,y) F(x,y,u, v) = 0 (x,y) ∈B((x0,y0), δ) (x0,y0) = (0, 0)

F

(1√10

,1√10

,1√10

,1√10

)= 0.

∂F(x,y,u, v)

∂(u, v)=

(∂F1(x,y,u,v)

∂u∂F1(x,y,u,v)

∂v∂F2(x,y,u,v)

∂u∂F2(x,y,u,v)

∂v

)= −2

(u v

−3u −4v

)

uv = 0(∂F(x,y,u, v)

∂(u, v)

)−1

= −1

2uv

(4v v−3u −u

)

δ, ε ∈(0, 1√

10

)

(x,y) ∈ B((

1√10

, 1√10

), δ)

F(x,y,u, v) = 0

(u(x,y), v(x,y)) ∈ B((

1√10

, 1√10

), ε)

(⇒ u(x,y), v(x,y) > 0)

(x,y) 4→ (u(x,y), v(x,y))

(∂u(x,y)∂x

∂u(x,y)∂y

∂v(x,y)∂x

∂v(x,y)∂y

)=

1

2u(x,y)v(x,y)

(4v(x,y) v(x,y)−3u(x,y) −u(x,y)

)(2x 2y2x 4y

)

=

(5x

u(x,y)6y

u(x,y)−4x

v(x,y)−5y

v(x,y)

), (x,y) ∈ B

((1√10

, 1√10

), δ).

U,W ⊂ Rn V ⊂ Rm W ⊂ U f : U× V → Rm

g : W → V f(x, g(x)) x ∈ W

U ⊂ Rn f : U → R x0 ∈ U f(x0) = 0ε > 0 f(x) = 0 ∀ x ∈ B(x0, ε) ⊂ U

U ⊂ Rn x0 ∈ U f : U → Rn Df(x0) ∈Rn×n U0 x0 ∈ U0 ⊂ U V :=B(f(x0), ε) ε > 0

f|U0: U0 → V

g := (f|U0)−1 : V → U0 Df(g(y)) ∈ Rn×n

y ∈ V

Dg(y) = (Df(g(y)))−1 ∈ Rn×n ∀ y ∈ V

Df(x) ∈ Rn×n x ∈ U0

Dg(f(x)) = (Df(x))−1 ∈ Rn×n ∀ x ∈ U0.

F : U× Rn → Rn, F(x, y) := f(x)− y,

F(x0, f(x0)) = 0∂F

∂x(x0, f(x0)) = Df(x0) ∈ Rn×n

∃ ε, δ > 0 ∀ y ∈ B(f(x0), ε) =: V ⊂ Rn ∃ ! g(y) ∈ B(x0, δ) ⊂ U : f(g(y)) = y,

g : V → U0 := {x ∈ B(x0, δ) : f(x) ∈ V}

Df(g(y)) ∈ Rn×n ∀ y ∈ V

Dg(y) = (Df(g(y)))−1 ∈ Rn×n ∀ y ∈ V .

f : U → Rn U ⊂ Rn

f−1(V) := {x ∈ U : f(x) ∈ V}

Rn

B(x0, δ)∩ f−1(V) = {x ∈ B(x0, δ) : f(x) ∈ V} = U0

Rn x0 ∈ U0 ⊂ B(x0, δ) ⊂ U

f(x) ∈ V ∀ x ∈ U0, f(U0) ⊂ V

∀ y ∈ V ∃ x := g(y) ∈ U0 : f(x) = y, V ⊂ f(U0).

f(U0) = V f|U0: U0 → V

y1 = f(x1) = f(x2) = y2, x1, x2 ∈ U0 ⇒ x1 = g(y1) = g(y2) = x2,

y1, y2 ∈ V f|U0: U0 → V

∀ y ∈ V : (f|U0)−1(y) = x ∈ U0 f(x) = y, x = g(y),

(f|U0)−1 = g : V → U0 ✷

f : U → Rn U ⊂ Rn

f−1 : f(U) → U f x ∈ U

f : (0,∞)× R → R2, f(r,ϕ) = (r cosϕ, r sinϕ),

Df(r,ϕ) =

(cosϕ −r sinϕsinϕ r cosϕ

), (r,ϕ) ∈ (0,∞)× R

(r,ϕ) ∈ (0,∞)× R detDf(r,ϕ) = r > 0

(Df(r,ϕ))−1 =

(cosϕ sinϕ

−sinϕ

r

cosϕ

r

).

f(r0,ϕ0) ∈ (0,∞)× R

g U ⊂ (0,∞)× R(r0,ϕ0) ∈ U V ⊂ R2 f(r0,ϕ0) ∈ V f|U : U → V

g : V → U

Dg(f(r,ϕ)) = (Df(r,ϕ))−1 =

(cosϕ sinϕ

−sinϕ

r

cosϕ

r

)∀ (r,ϕ) ∈ U.

(x,y) := f(r,ϕ) = (r cosϕ, r sinϕ)

r =√

x2 + y2,x

r=

y√x2 + y2

= cosϕ,y

r=

y√x2 + y2

= sinϕ

Dg(x,y) =

⎛

⎜⎝

x√x2 + y2

y√x2 + y2

−y

x2 + y2x

x2 + y2

⎞

⎟⎠ ∀ (x,y) ∈ V.

g fU := (0,∞)× (−π2 ,

π2 )

y

x= tanϕ, ϕ ∈ (−

π

2,π

2) ⇔ ϕ = arctan

y

x,

y

x∈ R

f|U : U → (0,∞)× R =: V

g(x,y) =(√

x2 + y2, arctany

x

)∀ (x,y) ∈ (0,∞)× R,

gf : (0,∞)× R → R2 \ {(0, 0)}

f(r,ϕ+ 2kπ) = f(r,ϕ) ∀ k ∈ Z

f : U → Rm U ⊂ Rn n,m ∈ N

f−1(V) := {x ∈ U : f(x) ∈ V} ⊂ Rn V ⊂ Rm

U ⊂ R2 (x,y) ∈ U

I = (α,β), J = (γ, δ) ⊂ R (x,y) ∈ I× J ⊂ U

I× J

f : R2 → R2 f(x,y) = (ex cosy, ex siny)(x,y) ∈ R2

f

f : R2 → R2 G1,G2 ⊂ R2

f(x,y) = (sin x coshy, cos x sinhy), G1 = R ×(0,π

2

), G2 = R ×

(π2,π).

f f(G1) f(G2) fG1 G2 G1 ∪G2

U = {(x,y, z) ∈ R3 : x+ y+ z+ 1 = 0} f : U → R3

f(x,y, z) =

⎛

⎜⎜⎜⎜⎜⎝

x

x+ y+ z+ 1y

x+ y+ z+ 1z

x+ y+ z+ 1

⎞

⎟⎟⎟⎟⎟⎠.

U

f

f 1− 1 f(U) f

f−1 : f(U) → R3

U ⊂ Rn f : U → R g = (g1, . . . ,gr) : U → Rr r < nf g(x) = 0

x0 ∈ M := {x ∈ U : g(x) = 0},

f|M x0

U ⊂ Rn f : U → R g = (g1, . . . ,gr) : U → Rr r < nf g(x) = 0

x0 g x0

Dg(x0) =

⎛

⎜⎜⎝

∂g1∂x1

(x0) . . . ∂g1∂xn

(x0)

∂gr∂x1

(x0) . . . ∂gr∂xn

(x0)

⎞

⎟⎟⎠ ∈ Rr×n

r

λj ∈ R, j = 1, . . . , r,

grad f(x0) = (λ1, . . . , λr)Dg(x0).

Dg(x0) ∈ Rr×n r rg r

x = (x1, . . . , xn)

y := (x1, . . . , xr), z := (xr+1, . . . , xn), x = (y, z),

y0 := (x(1)0 , . . . , x

(r)0 ), z0 := (x

(r+1)0 , . . . , x

(n)0 ), x0 = (y0, z0).

Dg(x0) ∈ Rr×n r

∂g

∂y(y0, z0) ∈ Rr×r

f g(x) = 0 x0

g(y0, z0) = 0.

gδ, ε > 0

V ×W := B(y0, ε)× B(z0, δ) ⊂ U ⊂ Rn = Rr × Rn−r,

∀ z ∈ W ∃ ! h(z) ∈ V : g(h(z), z) = 0

h : W → V

ϕ(z) := f(h(z), z), z ∈ W,

gradϕ(z)︸ ︷︷ ︸∈ R1×(n−r)

= grad f(h(z), z)︸ ︷︷ ︸∈ R1×n

(Dh(z)

I

)} ∈ Rr×(n−r)

} ∈ R(n−r)×(n−r)

=

(∂f

∂y(h(z), z)

︸ ︷︷ ︸∈ R1×r

,∂f

∂z(h(z), z)

︸ ︷︷ ︸∈ R1×(n−r)

)(Dh(z)

I

)

=∂f

∂y(h(z), z)Dh(z) +

∂f

∂z(h(z), z), ∀ z ∈ W,

I

∂f

∂y(y, z) :=

(∂f

∂x1(x), . . . ,

∂f

∂xr(x)

),

∂f

∂z(y, z) :=

(∂f

∂xn−r(x), . . . ,

∂f

∂xn(x)

).

(h(z), z) ∈ M ∀ z ∈ W,

f|M x0 = (y0, z0) = (h(z0), z0) ∈ Mϕ z0 ∈ W gradϕ(z0) = 0

∂f

∂y(y0, z0)Dh(z0) +

∂f

∂z(y0, z0) = 0.

ψ(z) := g(h(z), z) = 0 ∀ z ∈ W,

Dψ(z) = Dg(h(z), z)︸ ︷︷ ︸∈ Rr×n

(Dh(z)

I

)} ∈ Rr×(n−r)

} ∈ R(n−r)×(n−r)

=

(∂g

∂y(h(z), z)

︸ ︷︷ ︸∈ Rr×r

,∂g

∂z(h(z), z)

︸ ︷︷ ︸∈ Rr×(n−r)

)(Dh(z)

I

)

=∂g

∂y(h(z), z)Dh(z) +

∂g

∂z(h(z), z) = O ∈ Rr×(n−r), ∀ z ∈ W,

O

∂g

∂y(y, z) :=

⎛

⎜⎜⎝

∂g1∂x1

(x) . . . ∂g1∂xr

(x)

∂gr∂x1

(x) . . . ∂gr∂xr

(x)

⎞

⎟⎟⎠ ,∂g

∂z(y, z) :=

⎛

⎜⎜⎝

∂g1∂xr+1

(x) . . . ∂g1∂xn

(x)

∂gr∂xr+1

(x) . . . ∂gr∂xn

(x)

⎞

⎟⎟⎠ ,

z0 ∈ W h(z0) = y0

∂g

∂y(y0, z0)Dh(z0) +

∂g

∂z(y0, z0) = O ∈ Rr×(n−r).

−∂f

∂y(y0, z0)

︸ ︷︷ ︸∈ R1×r

(∂g

∂y(y0, z0)

)−1

︸ ︷︷ ︸∈ Rr×r

∂g

∂z(y0, z0)

︸ ︷︷ ︸∈ Rr×(n−r)

+∂f

∂z(y0, z0)

︸ ︷︷ ︸∈ R1×(n−r)

= 0.

λ := (λ1, . . . , λr) :=∂f

∂y(y0, z0)

(∂g

∂y(y0, z0)

)−1

∈ R1×r,

λ∂g

∂y(y0, z0) =

∂f

∂y(y0, z0)

λ∂g

∂z(y0, z0) =

∂f

∂z(y0, z0).

λDg(x0) = λ

(∂g

∂y(y0, z0),

∂g

∂z(y0, z0)

)

=

(∂f

∂y(y0, z0),

∂f

∂z(y0, z0)

)= grad f(x0),

✷

(∂f

∂x1(x0), . . . ,

∂f

∂xn(x0)

)= (λ1, . . . , λr)

⎛

⎜⎜⎝

∂g1∂x1

(x0) . . . ∂g1∂xn

(x0)

∂gr∂x1

(x0) . . . ∂gr∂xn

(x0)

⎞

⎟⎟⎠

grad f(x0) =r∑

j=1

λj gradgj(x0)

∂f

∂xi(x0) =

r∑

j=1

λj∂gj∂xi

(x0), i = 1, . . . ,n,

n r λj j = 1, . . . , r

f|M M = {x ∈ U : g(x) = 0}f g(x) = 0

(x, λ) = (x1, . . . , xn, λ1, . . . , λr) ∈ U× Rr

grad F(x, λ) = 0 ∈ Rn+r, F(x, λ) := λ · g(x)− f(x),

r∑

j=1

λj∂gj∂xi

(x)−∂f

∂xi(x) = 0, i = 1, . . . ,n,

gj(x) = 0, j = 1, . . . , r.

x (x, λ) Dg(x)r f|M

x ∈ M Dg(x)< r f|M

f|M

g(x) = 0r xi n− r

f|Mf g(x) = 0

f|M

∅ = A,B ⊂ Rn A B

∃ ξ ∈ A, η ∈ B : ∥ξ− η∥ ≤ ∥x− y∥ ∀ x ∈ A, y ∈ B.

d := inf{∥x− y∥ : x ∈ A, y ∈ B} ≥ 0,

(xν) ⊂ A (yν) ⊂ B ∥xν − yν∥ → d A(xkν) ⊂ (xν) xkν → ξ ∈ A

∥xkν∥ → ∥ξ∥ (ykν) ⊂ (yν)

∥ykν∥ ≤ ∥xkν − ykν∥+ ∥xkν∥

(yℓkν ) ⊂ (ykν)

yℓkν → η ∈ B xℓkν − yℓkν → ξ− η

∥xℓkν − yℓkν ∥ → ∥ξ− η∥ = d = min{∥x− y∥ : x ∈ A, y ∈ B},

✷

f : R2 → Rf(x,y) = xy

(0, 0) R2 S1 = {(x,y) ∈ R2 : x2 + y2 = 1}

f(x,y) = xy(x,y) ∈ R2 g(x,y) = x2 + y2 − 1 = 0 (x,y) ∈ R2

f|S1 S1 = {(x,y) ∈ R2 : x2 + y2 = 1} = {(x,y) ∈ R2 :g(x,y) = 0} = M U = R2 f,g : R2 → R

(x,y, λ) ∈ R3

0 = grad F(x,y, λ)

= grad (λg(x,y)− f(x,y))

= grad (λx2 + λy2 − λ− xy)

= (2λx− y, 2λy− x, x2 + y2 − 1)

⇔ y = 2λx, x = 2λy, 4λ2(y2 + x2) = 4λ2 = 1 ⇒ y = ±x, 2x2 = 1

⇒ (x,y) ∈{

1√2(1, 1),

1√2(1,−1),

1√2(−1,−1),

1√2(−1, 1)

}.

Dg(x,y) = grad (x2 + y2 − 1) = (2x, 2y)1 (x,y) ∈ R2 \ {(0, 0)} ⊃ S1 = {(x,y) ∈ R2 : x2 + y2 =

1} = M Dg(x,y) = (0, 0)f|S1

S1 f|S1

S1

f

(1√2(1, 1)

)=

1

2= f

(1√2(−1,−1)

),

f

(1√2(1,−1)

)= −

1

2= f

(1√2(−1, 1)

).

f S11√2(1, 1) 1√

2(−1,−1) 1

21√2(1,−1) 1√

2(−1, 1)

−12

f : R2 → Rf(x,y) = xy2 x2 + y2 = 1

f(x,y) = xy2 g(x,y) = x2 + y2 − 1R2 gradg(x,y) = 2(x,y) = (0, 0)

∀ (x,y) ∈ R2 g(x,y) = 0(x,y)

(x,y, λ) ∈ R3

grad F(x,y, λ) = grad (λx2 + λy2 − λ− xy2)

= (2λx− y2, 2λy− 2xy, x2 + y2 − 1) = 0.

y = 0 x = ±1 λ = 0 y = 0 x = λ y2 = 2x2 x = ± 1√3

y = ±√2|x| f|S1

(x,y) ∈{(±1, 0),

(1√3,±√

2

3

),

(−

1√3,±√

2

3

)}

f(±1, 0) = 0, f

(1√3,±√

2

3

)=

2

3√3, f

(−

1√3,±√

2

3

)= −

2

3√3.

S1 f|S1

(1, 0) f f|S1 (x,y) ∈ (0,∞)× Rf(x,y) ≥ 0 = f(1, 0) (−1, 0) f f|S1

(x,y) ∈ (−∞, 0)× R f(x,y) ≤ 0 = f(−1, 0) f(±1, 0) f|S1

y2 = 1− x2

f|M(x,y) = x(1− x2) =: h(x), x ∈ [−1, 1]

h(

1√3

)= f|M

(1√3,±√

23

)= 2

3√3, h

(− 1√

3

)= f|M

(− 1√

3,±√

23

)= − 2

3√3

h (−1) = f|M (−1, 0) = 0, h (1) = f|M (1, 0) = 0.

z = x+ y R3

(1, 0, 0) ∈ R3

f(x,y, z) = ∥(x,y, z)− (1, 0, 0)∥ =√

(x− 1)2 + y2 + z2, (x,y, z) ∈ R3

g(x,y, z) = z− x− y = 0, (x,y, z) ∈ R3.

ϕ := f2

g(x,y, z) = 0 R3 ϕ,g ϕ,gDg(x,y, z) =

gradg(x,y, z) = (−1,−1, 1) = (0, 0, 0) 1 ∀ (x,y, z) ∈ R3

ϕ g(x,y, z) = 0 (x,y, z)(x,y, z, λ) ∈ R4

grad F(x,y, z, λ) = grad (λz− λx− λy− (x− 1)2 − y2 − z2)

= (−λ− 2(x− 1),−λ− 2y, λ− 2z, z− x− y) = (0, 0, 0, 0)

⇔ x = 1−λ

2, y = −

λ

2, z =

λ

2,λ

2=

1

3⇔ (x,y, z, λ) =

1

3(2,−1, 1, 2)

f

(1

3(2,−1, 1)

)=

1√3.

(x,y, z) = 13 (2,−1, 1) f

z = x+yA = {(1, 0, 0)} B = {(x,y, z) ∈ R3 : z =

x+ y}

φ|M(x,y, z) = ϕ(x,y, x+ y) = (x− 1)2 + y2 + (x+ y)2 =: h(x,y), (x,y) ∈ R2.

∇h(x,y) = (2(x− 1) + 2(x+ y), 2y+ 2(x+ y)) = 2(2x+ y− 1, 2y+ x) = (0, 0)

(x,y) =(23 ,−

13

)

Hh(x,y) = 2

(2 11 2

),

(x,y) =(23 ,−

13

)

h (x,y, z) =(23 ,−

13 ,

13

)ϕ|M

z = αx+βy α,β ∈ R R3

M := {(x,y, z) ∈ R3 : g(x,y, z) := z−αx−βy = 0}

((xν,yν, zν)) ⊂ M (xν,yν, zν) →(x,y, z) ∈ R3 (x,y, z) ∈ M

g : R3 → R

0 = g(xν,yν, zν) → g(x,y, z) = 0, (x,y, z) ∈ M

f(x,y, z) = 5x + y − 3zx+ y+ z = 0

x2 + y2 + z2 = 1 R3

f

g(x,y, z) =

(x+ y+ z

x2 + y2 + z2 − 1

)= 0

Dg(x,y, z) =

(1 1 12x 2y 2z

)

2

(x,y, z) ∈ R3 \ {(x,y, z) = c(1, 1, 1) : c ∈ R}

⊃ {(x,y, z) ∈ R3 : g(x,y, z) = 0} = M.

f g R3

f|M (x,y, z) (x,y, z, λ1, λ2) ∈R5

grad F(x,y, z, λ1, λ2)

= grad (λ1x+ λ1y+ λ1z+ λ2x2 + λ2y

2 + λ2z2 − λ2 − 5x− y+ 3z)

= (λ1 + 2λ2x− 5, λ1 + 2λ2y− 1, λ1 + 2λ2z+ 3, x+ y+ z, x2 + y2 + z2 − 1)

= (0, 0, 0, 0, 0)

⇔ λ1 = 1, x =2

λ2, y = 0, z = −

2

λ2, λ2 = ±2

√2

⇒ (x,y, z) =

(± 1√

2, 0,∓ 1√

2

)

f

(± 1√

2, 0,∓ 1√

2

)= ±4

√2.

f f|M

f|M

f

(1√2, 0,−

1√2

)= 4

√2

f

(−

1√2, 0,

1√2

)= −4

√2

x + y + z = 0x2 + y2 + z2 = 1 R3

M = {(x,y, z) ∈ R3 : g(x,y, z) :=

(x+ y+ z

x2 + y2 + z2 − 1

)= 0}

M

(x,y, z) ∈ M ⇒ x2 + y2 + z2 = ∥(x,y, z)∥2 = 1.

M((xν,yν, zν)) ⊂ M (xν,yν, zν) → (x,y, z) ∈ R3

(x,y, z) ∈ M g : R3 → R2

0 = g(xν,yν, zν) → g(x,y, z) = 0

(x,y, z) ∈ M

f(x,y) = x2 + y2 (x,y) ∈ R2

S = {(x, 2) ∈ R2 : x ∈ R}

f|S(x,y) = f(x, 2) = x2+4 =: h(x) x ∈ R

hx = 0 h(0) = 4 f

S (x, 2) = (0, 2)f|S f|S(0, 2) = 4

S (x,y) ∈ R2

y = 2S = {(x,y) ∈ R2 : g(x,y) := y− 2 = 0}.

f|Sf g(x,y) = y− 2 = 0

f|S

f,gg r = 1 (x,y) ∈ R2

Dg(x,y) = gradg(x,y) = ∇g(x,y) =

(∂

∂xg(x,y),

∂

∂yg(x,y)

)= (0, 1) = (0, 0)

∀ (x,y) ∈ R2 ⊃ S,

∇(λg(x,y)− f(x,y)) = ∇(λ(y− 2)− x2 − y2) = (−2x, λ− 2y,y− 2) = (0, 0, 0)

(x,y, λ) = (0, 2, 4) f|S(x,y) = (0, 2)

f|S(0, 2) = f(0, 2) = 4 < x2 + 4 = f(x, 2) ∀ x = 0 ⇔ (x, 2) ∈ S \ {(0, 2)}.

(0, 2)f|S f|S

(x,y) λ ∈ R(x,y, λ) f|S

Rn nA ⊂ Rn

A = [α1,β1]× [α2,β2]× · · ·× [αn,βn]

= {x = (x1, x2, . . . , xn) : αi ≤ xi ≤ βi ∀ i = 1, . . . ,n},

αi,βi ∈ R αi < βi ∀ i = 1, . . . ,n

A ⊂ Rn

v(A) := (β1 −α1)(β2 −α2) · · · (βn −αn) =n∏

i=1

(βi −αi) > 0.

A ⊂ Rn

Rn

Rn B ⊂Rn

B = (α1,β1)× (α2,β2)× · · ·× (αn,βn),

αi,βi ∈ R αi < βi ∀ i = 1, . . . ,n

v(B) :=n∏

i=1

(βi −αi).

B Rn

A

B = A, A = B, ∂A = ∂B = A \B, v(A) = v(B).

n R

A ⊂ Rn B ⊂ Rm v(A× B) = v(A) ·v(B).

A = [α1,β1]× · · ·× [αn,βn] ⊂ Rn

P A

P = P1 × · · ·× Pn ={t(1)0 , . . . , t

(1)k1

}× · · ·×

{t(n)0 , . . . , t

(n)kn

}⊂ A,

i = 1, . . . ,n Pi ={t(i)0 , . . . , t

(i)ki

}ki ∈ N

[αi,βi]

αi = t(i)0 < t

(i)1 < . . . < t

(i)ki

= βi ∀ i = 1, . . . ,n.

P

∥P∥ := maxi=1,...,n

{∥Pi∥}, ∥Pi∥ := maxκ=1,...,ki

{t(i)κ − t

(i)κ−1

}.

A

P(A) := {P ⊂ A : P A}.

P A A k1 · k2 · . . . · kn ∈N Rn

S =[t(1)j1−1, t

(1)j1

]×[t(2)j2−1, t

(2)j2

]× . . .×

[t(i)ji−1, t

(i)ji

]× . . .×

[t(n)jn−1, t

(n)jn

],

ji = 1, . . . , ki i = 1, . . . ,n P

P SP

P′ ∈ P(A) P ∈ P(A) P′ ⊃ P

x1

x2

x1

x2

x1

x2

P ′′ ∈ P(A) P,P ′ ∈ P(A) P ′′ ⊃ P,P ′

AP = {α1,β1}× · · ·× {αn,βn}

A SP = {A}

S ∈ SP P A

S ⊂ A,⋃

S∈SP

S = A,∑

S∈SP

v(S) = v(A), S∩ S ′ = ∅, S = S ′.

A ⊂ Rn B ⊂ Rm P(A×B) = {PA × PB : PA ∈ P(A), PB ∈ P(B)}SP = {SA × SB : SA ∈ SPA

, SB ∈ SPB}

P ′ = P ′1 × · · ·× P ′

n ⊃ P = P1 × · · ·× Pn ⇔ P ′i ⊃ Pi ∀ i = 1, . . . ,n

P = P1 × · · · × Pn P ′ = P ′1 × · · · × P ′

n AP ′′ = (P1 ∪ P ′

1)× · · ·× (Pn ∪ P ′n)

A ⊂ Rn f : A → RP A

L(f,P) :=∑

S∈SP

inf f|S · v(S), inf f|S = inf {f(x) : x ∈ S},

U(f,P) :=∑

S∈SP

sup f|S · v(S), sup f|S = sup {f(x) : x ∈ S},

f P

z

y

x

inf f|S · v(S) SS 0xy L(f,P)

z

y

x

sup f|S · v(S) SS 0xy U(f,P)

f : A → R PA ⊂ Rn

k1 · . . . · kn ∈ NS P v(S)

S

∀ S ∈ SP : 0 ≤ v(S) ≤∑

S∈SP

v(S) = v(A) < ∞,

S ⊂ A f : A → R S ∈ SP

inf {f(x) : x ∈ A}︸ ︷︷ ︸= inf f > −∞

≤ inf {f(x) : x ∈ S}︸ ︷︷ ︸= inf f|S

≤ sup {f(x) : x ∈ S}︸ ︷︷ ︸= sup f|S

≤ sup {f(x) : x ∈ A}︸ ︷︷ ︸= sup f < ∞

.

A ⊂ Rn f : A → RL(f,P) U(f,P) f

P

∀ P ∈ P(A) −∞ < inf f · v(A) ≤ L(f,P) ≤ U(f,P) ≤ sup f · v(A) < ∞

∀ P,P′ ∈ P(A) P′ ⊃ P L(f,P) ≤ L(f,P′) U(f,P′) ≤ U(f,P)

P′ P P′

P P′

P

∀ P,P′ ∈ P(A) L(f,P′) ≤ U(f,P)

v(A) =∑

S∈SP

v(S) inf f ≤ inf f|S ≤ sup f|S ≤ sup f ∀ S ∈ SP

∑

S∈SP

inf f · v(S)

︸ ︷︷ ︸= inf f · v(A)

≤∑

S∈SP

inf f|S · v(S)

︸ ︷︷ ︸= L(f,P)

≤∑

S∈SP

sup f|S · v(S)

︸ ︷︷ ︸= U(f,P)

≤∑

S∈SP

sup f · v(S)

︸ ︷︷ ︸= inf f · v(A)

.

S ∈ SP ℓS ∈ N T(S)i ∈ SP′ i = 1, . . . , ℓS

S =ℓS⋃

i=1

T(S)i , v(S) =

ℓS∑

i=1

v(T(S)i )

inf f|S ≤ inf f|T(S)i

≤ sup f|T(S)i

≤ sup f|S ∀ i = 1, . . . , ℓS.

inf f|S · v(S) =ℓS∑

i=1

inf f|S · v(T (S)i ) ≤

ℓS∑

i=1

inf f|T(S)i

· v(T (S)i )

≤ℓS∑

i=1

sup f|T(S)i

· v(T (S)i ) ≤

ℓS∑

i=1

sup f|S · v(T (S)i ) = sup f|S · v(S)

SP′ = {T(S)i ∈ SP′ : i = 1, . . . , ℓS, S ∈ SP},

∑

S∈SP

inf f|S · v(S)

︸ ︷︷ ︸= L(f,P)

≤∑

S∈SP

ℓS∑

i=1

inf f|T(S)i

· v(T (S)i ) =

∑

T∈SP′

inf f|T · v(T)

︸ ︷︷ ︸= L(f,P′)

≤∑

T∈SP′

sup f|T · v(T)

︸ ︷︷ ︸= U(f,P′)

=∑

S∈SP

ℓS∑

i=1

sup f|T(S)i

· v(T (S)i ) ≤

∑

S∈SP

sup f|S · v(S)

︸ ︷︷ ︸= U(f,P)

.

P′′ ∈ P(A) P,P′

L(f,P′) ≤ L(f,P′′) ≤ U(f,P′′) ≤ U(f,P).

✷

{L(f,P) : P ∈ P(A)} {U(f,P) : P ∈ P(A)}

R

A ⊂ Rn f : A → RL(f,P) U(f,P) f

P

Lf := sup {L(f,P) : P ∈ P(A)} Uf := inf {U(f,P) : P ∈ P(A)}

f

A ⊂ Rn f : A → RLf Uf f

Lf ≤ Uf.

P ∈ P(A) L(f,P′) ≤ U(f,P)∀ P′ ∈ P(A) Lf ≤ U(f,P) P ∈ P(A)

Lf ≤ Uf ✷

A ⊂ Rn f : A → R

Lf = Uf.

∫

Af :=

∫

Af(x)dx =

∫

Af(x1, . . . , xn)d(x1, . . . , xn) := Lf = Uf

f

f : C → R C ⊂ Rn A ⊂ C fA f|A

∫

Af :=

∫

Af|A

f A

A ⊂ Rn f : A → Rf(x) = c ∀ x ∈ A

inf f|S = sup f|S = c ∀ S ∈ SP, ∀ P ∈ P(A)

⇒ L(f,P) = U(f,P) = c∑

S∈SP

v(S) = c · v(A) ∀ P ∈ P(A)

⇒ Lf = Uf = c · v(A)

⇒∫

Ac :=

∫

Ac dx =

∫

Af = c · v(A).

c = 1 ∫

A1 = v(A)

c = 0 ∫

A0 = 0.

f : [0, 1]× [0, 1] → R

f(x,y) =

{0, x ∈ Q

1, x ∈ R \ Q.

P ∈ P(A) S = [α,β]× [γ, δ] ∈ SP v(S) > 0x1 ∈ [α,β]∩ Q x2 ∈ [α,β]∩ (R \ Q)

inf f|S = 0, sup f|S = 1 ∀ S ∈ SP v(S) > 0, ∀ P ∈ P(A)

⇒ L(f,P) = 0, U(f,P) =∑

S∈SP

v(S) = v([0, 1]× [0, 1]) = 1 ∀ P ∈ P(A)

⇒ Lf = 0, Uf = 1,

f

A ⊂ Rn f : A → RL(f,P) U(f,P) f P

f ⇐⇒ ∀ ε > 0 ∃ P ∈ P(A) : U(f,P)− L(f,P) < ε.

⇒: ε > 0 Lf Uf

P′,P′′

U(f,P′) < Uf +ε

2L(f,P′′) > Lf −

ε

2.

P P′,P′′

U(f,P)− L(f,P) ≤ U(f,P′)− L(f,P′′) < Uf +ε

2−(Lf −

ε

2

)= ε.

⇐: Lf,Uf

0 ≤ Uf − Lf ≤ U(f,P′)− L(f,P′) ∀ P′ ∈ P(A).

∀ ε > 0 : 0 ≤ Uf − Lf < ε

Uf = Lf ✷

A ⊂ Rn f : A → RL(f,P) U(f,P) f P

∥P∥ P

f

⇐⇒ ∀ ε > 0 ∃ δ > 0 ∀ P ∈ P(A) ∥P∥ < δ : U(f,P)− L(f,P) < ε.

⇐f

A = [α1,β1]× · · ·× [αn,βn] ⊂ Rn f :A → R |f(x)| ≤ M ∀ x ∈ A P = P1 × · · ·× Pn

P ′ = P ′1 × · · ·× P ′

n A P ′j rj

Pj j = 1, . . . ,n

L(f,P ′) ≤ L(f,P) +C, U(f,P)−C ≤ U(f,P ′), C := 2M∥P∥v(A)n∑

j=1

rjβj −αj

.

ri = 1 i = 1, . . . ,n rj = 0j = 1, . . . ,n j = i

L(f,P ′) ≤ L(f,P) +2M∥P∥v(A)

βi −αiU(f,P)−

2M∥P∥v(A)

βi −αi≤ U(f,P ′),

rj j = 1, . . . ,nP ′ ⊂ P ⇒ ∥P ′∥ ≤ ∥P∥

s ∈(t(i)κ−1, t

(i)κ

)

Pi ={αi = t

(i)0 < t

(i)1 . . . < t

(i)κ−1 < t

(i)κ < . . . < t

(i)ki

= βi}.

S ∈ SP

S =[t(1)κ1−1, t

(1)κ1

]× · · ·×

[t(i)κ−1, t

(i)κ

]× · · ·×

[t(n)κn−1, t

(n)κn

],

κj = 1, . . . , kj j = 1, . . . ,n j = i t(j)0 = αj t

(j)kj

= βj

Sℓ =[t(1)κ1−1, t

(1)κ1

]× · · ·×

[t(i)κ−1, s

]× · · ·×

[t(n)κn−1, t

(n)κn

]

Sr =[t(1)κ1−1, t

(1)κ1

]× · · ·×

[s, t

(i)κ

]× · · ·×

[t(n)κn−1, t

(n)κn

],

P L(f,P ′)U(f,P ′) inf f|Sv(S) sup f|Sv(S)

L(f,P) U(f,P)

inf f|Sℓv(Sℓ) + inf f|Srv(Sr)

= inf f|Sv(S) +(inf f|Sℓ − inf f|S

)v(Sℓ) + (inf f|Sr − inf f|S) v(Sr)

≤ inf f|Sv(S) + 2Mv(S)

sup f|Sℓv(Sℓ) + sup f|Srv(Sr)

= sup f|Sv(S)−(sup f|S − sup f|Sℓ

)v(Sℓ)− (sup f|S − sup f|Sr) v(Sr),

≥ sup f|Sv(S)− 2Mv(S)

S

L(f,P ′) ≤ L(f,P) + 2Mv(S ′) U(f,P ′) ≥ U(f,P)− 2Mv(S ′),

S ′ = [α1,β1]× · · ·×[t(i)κ−1, t

(i)κ

]× · · ·× [αn,βn],

v(S ′) =(t(i)κ − t

(i)κ−1

) n∏

j=1j =i

(βj −αj) ≤∥P∥

βi −αiv(A),

✷

A = [α1,β1]× · · ·× [αn,βn] ⊂ Rn

f : A → R |f(x)| ≤ M ∀ x ∈ A

∀ ε > 0 ∃ δ > 0 ∀ P ∈ P(A) ∥P∥ < δ : Lf − ε < L(f,P), U(f,P) < Uf + ε.

ε > 0 P ′ = P ′1 × · · ·× P ′

n

L(f,P ′) > Lf −ε

2,

P ′j r ′j = αj,βj j = 1, . . . ,n

P

L(f,P ′ ∪ P) ≥ L(f,P ′)

L(f,P ′ ∪ P) ≤ L(f,P) + ∥P∥c ′ c ′ := 2Mv(A)n∑

j=1

r ′jβj −αj

.

Lf −ε

2< L(f,P) + ∥P∥c ′.

P ′′ = P ′′1 × · · ·×P ′′

n P ′′j

r ′′j = αj,βj j = 1, . . . ,n P

U(f,P)− ∥P∥c ′′ < Uf +ε

2c ′′ := 2Mv(A)

n∑

j=1

r ′′jβj −αj

.

c ′ + c ′′ = 0 Pc ′ + c ′′ > 0 P

∥P∥ <ε

2(c ′ + c ′′)=: δ.

✷

f Lf = Uf

✷

Rn

A ⊂ Rn f : A → Rf

∫A f

f ε > 0 δ > 0P ∈ P(A) A ∥P∥ < δ

ξP := (ξS)S∈SPξS ∈ S

∣∣∣∣∣∣

∑

S∈SP

f(ξS) · v(S)−∫

Af

∣∣∣∣∣∣< ε.

S(f,P, ξP) =∑

S∈SP f(ξS) · v(S) f P

ξP

A ⊂ Rn f : A → R

f

⇐⇒ f

f

⇒∫A f ε > 0

δ > 0P ∈ P(A) ∥P∥ < δ ξP

∫

Af− ε ≤ U(f,P)− ε < L(f,P)

≤∑

S∈SP

f(ξS) · v(S) ≤ U(f,P) < L(f,P) + ε ≤∫

Af+ ε.

⇐∫A f ε > 0

δ > 0 P ∈ P(A) ∥P∥ < δξP

−ε <∑

S∈SP

f(ξS) · v(S)−∫

Af < ε.

−ε ≤ L(f,P)−

∫

Af ≤ Lf −

∫

Af ≤ Uf −

∫

Af ≤ U(f,P)−

∫

Af ≤ ε.

ε > 0 Lf =∫A f = Uf ✷

A ⊂ Rn f,g : A → R α ∈ R

f+ g∫A(f+ g) =

∫A f+

∫A g

αf∫A(αf) = α

∫A f

f ≤ g =⇒∫A f ≤

∫A g

|f|∣∣∫

A f∣∣ ≤

∫A |f|

fg

f : A → R A ⊂ Rn

L(f,P) =∑

S∈SP

inf f|S · v(S) ≤ Lf ≤ Uf ≤ U(f,P) =∑

S∈SP

sup f|S · v(S)

∀ P ∈ P(A).

f,g : A → R

inf f|S + inf g|S ≤ f(x) + g(x) ≤ sup f|S + supg|S∀ x ∈ S, S ∈ SP, P ∈ P(A),

inf f|S + inf g|S ≤ inf(f+ g)|S ≤ sup(f+ g)|S ≤ sup f|S + supg|S∀ S ∈ SP, P ∈ P(A).

P = {α1,β1} × . . . × {αn,βn}A = [α1,β1] × . . . × [αn,βn] SP = {A} f + g

L(f,P) + L(g,P) ≤ L(f+ g,P) ≤ Lf+g

≤ Uf+g ≤ U(f+ g,P) ≤ U(f,P) +U(g,P) ∀ P ∈ P(A).

f,g : A → R

∀ ε > 0 ∃ P′,P′′ ∈ P(A) : U(f,P′)− L(f,P′) <ε

2, U(g,P′′)− L(g,P′′) <

ε

2,

P ∈ P(A) P′,P′′

∀ ε > 0 ∃ P ∈ P(A) : U(f,P)− L(f,P) <ε

2, U(g,P)− L(g,P) <

ε

2.

Lf = Uf Lg = Ug

∀ ε > 0 ∃ P ∈ P(A) : −ε < L(f,P) + L(g,P)−U(f,P)−U(g,P)

≤ Lf+g −Uf −Ug

≤ Uf+g − Lf − Lg

≤ U(f,P) +U(g,P)− L(f,P)− L(g,P)

< ε,

Lf+g = Uf +Ug = Lf + Lg = Uf+g

α = 0

α > 0

inf f|S ≤ f(x) ≤ sup f|S ∀ x ∈ S, S ∈ SP, P ∈ P(A)

⇒ α inf f|S ≤ αf(x) ≤ α sup f|S ∀ x ∈ S, S ∈ SP, P ∈ P(A)

⇒ α inf f|S ≤ inf(αf)|S ≤ sup(αf)|S ≤ α sup f|S ∀ S ∈ SP, P ∈ P(A),

S = A αf

αL(f,P) ≤ L(αf,P) ≤ Lαf ≤ Uαf ≤ U(αf,P) ≤ αU(f,P) ∀ P ∈ P(A),

αLf ≤ Lαf ≤ Uαf ≤ αUf,

Lf = Uf

α < 0

inf f|S ≤ f(x) ≤ sup f|S ∀ x ∈ S, S ∈ SP, P ∈ P(A)

⇒ α sup f|S ≤ αf(x) ≤ α inf f|S ∀ x ∈ S, S ∈ SP, P ∈ P(A)

⇒ α sup f|S ≤ inf(αf)|S ≤ sup(αf)|S ≤ α inf f|S ∀ S ∈ SP, P ∈ P(A),

S = A αf

αU(f,P) ≤ L(αf,P) ≤ Lαf ≤ Uαf ≤ U(αf,P) ≤ αL(f,P) ∀ P ∈ P(A),

αUf ≤ Lαf ≤ Uαf ≤ αLf,

Lf = Uf

inf f|S ≤ f(x) ≤ g(x) ∀ x ∈ S, S ∈ SP, P ∈ P(A)

⇒ inf f|S ≤ inf g|S ∀ S ∈ SP, P ∈ P(A)

⇒ L(f,P) ≤ L(g,P) ≤ Lg ∀ P ∈ P(A)

⇒ Lf ≤ Lg.

B ⊂ R

sup {|x− y| : x,y ∈ B} = supB− inf B.

inf B ≤ x ≤ supB ∀ x ∈ B − supB ≤ −y ≤ − inf B ∀ y ∈ B

⇒ − (supB− inf B) ≤ x− y ≤ supB− inf B ∀ x,y ∈ B

⇒ |x− y| ≤ supB− inf B ∀ x,y ∈ B

⇒ sup {|x− y| : x,y ∈ B} ≤ supB− inf B

∀ ε > 0 ∃ x,y ∈ B : x > supB−ε

2, y < inf B+

ε

2⇒ ∀ ε > 0 ∃ x,y ∈ B : |x− y| ≥ x− y > supB− inf B− ε

⇒ ∀ ε > 0 : sup {|x− y| : x,y ∈ B} > supB− inf B− ε

⇒ sup {|x− y| : x,y ∈ B} ≥ supB− inf B.

f |f|

∣∣|f(x)|− |f(y)|∣∣ ≤ |f(x)− f(y)| ≤ sup f|S − inf f|S

∀ x, y ∈ S, S ∈ SP, P ∈ P(A),

sup |f|∣∣S − inf |f|

∣∣S ≤ sup f|S − inf f|S ∀ S ∈ SP, P ∈ P(A),

U(|f|,P)− L(|f|,P) ≤ U(f,P)− L(f,P) ∀ P ∈ P(A),

|f|

±f ≤ |f|

±∫

Af =

∫

A(±f) ≤

∫

A|f|

∣∣∣∣∫

Af

∣∣∣∣ ≤∫

A|f|.

f,g fg

|(fg)(x)− (fg)(y)| = |f(x)(g(x)− g(y)) + g(y)(f(x)− f(y))|

≤ |f(x)| |g(x)− g(y)|+ |g(y)| |f(x)− f(y)|

≤ sup |f| · |g(x)− g(y)|+ sup |g| · |f(x)− f(y)| ∀ x, y ∈ A,

|g(x)− g(y)| ≤ supg|S − inf g|S, |f(x)− f(y)| ≤ sup f|S − inf f|S∀ x, y ∈ S, S ∈ SP, P ∈ P(A),

|(fg)(x)− (fg)(y)| ≤ sup |f| (supg|S − inf g|S) + sup |g| (sup f|S − inf f|S)

∀ x, y ∈ S, S ∈ SP, P ∈ P(A),

sup(fg)|S − inf(fg)|S ≤ sup |f| (supg|S − inf g|S)

+ sup |g| (sup f|S − inf f|S) ∀ S ∈ SP, P ∈ P(A),

U(fg,P)− L(fg,P) ≤ sup |f| (U(g,P)− L(g,P))

+ sup |g| (U(f,P)− L(f,P)) ∀ P ∈ P(A).

f g fgsup |f|, sup |g| > 0

∀ ε > 0 ∃ P′,P′′ ∈ P(A) :

U(g,P′)− L(g,P′) <ε

2 sup |f|, U(f,P′′)− L(f,P′′) <

ε

2 sup |g|,

P ∈ P(A) P′,P′′

∀ ε > 0 ∃ P ∈ P(A) : U(fg,P)− L(fg,P) < ε,

sup |f| sup |g| = 0 fg = 0

✷

A ⊂ Rn f : A → R

inf f · v(A) ≤∫

Af ≤ sup f · v(A).

A ⊂ Rn P ∈ P(A) f : A → R

f ⇐⇒ f|S ∀ S ∈ SP

∫

Af =

∑

S∈SP

∫

Sf|S.

f ⇐⇒ f|S ∀ S ∈ SP

P = P1 × · · ·× Pn A R = R1 × · · ·× Rn ∈P(A) Q = (R1 ∪ P1)× · · ·× (Rn ∪ Pn) ∈ P(A)

R Q∩ S ∈ P(S) ∀ S ∈ SP

L(f,R) ≤ L(f,Q) =∑

S∈SP

L(f|S,Q∩ S) ≤∑

S∈SP

Lf|S ,

U(f,R) ≥ U(f,Q) =∑

S∈SP

U(f|S,Q∩ S) ≥∑

S∈SP

Uf|S ,

Lf ≤∑

S∈SP

Lf|S Uf ≥∑

S∈SP

Uf|S ,

Uf − Lf ≥∑

S∈SP

(Uf|S − Lf|S

)≥ 0

S ∈ SP P(S) = P(S)1 × · · ·×

P(S)n ∈ P(S) R =

(⋃S∈SP

P(S)1

)× · · · ×

(⋃S∈SP

P(S)n

)∈ P(A)

R∩ S ∈ P(S) P(S)

Lf ≥ L(f,R) =∑

S∈SP

L(f|S,R∩ S) ≥∑

S∈SP

L(f|S,P(S)),

Uf ≤ U(f,R) =∑

S∈SP

U(f|S,R∩ S) ≤∑

S∈SP

U(f|S,P(S)),

Lf ≥∑

S∈SP

Lf|S Uf ≤∑

S∈SP

Uf|S ,

0 ≤ Uf − Lf ≤∑

S∈SP

(Uf|S − Lf|S

).

✷

nA Rn [α,β]

A x ∈ [α,β]n− 1

n− 1 x ∈ [α,β]n = 2

A = [α,β]× [γ, δ] f : A → R[α,β] [ti−1, ti] i = 1, . . . , k

t0 = α tk = β ti−1 < ti xi

{xi}× [γ, δ]f(xi, ·) : [γ, δ] → R

∫δ

γf(xi,y)dy,

ti− ti−1 [ti−1, ti]× [γ, δ]f

f

∫

[ti−1,ti]×[γ,δ]f(x,y)d(x,y) ≈ (ti − ti−1)

∫δ

γf(xi,y)dy,

f A

∫

Af(x,y)d(x,y) =

k∑

i=1

∫

[ti−1,ti]×[γ,δ]f(xi,y)d(x,y)

≈k∑

i=1

(ti − ti−1)

∫δ

γf(xi,y)dy,

≈∫β

α

(∫δ

γf(x,y)dy

)dx,

z

y

xγ = 0

α ti−1xi ti β

δ

{(xi,y, z) : γ ≤ y ≤ δ, 0 ≤ z ≤ f(xi,y)}

xi ∈ [ti−1, ti] ⊂ [α,β]∫δγ f(xi,y)dy

f

A ⊂ Rn B ⊂ Rm f : A× B → R

A ∋ x 4→ Lf(x,·) ∈ R, A ∋ x 4→ Uf(x,·) ∈ R,

Lf(x,·) Uf(x,·)

f(x, ·) : B ∋ y 4→ f(x, y) ∈ R, x ∈ A,

∫

A×Bf =

∫

ALf(x,·) dx =

∫

AUf(x,·) dx.

f : A× B → Rf(x, ·) : B → R x ∈ A

Lf(x,·) Uf(x,·) Rx ∈ A

L : A ∋ x 4→ Lf(x,·) ∈ R, U : A ∋ x 4→ Uf(x,·) ∈ R

∀ x ∈ A : inf f · v(B) ≤ inf f(x, ·) · v(B) ≤ Lf(x,·) = L(x)≤ U(x) = Uf(x,·) ≤ sup f(x, ·) · v(B) ≤ sup f · v(B).

L, U : A → R

∫

AL =

∫

A×Bf =

∫

AU .

L U

P ∈ P(A× B) P = PA × PB PA ∈ P(A) PB ∈ P(B)

SP = {S = SA × SB : SA ∈ SPA, SB ∈ SPB

}.

L(f,P) =∑

S∈SP

inf f|S · v(S) =∑

SA∈SPA

∑

SB∈SPB

inf f|SA×SB· v(SB) · v(SA).

inf f|SA×SB≤ f(x, y) ∀ (x, y) ∈ SA × SB

⇒ inf f|SA×SB≤ inf f(x, ·)|SB

∀ x ∈ SA,

∑

SB∈SPB

inf f|SA×SB· v(SB) ≤

∑

SB∈SPB

inf f(x, ·)|SB· v(SB) = L(f(x, ·),PB)

≤ Lf(x,·) = L(x) ∀ x ∈ SA,

∑

SB∈SPB

inf f|SA×SB· v(SB) ≤ inf L|SA

,

L(f,P) ≤∑

SA∈SPA

inf L|SA· v(SA) = L(L,PA) ≤ LL.

UL ≤ U(L,PA) =∑

SA∈SPA

supL|SA· v(SA),

∀ x ∈ SA : L(x) ≤ U(x) ≤ U(f(x, ·),PB) =∑

SB∈SPB

sup f(x, ·)|SB· v(SB)

≤∑

SB∈SPB

sup f|SA×SB· v(SB),

supL|SA≤

∑

SB∈SPB

sup f|SA×SB· v(SB),

UL ≤∑

SA∈SPA

∑

SB∈SPB

sup f|SA×SB· v(SB) · v(SA) =

∑

S∈SP

sup f|S · v(S) = U(f,P)

L(f,P) ≤ LL ≤ UL ≤ U(f,P).

P ∈ P(A× B)

Lf ≤ LL ≤ UL ≤ Uf,

Lf = Uf =∫A×B f L ✷

f : A× B → R A ⊂ Rn B ⊂Rm B ∋ y 4→ Lf(·,y), Uf(·,y)

∫

A×Bf =

∫

BLf(·,y) dy =

∫

BUf(·,y) dy.

A ⊂ Rn B ⊂ Rm f : A×B → Rf(x, ·) : B → R x ∈ A

A ∋ x 4→∫

Bf(x, y)dy ∈ R

∫

A×Bf(x, y)d(x, y) =

∫

A

(∫

Bf(x, y)dy

)dx.

f(x, ·) : B → R x ∈ A

Lf(x,·) = Uf(x,·) =∫

Bf(x, ·) =

∫

Bf(x, y)dy ∀ x ∈ A,

✷

f : A× B → R A ⊂ Rn B ⊂ Rm

f(·, y) : A → R y ∈ B B ∋ y 4→∫A f(x, y)dx

∫

A×Bf(x, y)d(x, y) =

∫

B

(∫

Af(x, y)dx

)dy.

f

A ⊂ Rn B ⊂ Rm f : A×B → Rf(x, ·) : B → R f(·, y) : A → R x ∈ A

y ∈ B A ∋ x 4→∫B f(x, y)dy B ∋ y 4→

∫A f(x, y)dx

∫

A×Bf(x, y)d(x, y) =

∫

A

(∫

Bf(x, y)dy

)dx =

∫

B

(∫

Af(x, y)dx

)dy.

Rn

Rn

f : A → R A = [α1,β1]× · · ·× [αn,βn] ⊂ Rn

∫

Af(x1, . . . , xn)d(x1, . . . , xn) =

∫β1

α1

(. . .

(∫βn

αn

f(x1, . . . , xn)dxn

). . .

)dx1,

∫

Af(x1, . . . , xn)d(x1, . . . , xn) =

∫β1

α1

∫β2

α2

. . .

∫βn

αn

f(x1, . . . , xn)dxn . . . dx2 dx1.

dxi∫βiαi

∫

Af(x1, . . . , xn)d(x1, . . . , xn) =

∫β1

α1

dx1

∫β2

α2

dx2 . . .

∫βn

αn

dxn f(x1, . . . , xn),

A ⊂ Rn

n µ(A) = 0

∀ ε > 0 ∃ (Ui)i∈N ⊂ Rn :∞⋃

i=1

Ui ⊃ A∞∑

i=1

v(Ui) < ε,

n v(A) = 0

∀ ε > 0 ∃ (Ui)ki=1 ⊂ Rn, k ∈ N :

k⋃

i=1

Ui ⊃ Ak∑

i=1

v(Ui) < ε.

A ⊂ Rn

ε > 0

< ε

S ⊂ Rn

S

Rn

Rn

A ⊂ Rn

A

∀ (xν) ⊂ A ∃ (xkν) ⊂ (xν) x ∈ A : xkν → x

∀ (Oi)i∈I Oi ⊂ Rn ⋃i∈IOi ⊃ A ∃ i1, . . . , ik ∈ I k ∈ N⋃k

κ=1Oiκ ⊃ A

U = [α1,β1]× · · ·× [αn,βn] v(U) >0

V =

(α1 −

( n√2− 1)(β1 −α1)

2,β1 +

( n√2− 1)(β1 −α1)

2

)× · · ·

· · ·×(αn −

( n√2− 1)(βn −αn)

2,βn +

( n√2− 1)(βn −αn)

2

)

v(V) =n∏

i=1

n√2 (βi −αi) = 2 v(U).

A ⊂ Rn µ(A) = 0 v(A) = 0 ε > 0Ui

⋃

i

Ui ⊃ A∑

i

v(Ui) <ε

2.

Ui Vi

⋃

i

Vi ⊃ A∑

i

v(Vi) < ε.

Vi

Ui = Vi ⊃ Vi v(Vi) = v(Vi) = v(Ui)

AB ⊂ A

Aκ ⊂ Rn κ = 1, . . . , k k ∈ N

ε > 0 κ = 1, . . . , k U(κ)i

i = 1, . . . , ℓκ ℓκ ∈ N

ℓκ⋃

i=1

U(κ)i ⊃ Aκ

ℓκ∑

i=1

v(U

(κ)i

)<ε

k∀ κ = 1, . . . , k.

k⋃

κ=1

ℓκ⋃

i=1

U(κ)i ⊃

k⋃

κ=1

Aκ

k∑

κ=1

ℓκ∑

i=1

v(U

(κ)i

)<

k∑

κ=1

ε

k= ε.

(Ak)k∈N Rn

ε > 0 k ∈ N(U

(k)i

)i∈N

∞⋃

i=1

U(k)i ⊃ Ak

∞∑

i=1

v(U

(k)i

)<ε

2k∀ k ∈ N.

∞⋃

k=1

∞⋃

i=1︸ ︷︷ ︸

=:∞⋃

k,i=1

U(k)i ⊃

∞⋃

k=1

Ak

∞∑

k=1

∞∑

i=1︸ ︷︷ ︸

=:∞∑

k,i=1

v(U

(k)i

)<

∞∑

k=1

ε

2k= ε,

⋃∞k=1Ak{

U(k)i : i, k ∈ N

}Q N × N

N

v(A) = 0 ε > 0 k ∈ N Ui

i = 1, . . . , kk⋃

i=1

Ui ⊃ Ak∑

i=1

v(Ui) <ε

2.

Ui :=

[−1

2

( ε

2i−k+1

) 1n,1

2

( ε

2i−k+1

) 1n

]n, i ∈ N, i ≥ k+ 1,

[α,β]n := [α,β]× · · ·× [α,β]︸ ︷︷ ︸n

∞⋃

i=1

Ui ⊃ A∞∑

i=1

v(Ui) <ε

2+

∞∑

i=k+1

ε

2i−k+1= ε.

(Ui)i∈N A∑∞i=1 v(Ui) < ε A

ε

A = [α1,β1]× · · ·× [αn,βn] ⊂ Rn v(A) :=∏n

i=1(βi − αi) > 0

Uκ =[α(κ)1 ,β

(κ)1

]× · · ·×

[α(κ)n ,β

(κ)n

], κ = 1, . . . , k, k ∈ N,

A ⊂ ⋃kκ=1Uκ i = 1, . . . ,n αi,βi

α(κ)i ,β

(κ)i (αi,βi)

Pi [αi,βi] S ∈ SP P = P1 × · · ·× Pn ∈ P(A)Uκ Uκ ∩ A = ∅ v(Uκ) ≥ v(Uκ ∩A) ≥ v(S)

Uκ Uκ ∩A S

k∑

κ=1

v(Uκ) ≥∑

S∈SP

v(S) = v(A) > 0,

AA 0

A ⊂ Rn ε > 0Ui i = 1, . . . ,k k ∈ N

Ui i δi > 0 Ui ⊂ B(0, δi)

A ⊂k⋃

i=1

Ui ⊂ B(0, δ) δ := max{δi : i = 1, . . . ,k}.

✷

∅ ⊂ Rn 0

{x} ⊂ Rn x = (x1, . . . , xn) ∈ Rn 0ε > 0

{x} ⊂ U :=

[x1, x1 +

(ε2

) 1n

]× . . .×

[xn, xn +

(ε2

) 1n

]v(U) =

ε

2< ε.

Rn 0Rn 0

Q ⊂ R Q ∩ [0, 1] ⊂ R 00

Q Q

Q ∩ [0, 1]R [αi,βi] i = 1, . . . , k

k ∈ N

Q ∩ [0, 1] ⊂k⋃

i=1

[αi,βi].

Q ∩ [0, 1] = [0, 1] ⊂k⋃

i=1

[αi,βi],

k∑

i=1

v([αi,βi]) ≥ v([0, 1]) = 1.

Q ∩ [0, 1] 0

xi = c i = 1, . . . ,n c ∈ R Rn

H = {x = (x1, . . . , xn) ∈ Rn : xi = c} = R × · · ·× R × {c}︸︷︷︸i

× · · ·× R

0 H 0

ε > 0

Uk = [−k, k]× · · ·× [−k, k]×[c−

ε

2k+2(2k)n−1, c+

ε

2k+2(2k)n−1

]

︸ ︷︷ ︸i

×

× [−k, k]× · · ·× [−k, k], k ∈ N.

H ⊂∞⋃

k=1

Uk

∞∑

k=1

v(Uk) =∞∑

k=1

(2k)n−1 2ε

2k+2(2k)n−1=ε

2< ε,

A ⊂ H ∥x∥ =√∑n

i=1 x2i ≤ C ∀ x ∈ A

A ⊂ Uk0k0 ≥ C v(Uk0

) =ε

2k0+1.

A = [α1,β1]× · · · × [αn,βn]

∂A =n⋃

i=1

[α1,β1]× · · ·× {αi,βi}× · · ·× [αn,βn]

0

A ⊂ Rn f : A → R

f ⇐⇒ f

A ⊂ Rn f : A → Rf

A ⊂ Rn f : A → R a ∈ A

o(f, a) := limδ→0

g(δ), g(δ) := sup {|f(x)− f(y)| : x, y ∈ A∩ B(a, δ)}, δ > 0,

B(a, δ) = {x ∈ Rn : ∥x− a∥ < δ} ∥ · ∥f a

[0, 2 sup |f|]g : (0,∞) → [0, 2 sup |f|]

g(δ) = sup f|A∩B(a,δ) − inf f|A∩B(a,δ) ∀ δ > 0.

A ⊂ Rn f : A → R a ∈ A

f a ⇐⇒ o(f, a) = 0.

∀ ε > 0 ∃ δ0 > 0 ∀ x ∈ A∩ B(a, δ0) : |f(x)− f(a)| < ε

⇐⇒ ∀ ε > 0 ∃ δ1 > 0 ∀ x, y ∈ A∩ B(a, δ1) : |f(x)− f(y)| < ε

⇐⇒ ∀ ε > 0 ∃ δ2 > 0 : g(δ2) < ε

⇐⇒ ∀ ε > 0 ∃ δ3 > 0 ∀ δ ∈ (0, δ3) : 0 ≤ g(δ) < ε

✷

A ⊂ Rn f : A → R

f ⇐⇒ µ(B) = 0 B := {x ∈ A : f x}.

⇐: ε > 0 (Ui)i∈N ⊂ Rn

∞⋃

i=1

Ui ⊃ B∞∑

i=1

v(Ui) < ε.

x ∈ A \ B f ∥ · ∥∥ · ∥∞ Vx ⊂ Rn

x ∈ Vx sup f|Vx∩A − inf f|Vx∩A < ε.

A ⊂⋃

x∈A\B

Vx ∪∞⋃

i=1

Ui

A ⊂ Rn

x1, . . . , xℓ ∈ A \ B ℓ ∈ N i1, . . . , ik ∈ N k ∈ N

A ⊂ Vx1 ∪ . . .∪ Vxℓ ∪Ui1 ∪ . . .∪Uik .

P ∈ P(A) S ∈ SP

Vxλ λ = 1, . . . , ℓ Uiκ κ = 1, . . . , k

U(f,P)− L(f,P) =∑

S∈SP

(sup f|S − inf f|S) · v(S)

=∑

S⊂Vxλ

(sup f|S − inf f|S) · v(S) +∑

S⊂Uiκ

(sup f|S − inf f|S) · v(S),

≤ ε∑

S⊂Vxλ

v(S) + 2 sup |f|∑

S⊂Uiκ

v(S),

≤ (v(A) + 2 sup |f|) ε,∑

S⊂VxλS ∈ SP

Vxλ

∑S⊂Uiκ

∀ ε > 0 ∃ P ∈ P(A) : 0 ≤ Uf − Lf ≤ (v(A) + 2 sup |f|) · ε

Uf = Lf f⇒:

B = {x ∈ A : o(f, x) > 0} =∞⋃

k=1

B 1k, B 1

k:=

{x ∈ A : o(f, x) ≥ 1

k

}, k ∈ N,

B 1k

k ∈ N ε > 0 P ∈ P(A) U(f,P)− L(f,P) < ε2k

B 1k⊂

⋃

S∈SP

∂S∪⋃

S∈SS, S := {S ∈ SP : S∩ B 1

k= ∅}.

S ∈ S x ∈ S

a := limδ→0

(sup f|A∩B(x,δ) − inf f|A∩B(x,δ)

)≥ 1

k,

∀ ε ′ > 0 ∃ δ > 0 : B(x, δ) ⊂ S ⊂ S ⊂ A

sup f|S − inf f|S ≥ sup f|B(x,δ) − inf f|B(x,δ) ≥ a− ε ′ ≥ 1

k− ε ′,

sup f|S − inf f|S ≥ 1

k.

1

k

∑

S∈Sv(S) ≤

∑

S∈S(sup f|S − inf f|S) · v(S) ≤ U(f,P)− L(f,P) <

ε

2k,

∑

S∈Sv(S) <

ε

2.

m ∈ N ∂S S ∈ SP

ℓ ∈ N T(µ)λ ⊂ Rn µ = 1, . . . ,m λ = 1, . . . , ℓ

v(T(µ)ℓ

)= ε

4ℓm

⋃

S∈SP

∂S ⊂m⋃

µ=1

ℓ⋃

λ=1

T(µ)λ

m∑

µ=1

ℓ∑

λ=1

v(T(µ)λ

)=

m∑

µ=1

ℓ∑

λ=1

ε

4ℓm=ε

4<ε

2.

✷

A ⊂ Rn f : A → Rf(x0) > 0 x0 ∈ A

∫A f > 0

f x0 f(x0) > 0 δ > 0

x ∈ A ∩ B(x0, δ) f(x) ≥ f(x0)2

S0 = {x ∈ A : ∥x− x0∥∞ ≤ δ2√n} inf f|S0

≥ f(x0)2 > 0

inf f ≥ 0 P ∈ P(A) S0 ∈ SP

0 < inf f|S0· v(S0) ≤ L(f,P) ≤ Lf =

∫

Af.

✷

A ⊂ Rn f : A → R∫A f > 0

f x ∈ A \ BB ⊂ A f A \B

x0 ∈ A \ Bf(x0) > 0 f

✷

A ⊂ Rn f : A → R

f = 0 ⇐⇒∫

Af = 0.

⇒: B ⊂ A µ(B) = 0 f(x) = 0 ∀ x ∈ A \ BP ∈ P(A) S ∈ SP S ⊂ B

S ∩ (A \ B) = ∅ inf f|S = 0S ∈ SP L(f,P) = 0 P ∈ P(A) Lf = 0⇐: {x ∈ A : f(x) > 0}

f 0 ✷

A ⊂ Rn f,g : A → R

f = g ⇐⇒∫

A|f− g| = 0.

f,g |f− g|f = g ⇐⇒ |f− g| = 0

✷

A ⊂ Rn f,g : A → R∫A f =

∫A g.

∫A |f− g| = 0

∫

A(f− g) =

∫

Af−

∫

Ag ≤ 0 −

∫

A(f− g) = −

∫

Af+

∫

Ag ≤ 0.

✷

f,g : A → R∫A f =

∫A g ⇒ f = g A = [−1, 1]

f(x) = x g = 0

A ⊂ Rn f,g : A → RA

∫A f =

∫A g.

f = g A f g∂A

✷

A ⊂ Rn f : A → R{x ∈ A : f(x) = 0} ⊂ ∂A f

∫A f = 0.

∂A0 f g ≡ 0

∫A g = 0

✷

f : [0, 1]× [0, 1] → R

f(x,y) =

⎧⎪⎨

⎪⎩

0, 0 ≤ x <1

2,

1,1

2≤ x ≤ 1.

f

∫

[0,1]×[0,1]f =

1

2.

A ⊂ Rn f : A → Rg : A → R f(x) = g(x) x ∈ A

g∫

Af =

∫

Ag.

f : [0, 1]× [0, 1] → R

f(x,y) =

⎧⎨

⎩

0, x y1

q, y =

p

qp,q

f∫

[0,1]×[0,1]f = 0.

Rn

B ⊂ Rn ∂B

B ⊂ Rn

∂B

A ⊂ Rn f : A → R∫A f =

0 {x ∈ A : f(x) = 0}

f : A × B → R A ⊂ Rn B ⊂ Rm

f(x, ·) : B → R f(·, y) : A → Rx ∈ A y ∈ B

∫

A(2x+ 3y)d(x,y) A = [0, 2]× [3, 4]

∫

A(xy+ y2)d(x,y) A = [0, 1]× [0, 1]

∫

Aex+y d(x,y) A = [1, 2]× [1, 2]

∫

Asin(x+ y)d(x,y) A = [0, π2 ]× [0, π2 ]

∫

A

2z

(x+ y)2d(x,y, z) A = [1, 2]× [2, 3]× [0, 2]

∫

A

x2z3

1+ y2d(x,y, z) A = [0, 1]× [0, 1]× [0, 1]

f : [a,b] → R g : [c,d] → R

∫

[a,b]×[c,d]f(x)g(y)d(x,y) =

(∫b

af(x)dx

)(∫d

cg(y)dy

).

f : [a,b] → R(∫b

af(x)dx

)(∫b

a

1

f(x)dx

)≥ (b− a)2.

z+1

z≥ 2 ∀ z > 0.

f : A → RA ⊂ Rn

f ≡ 1 : A → R f(x) := 1 ∀ x ∈ A∫

A1 = v(A) =

∑

S∈SP

sup 1|S · v(S) =∑

S∈SP

inf 1|S · v(S) ∀ P ∈ P(A).

B ⊂ Rn

B = A

∫

B1 = v(B),

f ≡ 1 : B → RB

B

A ⊂ Rn B ⊂ A

A f ≡ 1 :A → R A T

C ⊃ Ax ∈ C \A f(x) = 0

C

B ⊂ Rn f : B → RfB : Rn → R

fB(x) :=

{f(x), x ∈ B,

0, x ∈ Rn \B

A ⊂ Rn B ⊂ AfB A f

∫

Bf :=

∫

Bf(x)dx =

∫

Bf(x1, . . . , xn)d(x1, . . . , xn) :=

∫

AfB.

fB AfB|A

fB A

∫

AfB :=

∫

AfB|A.

fA ⊂ Rn B ⊂ A

C ⊂ Rn B ⊂ C v(A∩C) > 0 A\C = ∅C\A = ∅ P ∈ P(A) Q ∈ P(C) A ∩ C ∈ SP ∩ SQ

∫

AfB =

∫

A∩CfB +

∑

S∈SP\{A∩C}

∫

SfB =

∫

A∩CfB +

∑

T∈SQ\{A∩C}

∫

TfB =

∫

CfB

S, T = A ∩ C

v(A ∩ C) > 0 A\C = ∅ C\A = ∅S, T = A∩C v(A∩C) = 0∫

A fB = 0 =∫C fB

B = A ⊂ Rn fB|A = f

B ⊂ Rn

f ≡ 1 : B → R f(x) = 1 ∀ x ∈ B

v(B) :=

∫

B1 ≥ 0

B∅ v(∅) := 0

∅ = B ⊂ Rn B ⊂ AA ⊂ Rn

B

χB(x) :=

{1, x ∈ B,

0, x ∈ Rn \B,

A

v(B) =

∫

B1 =

∫

AχB.

Rn

Rn

∅ ⊂ Rn

∅ = B ⊂ Rn B ⊂ A A ⊂ Rn

χBA

χB A

{x ∈ A : χB x} = ∂B,

x ∈ B ∪ (A\B) χBx ∈ ∂B χB B

∂B 0 0 ∂B ✷

f : B → RB ⊂ Rn

∅ = B ⊂ Rn f : B → R

f ⇐⇒ f .

A ⊂ Rn B ⊂ A fB

f ⇐⇒ fB A,

f ⇐⇒ fB|A .

{x ∈ B : f x}

⊂ {x ∈ A : fB|A x}

⊂ {x ∈ B : f x}∪ ∂B,

∂B

µ({x ∈ B : f x}) = 0 ⇐⇒ µ({x ∈ A : fB|A x}) = 0,

✷

∅ = B ⊂ Rn f : B → Rf

∅ = B ⊂ Rn f,g : B → R α ∈ R

f+ g∫B(f+ g) =

∫B f+

∫B g

αf∫B(αf) = α

∫B f

f ≤ g =⇒∫B f ≤

∫B g

|f|∣∣∫

B f∣∣ ≤

∫B |f|

fg

∅ = B ⊂ Rn f : B → R

inf f · v(B) ≤∫

Bf ≤ sup f · v(B).

f : B → R B ⊂ Rn

B

∅ = B ⊂ Rn f : B → R∅ = D ⊂ B f|D f

D∫

Df :=

∫

Bf|D

f D

A,B ⊂ Rn

A∪ B A∩ B A \B B \A

A,BA,B = ∅

∂A∪ ∂B

∂(A∪ B) = A∪ B \ (A∪ B)˚ ⊂ (A∪ B) \ (A∪ B) ⊂ (A \ A)∪ (B \ B) = ∂A∪ ∂B,∂(A∩ B) = ∂((A∩ B)c) = ∂(Ac ∪ Bc) ⊂ ∂(Ac)∪ ∂(Bc) = ∂A∪ ∂B,∂(A \B) = ∂(A∩ Bc) ⊂ ∂A∪ ∂(Bc) = ∂A∪ ∂B,

∂A = A \ A = A∩ (A)c = A∩Ac = ∂(Ac),

A ⊂ A ⇒ (A)c ⊃ Ac ⇒ (A)c ⊃ Ac

Ac ⊃ Ac ⇒ (Ac)c ⊂ A ⇒ (Ac)c ⊂ A ⇒ Ac ⊃ (A)c.

✷

f A,B ⊂ Rn

f A∪B A∩B∫A∩B f := 0 A∩B = ∅

∫

A∪Bf =

∫

Af+

∫

Bf−

∫

A∩Bf.

A ∪ B A ∩ Bf

A∪B f A∩B A∩B = ∅

A ∩ B = ∅fA∪B = fA + fB C ⊂ Rn

A∪ B ⊂ C∫

Af+

∫

Bf =

∫

CfA +

∫

CfB =

∫

C(fA + fB) =

∫

CfA∪B =

∫

A∪Bf.

A∩B = ∅ A,B,A∪B

A = (A∩ B)∪ (A∩ Bc) = (A∩ B)∪ (A \B),

B = (B∩A)∪ (B ∩Ac) = (A∩ B)∪ (B \A),

A∪ B = (A∩ B)∪ (A \B)∪ (B \A),

∫

Af =

∫

A∩Bf+

∫

A\Bf

∫

Bf =

∫

A∩Bf+

∫

B\Af,

∫

Af+

∫

Bf =

∫

A∩Bf+

∫

A\Bf+

∫

A∩Bf+

∫

B\Af =

∫

A∩Bf+

∫

A∪Bf.

✷

A,B ⊂ Rn

v(A∪ B) = v(A) + v(B)− v(A∩ B)

A ⊂ B v(A) ≤ v(B)

A,Bv(∅) = 0 A,B = ∅f = χA∪B

B = A∪ (B \A) A ⊂ B A∩ (B \A) = ∅ v(B \A) ≥ 0. ✷

A ⊂ Rn v(A) = 0A

ε > 0A < ε

A A = ∅∫A 1 = 0

B ⊂ Rn

B

⇐⇒ B v(B) = 0 .

B = ∅v(B) = 0 B = ∅

⇒: ε > 0 Ui ⊂ Rn i = 1, . . . , kk ∈ N

B ⊂k⋃

i=1

Ui

k∑

i=1

v(Ui) < ε.

∂B ⊂ B ⊂k⋃

i=1

Ui

k∑

i=1

v(Ui) < ε,

∂BB

0 ≤ v(B) ≤ v

(k⋃

i=1

Ui

)≤

k∑

i=1

v(Ui) < ε.

ε > 0 v(B) = 0⇐: A ⊂ Rn B ⊂ A

χB A∫A χB = 0

ε > 0 P ∈ P(A)

U(χB,P)− L(χB,P) < ε.

0 ≤ L(χB,P) =∑

S∈SP

inf χB|S · v(S) ≤ LχB = 0

U(χB,P) =∑

S∈SP

supχB|S · v(S) < ε.

S = {S ∈ SP : supχB|S = 1}

B ⊂⋃

S∈SS U(χB,P) =

∑

S∈Sv(S) < ε,

B ✷

B ⊂ Rn v(B) = 0f : B → R f

∫B f = 0

Bf

f : B → Rf ✷

f : A ∪ B → RA,B ⊂ Rn A ∩ B = ∅

∫

A∪Bf =

∫

Af+

∫

Bf.

A,BA,B A ∩ B ⊂ ∂A ∪ ∂B

A,B

0

A,B ⊂ Rn A ∩ B = ∅f : A∪ B → R

f ⇐⇒ f|A, f|B .

∫

A∪Bf =

∫

Af+

∫

Bf.

A ∪ Bf f|A, f|B

∫

A∩Bf = 0 A∩ B = ∅.

A ∩ B f|A∩B A ∩ B = ∅A ∩ B ⊂ ∂A ∪ ∂B A,B ∂A,∂B

0 ∂A ∪ ∂BA∩ B 0 ✷

A,B ⊂ Rn A∩ B = ∅

v(A∪ B) = v(A) + v(B).

A,BA,B = ∅ f ≡ 1 :

A∪ B → R ✷

B ⊂ Rn f : B → RN ⊂ B v(N) = 0 g : B → R f = gB \N g

∫B g =

∫B f

N = ∅ N = Bg

∫B g =

∫B f = 0

∅ = N = B B \N

f B \N N∫N f = 0

g N∫N g = 0 (B \

N)∩N = ∅ g

∫

Bg =

∫

(B\N)∪Ng =

∫

B\Ng+

∫

Ng =

∫

B\Nf+

∫

Nf =

∫

Bf.

✷

B ⊂ Rn f : B → Rf

Γf = {(x, f(x)) : x ∈ B} ⊂ Rn+1

(n+ 1)

A ⊂ Rn B ⊂ A fB : Rn → RfB|A : A → R

ε > 0P ∈ P(A)

U(fB|A,P)− L(fB|A,P) =∑

S∈SP

(sup fB|S − inf fB|S) · v(S) < ε.

Γf = {(x, f(x)) : x ∈ B} ⊂ {(x, f(x)) : x ∈ B}∪ {(x, 0) : x ∈ A \B}

= {(x, fB(x)) : x ∈ A} =⋃

S∈SP

{(x, fB(x)) : x ∈ S} ⊂⋃

S∈SP

S× [inf fB|S, sup fB|S],

AS := S× [inf fB|S, sup fB|S] ⊂ Rn+1 v(AS) = (sup fB|S − inf fB|S) · v(S)

Γf ⊂⋃

S∈SP

AS

∑

S∈SP

v(AS) < ε,

Γf AS

{(x, f(x)) : x ∈ B ∩ S} ΓfAS < ε

Γf ✷

Rn

1

B ⊂ Rn f1, f2 : B → Rf1 ≤ f2

M = {(x,y) : x ∈ B, f1(x) ≤ y ≤ f2(x)} ⊂ Rn+1

v(M) =

∫

B(f2 − f1).

A ⊂ Rn B ⊂ Rn a < inf f1 sup f2 < b

M ⊂ B× [a,b] ⊂ A× [a,b] ⊂ Rn+1

MM ∂M

(n+ 1) ∂M (n+ 1)

(x,y) = (x1, . . . , xn,y) ∈ M

x ∈ B, y ∈ (f1(x), f2(x)) f1, f2 x

M ε, δ > 0

(y− ε,y+ ε) ⊂ [f1(ξ), f2(ξ)] ∀ ξ ∈ (x1 − δ, x1 + δ)× · · ·× (xn − δ, xn + δ) ⊂ B,

(x,y) ∈ (x1 − δ, x1 + δ)× · · ·× (xn − δ, xn + δ)× (y− ε,y+ ε) ⊂ M,

(x,y) ∈ M M(x,y) ∈ B× [a,b]∂M

T := {(x,y) : x ∈ ∂B, y ∈ [a,b]},

Γ1 := {(x, f1(x)) : x ∈ B},

Γ2 := {(x, f2(x)) : x ∈ B},

A1 := {(x,y) : x ∈ B f1, y ∈ [a,b]},

A2 := {(x,y) : x ∈ B f2, y ∈ [a,b]}.

(n + 1)Γ1 Γ2 T

ε > 0 Ui ⊂ Rn i = 1, . . . , k

∂B ⊂k⋃

i=1

Ui

k∑

i=1

v(Ui) <ε

b− a.

Ui × [a,b] ⊂ Rn+1 T = ∂B× [a,b]< ε Aℓ ℓ = 1, 2 ε > 0

Vℓi ⊂ Rn i ∈ N

{x ∈ B : fℓ x} ⊂∞⋃

i=1

Vℓi

∞∑

i=1

v(Vℓi ) <ε

b− a.

Vℓi × [a,b] ⊂ Rn+1 Aℓ< ε M

M M ⊂ A× [a,b] ⊂ Rn+1

A× [a,b]χM : Rn+1 → R A× [a,b]

v(M) =

∫

M1 =

∫

A×[a,b]χM(x,y)d(x,y).

x ∈ B χM(x, ·) : [a,b] → R

χM(x,y) =

{1, y ∈ [f1(x), f2(x)],

0, y ∈ [a, f1(x))∪ (f2(x),b]

χM(x, ·) : [a,b] → R f1(x)f2(x) 0

∫b

aχM(x,y)dy =

∫f2(x)

f1(x)1 dy = f2(x)− f1(x) ∀ x ∈ B,

x ∈ A \ B χM(x, ·) : [a,b] → R

∫b

aχM(x,y)dy = 0 ∀ x ∈ A \B.

∫

A×[a,b]χM(x,y)d(x,y) =

∫

A

(∫b

aχM(x,y)dy

)dx,

∫

A

(∫b

aχM(x,y)dy

)dx =

∫

B(f2(x)− f1(x)) dx.

✷

R2 (x0,y0) ∈ R2 r ≥ 0

∆ :={(x,y) ∈ R2 : (x− x0)

2 + (y− y0)2 ≤ r2

}.

(y− y0)2 ≤ r2 − (x− x0)

2

⇔ |y− y0| ≤√

r2 − (x− x0)2

⇔ −√

r2 − (x− x0)2 ≤ y− y0 ≤√

r2 − (x− x0)2

⇔ y0 −√

r2 − (x− x0)2 ≤ y ≤ y0 +√

r2 − (x− x0)2

(x− x0)2 ≤ r2

⇔ |x− x0| ≤ r

⇔ − r ≤ x− x0 ≤ r

⇔ x0 − r ≤ x ≤ x0 + r,

∆ = {(x,y) ∈ R2 : x ∈ B, f1(x) ≤ y ≤ f2(x)}

B := [x0 − r, x0 + r],

f1(x) := y0 −√

r2 − (x− x0)2,

f2(x) := y0 +√

r2 − (x− x0)2.

B R f1, f2 :B → R

Rn f1(x) ≤ f2(x) ∀ x ∈ B

v(∆) =

∫x0+r

x0−r(f2(x)− f1(x))dx = 2

∫x0+r

x0−r

√r2 − (x− x0)2 dx

= 2

∫r

−r

√r2 − t2 dt = 4

∫r

0

√r2 − t2 dt = 4 r2

∫1

0

√1− s2 ds = π r2,

∫1

0

√1− s2 ds = s

√1− s2

∣∣∣1

s=0+

∫1

0

s2√1− s2

ds

= −

∫1

0

√1− s2 ds+

∫1

0

1√1− s2

ds =1

2

∫1

0

1√1− s2

ds =1

2arcsin s

∣∣∣1

s=0=π

4.

∫r

0

√r2 − t2 dt =

π

4r2 r ≥ 0

M ⊂ Rn

M x1 = a x1 = bx1 ∈ [a,b] ∀ x = (x1, . . . , xn) ∈ M

ξ ∈ [a,b] x1 = ξ M(n− 1) q(ξ)

q : [a,b] → R

v(M) =

∫b

aq(ξ)dξ.

M ⊂ Rn

M ⊂ [a,b]×A A ⊂ Rn−1

v(M) =

∫

M1 =

∫

[a,b]×AχM(x1, x

′)d(x1, x′), x ′ := (x2, . . . , xn).

Q(ξ) = {x ′ ∈ Rn−1 : (ξ, x ′) ∈ M}, ξ ∈ [a,b],

ξ ∈ [a,b]

x ′ ∈ Q(ξ) ⇔ (ξ, x ′) ∈ M ⊂ [a,b]×A ⇒ (ξ, x ′) ∈ {ξ}×A ⇔ x ′ ∈ A,

Q(ξ) ⊂ A

q(ξ) = v(Q(ξ)) =

∫

AχQ(ξ)(x

′)dx ′.

χQ(ξ)(x′) =

{1, x ′ ∈ Q(ξ)

0, x ′ ∈ Rn−1 \Q(ξ)=

{1, (ξ, x ′) ∈ M

0, (ξ, x ′) ∈ Rn \M= χM(ξ, x ′),

q(ξ) =

∫

AχM(ξ, x ′)dx ′ ∀ ξ ∈ [a,b].

✷

R3 (x0,y0, z0) ∈ Rn r ≥ 0

M :={(x,y, z) ∈ R3 : (x− x0)

2 + (y− y0)2 + (z− z0)

2 ≤ r2}.

M

M ={(x,y, z) ∈ R3 : (x,y) ∈ ∆, χ1(x,y) ≤ z ≤ χ2(x,y)

}

∆ :={(x,y) ∈ R2 : (x− x0)

2 + (y− y0)2 ≤ r2

},

χ1(x,y) := z0 −√

r2 − (x− x0)2 − (y− y0)2,

χ2(x,y) := z0 +√

r2 − (x− x0)2 − (y− y0)2,

∆ ⊂ R2

χ1,χ2 : ∆ → R∆

M

v(M) = 2

∫

∆

√r2 − (x− x0)2 − (y− y0)2 d(x,y).

y = y0 z = z0 (x− x0)2 ≤ r2

x = x0 − r x = x0 + rx = ξ ξ ∈ [x0 − r, x0 + r]

Q(ξ) = {(y, z) ∈ R2 : (y− y0)2 + (z− z0)

2 ≤ r2 − (ξ− x0)2},

R2 (y0, z0) ∈ R2√

r2 − (ξ− x0)2 ≥ 0Q(ξ) q(ξ) = π (r2 − (ξ − x0)

2)ξ ∈ [x0 − r, x0 + r]

v(M) =

∫x0+r

x0−rq(ξ)dξ = π

∫x0+r

x0−r(r2 − (ξ− x0)

2)dξ

= 2π

∫r

0(r2 − t2)dt = 2π

(r2t−

t3

3

)∣∣∣r

t=0=

4π

3r3.

ϕ1,ϕ2 : [a,b] → R ϕ1 ≤ ϕ2

B ⊂ R2

B = {(x,y) ∈ R2 : a ≤ x ≤ b, ϕ1(x) ≤ y ≤ ϕ2(x)}

0x ψ1,ψ2 : [c,d] →R ψ1 ≤ ψ2 C ⊂ R2

C = {(x,y) ∈ R2 : c ≤ y ≤ d, ψ1(y) ≤ x ≤ ψ2(y)}

0yR2

B = C ⊂ R2 0x 0y

0x 0y

B,C ⊂ R2

B,C[a,b], [c,d]

R ϕ1,ϕ2 ψ1,ψ2

B,C[a,b] ϕ1,ϕ2

((xν,yν)) ⊂ B (xν,yν) → (x,y) ∈ R2 (xν) ⊂[a,b] xν → x x ∈ [a,b] ϕ1(xν) ≤ yν ≤ ϕ2(xν)ϕ1(xν) → ϕ1(x) yν → y ϕ2(xν) → ϕ2(x) ϕ1(x) ≤ y ≤ ϕ2(x)(x,y) ∈ B C ✷

B,C ⊂ R2

f : B → R g : C → R f,g

∫

Bf(x,y)d(x,y) =

∫b

a

(∫ϕ2(x)

ϕ1(x)f(x,y)dy

)dx

∫

Cg(x,y)d(x,y) =

∫d

c

(∫ψ2(y)

ψ1(y)g(x,y)dx

)dy.

B,C f,g

f gm := minϕ1 M := maxϕ2 B ⊂ A := [a,b]× [m,M]

∫

Bf(x,y)d(x,y) =

∫

AfB(x,y)d(x,y)

fB x ∈ [a,b]

∫M

mfB(x,y)dy =

∫ϕ2(x)

ϕ1(x)f(x,y)dy

✷

B = C ⊂ R2 B,C0x 0y f : B → R

∫b

a

(∫ϕ2(x)

ϕ1(x)f(x,y)dy

)dx =

∫d

c

(∫ψ2(y)

ψ1(y)f(x,y)dx

)dy.

M

v(M) = 2

∫

∆

√r2 − (x− x0)2 − (y− y0)2 d(x,y),

∆x

B = [x0 − r, x0 + r] f1, f2 : B → R

f1(x) = y0 −√

r2 − (x− x0)2, f2(x) = y0 +√

r2 − (x− x0)2.

v(M) = 2

∫x0+r

x0−r

(∫y0+√

r2−(x−x0)2

y0−√

r2−(x−x0)2

√r2 − (x− x0)2 − (y− y0)2 dy

)dx

= 4

∫x0+r

x0−r

(∫√r2−(x−x0)2

0

√r2 − (x− x0)2 − η2 dη

)dx

= 8

∫r

0

⎛

⎝∫√r2−ξ2

0

√r2 − ξ2 − η2 dη

⎞

⎠ dξ,

v(M) = 8

∫r

0

π

4

(r2 − ξ2

)dξ = 8

π

4

(r3 −

r3

3

)=

4π

3r3,

B ⊂ R2

χ1,χ2 : B → R χ1 ≤ χ2 M ⊂ R3

M = {(x,y, z) ∈ R3 : (x,y) ∈ B, χ1(x,y) ≤ z ≤ χ2(x,y)}

0xyf : M → R f

∫

Mf(x,y, z)d(x,y, z) =

∫

B

(∫χ2(x,y)

χ1(x,y)f(x,y, z)dz

)d(x,y).

B ⊂ R2 0x0y∫

Mf(x,y, z)d(x,y, z) =

∫b

a

(∫ϕ2(x)

ϕ1(x)

(∫χ2(x,y)

χ1(x,y)f(x,y, z)dz

)dy

)dx

∫

Mf(x,y, z)d(x,y, z) =

∫d

c

(∫ψ2(y)

ψ1(y)

(∫χ2(x,y)

χ1(x,y)f(x,y, z)dz

)dx

)dy,

M(x,y)

v(M) =

∫

M1d(x,y, z)

=

∫

∆

(∫z0+√

r2−(x−x0)2−(y−y0)2

z0−√

r2−(x−x0)2−(y−y0)21dz

)d(x,y)

=

∫x0+r

x0−r

(∫y0+√

r2−(x−x0)2

y0−√

r2−(x−x0)2

(∫z0+√

r2−(x−x0)2−(y−y0)2

z0−√

r2−(x−x0)2−(y−y0)21 dz

)dy

)dx,

∆ ⊂ R2 0x

B ⊂ Rn∫B 1

B ⊂ Rn∫B 1 = 0

A ⊂ Rn B ⊂ A

B ⇐⇒ ∀ ε > 0 ∃ P ∈ P(A) :∑

S∈S1

v(S)−∑

S∈S2

v(S) < ε,

S1 := {S ∈ SP : S∩ B = ∅} S2 := {S ∈ SP : S ⊂ B}

B ⊂ Rn ε > 0 C ⊂ B∫B\C 1 < ε

B ⊂ Rn f : B → RintB f|intB

∫

intBf =

∫

Bf.

B ⊂ Rn f : B → Rf : B → R f B f

∫

Bf =

∫

Bf.

z = x+ y[0, 1]× [0, 2] 3

z = x2 + y2 [0, 1]× [0, 1] 23

z = xy2 + y3 [0, 2]× [0, 2] 403

x2

a2+

y2

b2≤ 1 a,b > 0

πab

x2

a2+

y2

b2+

z2

c2≤ 1 a,b, c >

0 43πabc

∫1

0

∫x

0f(x,y)dydx,

∫4

1

∫2√xf(x,y)dydx,

∫ π2

0

∫ siny

0f(x,y)dxdy,

∫1

0

∫√1−y2

1−y2f(x,y)dxdy.

∫

Bx2yd(x,y), B

(0, 0) 2 6415∫

B(x+ y2)d(x,y), B (0, 0), (1, 0), (0, 1) 1

4∫

B(x2 + y2)d(x,y), B (0, 0), (1, 0), (12 ,

12 )

112

∫

Bxyd(x,y), B

y = x y = x2 124∫

B

√xyd(x,y), B

y =√x y = x2 6

55

z = 2− 2x− y 23

z = xy (0, 0, 0), (1, 0, 0), (0, 1, 0)x+ y = 1 1

24

z = 0 z =

αx+ βy+ γ x2

a2 + y2

b2 ≤ 1 a,b > 0πabγ

xy z = 1− x2 − y2π2

∫B(x

2 + 3y2 + 1)d(x,y) B(0, 0) 2 32π

∫

B

sin x

xd(x,y) B

(0, 0), (1, 0), (1, 1) 1− cos 1

A ⊂ Rn g : A → Rn 1− 1

detDg(y) > 0 ∀ y ∈ A detDg(y) < 0 ∀ y ∈ A,

T ⊂ A f : g(T) → Rg(T) ⊂ Rn ∂g(T) = g(∂T)

∫

g(T)f(x)dx =

∫

Tf(g(y)) | detDg(y)|dy.

detDg g1− 1 N ⊂ T

✷

g : Rn → Rn, x = g(y) = Ay+ b, A ∈ Rn×m detA = 0, b ∈ Rn.

Dg(y) = A ∀ y ∈ Rn y = g−1(x) = A−1x−A−1b ∀ x ∈ Rn,

∫

AT+bf(x)dx = |detA|

∫

Tf(Ay+ b)dy,

AT + b = {Ay+ b : y ∈ T }A = I ∈ Rn×n g b = 0

g

g : (0,∞)× (0, 2π) =: Aπ → R2,

(xy

)= g(r,ϕ) =

(r cosϕr sinϕ

).

Dg(r,ϕ) =

(cosϕ −r sinϕsinϕ r cosϕ

)detDg(r,ϕ) = r > 0 ∀ (r,ϕ) ∈ Aπ

A \N g 1− 1

(rϕ

)= g−1(x,y) =

(√x2 + y2

φ(x,y)

)∀ (x,y) ∈ g(Aπ) = R2 \ {(x, 0) : x ≥ 0},

φ(x,y) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

arctany

x, x > 0, y > 0,

π

2, x = 0, y > 0,

π+ arctany

x, x < 0, y ∈ R,

3π

2, x = 0, y < 0,

2π+ arctany

x, x > 0, y < 0,

arctan : R →(−π2 ,

π2

)

∫

g(T)f(x,y)d(x,y) =

∫

Tf(r cosϕ, r sinϕ) r d(r,ϕ),

T = [r1, r2]× [ϕ1,ϕ2] ⊂ Aπ∫

g(T)f(x,y)d(x,y) =

∫ϕ2

ϕ1

∫r2

r1

f(r cosϕ, r sinϕ) r dr dϕ

gE : R2 → R2

1− 1 T0 =[0, r0]× [0, 2π]

gE(0,φ) = (0, 0) ∀ φ ∈ R gE(r, 0) = gE(r, 2π) = (r, 0) ∀ r ∈ R.

T0

∆0 = {(x,y) ∈ R2 : x2 + y2 ≤ r20} = gE(T0), T0 = [0, r0]× [0, 2π],

f : ∆0 → R ∆0 ⊂ R2 T0 ⊂ R2

gE : R2 → R2

∫

∆0

f(x,y)d(x,y)

∫2π

0

∫r0

0f(r cosϕ, r sinϕ) r dr dϕ.

T ′ := [ρ, r0]× [ϕ1,ϕ2] 0 < ρ < r0, 0 < ϕ1 < ϕ2 < 2π

∫

g(T ′)f(x,y)d(x,y) =

∫ϕ2

ϕ1

∫r0

ρf(r cosϕ, r sinϕ) r dr dϕ,

lim(ρ,ϕ1,ϕ2)→(0,0,2π)

∫

g(T ′)f(x,y)d(x,y) =

∫

∆0

f(x,y)d(x,y),

lim(ρ,ϕ1,ϕ2)→(0,0,2π)

∫ϕ2

ϕ1

∫r0

ρf(r cosϕ, r sinϕ) r dr dϕ

=

∫2π

0

∫r0

0f(r cosϕ, r sinϕ) r dr dϕ.

∫

∆0

f(x,y)d(x,y) =

∫2π

0

∫r0

0f(r cosϕ, r sinϕ) r dr dϕ.

T = [r1, r2]× [ϕ1,ϕ2] ⊂ [0,∞)× [0, 2π]f : gE(T) → R gE(r,ϕ) = (r cosϕ, r sinϕ)∫

gE(T)f(x,y)d(x,y) =

∫ϕ2

ϕ1

∫r2

r1

f(r cosϕ, r sinϕ) r dr dϕ

∆ := {(x,y) ∈ R2 : (x− x0)2 + (y− y0)

2 ≤ r2}

∆ ∋ (x,y) 4→ (ξ,η) := (x− x0,y− y0) ∈ ∆0 := {(ξ,η) ∈ R2 : ξ2 + η2 ≤ r2}

[0, r]× [0, 2π] ∋ (r,ϕ) 4→ (ξ,η) := (r cosϕ, r sinϕ) ∈ ∆0

v(∆) =

∫

∆1d(x,y) =

∫

∆0

1d(ξ,η) =

∫2π

0

∫r

0ρdρdϕ = 2π

r2

2= π r2.

y

x

z

r

y

xϕ

⎛

⎝xyz

⎞

⎠ =

⎛

⎝r cosϕr sinϕ

0

⎞

⎠

⎛

⎝xyz

⎞

⎠ =

⎛

⎝r cosϕr sinϕ

z

⎞

⎠

(x,y, z) (r,ϕ, z)

g : (0,∞)× (0, 2π)× R =: Aκ → R3,

⎛

⎝xyz

⎞

⎠ = g(r,ϕ, z) =

⎛

⎝r cosϕr sinϕ

z

⎞

⎠ .

Dg(r,ϕ, z) =

⎛

⎝cosϕ −r sinϕ 0sinϕ r cosϕ 00 0 1

⎞

⎠ detDg(r,ϕ, z) = r > 0

∀ (r,ϕ, z) ∈ Aκ

⎛

⎝rϕz

⎞

⎠ = g−1(x,y, z) =

⎛

⎝

√x2 + y2

φ(x,y)z

⎞

⎠ , φ(x,y) ,

∀ (x,y) ∈ g(Aκ) = R3 \ {(x, 0, z) : x ≥ 0, z ∈ R},

∫

g(T)f(x,y, z)d(x,y, z) =

∫

Tf(r cosϕ, r sinϕ, z) r d(r,ϕ, z),

T = [r1, r2]× [ϕ1,ϕ2]× [z1, z2] ⊂ Aκ∫

g(T)f(x,y, z)d(x,y, z) =

∫z2

z1

∫ϕ2

ϕ1

∫r2

r1

f(r cosϕ, r sinϕ, z) r dr dϕdz

(r1,ϕ1,ϕ2) → (0, 0, 2π) r2 = r0

(x,y)

K0 ={(x,y, z) ∈ R3 : x2 + y2 ≤ r20, z1 ≤ z ≤ z2

}= gE(T0),

T0 = [0, r0]× [0, 2π]× [z1, z2] gE : R3 → R3 gf : K → R

∫

K0

f(x,y, z)d(x,y, z) =

∫z2

z1

∫2π

0

∫r0

0f(r cosϕ, r sinϕ, z) r dr dϕdz.

T = [r1, r2]× [ϕ1,ϕ2]× [z1, z2] ⊂ [0,∞)× [0, 2π]×Rf : gE(T) → R gE(r,ϕ) = (r cosϕ, r sinϕ, z)

∫

gE(T)f(x,y, z)d(x,y, z) =

∫z2

z1

∫ϕ2

ϕ1

∫r2

r1

f(r cosϕ, r sinϕ, z) r dr dϕdz

v(K0) =

∫

K0

1d(x,y, z) =

∫z2

z1

∫2π

0

∫r0

0r dr dϕdz = π (z2 − z1) r

20.

z

y

x

r

ϕ

ϑ

⎛

⎝xyz

⎞

⎠ =

⎛

⎝r sin ϑ cosϕr sin ϑ sinϕ

0

⎞

⎠

⎛

⎝xyz

⎞

⎠ =

⎛

⎝r sin ϑ cosϕr sin ϑ sinϕ

r cos ϑ

⎞

⎠

(x,y, z) (r, ϑ,ϕ)

g : (0,∞)× (0,π)× (0, 2π) =: Aσ → R3,

⎛

⎝xyz

⎞

⎠ = g(r, ϑ,ϕ) =

⎛

⎝r sin ϑ cosϕr sin ϑ sinϕ

r cos ϑ

⎞

⎠ .

Dg(r, ϑ,ϕ) =

⎛

⎝sin ϑ cosϕ r cos ϑ cosϕ −r sin ϑ sinϕsin ϑ sinϕ r cos ϑ sinϕ r sin ϑ cosϕ

cos ϑ −r sin ϑ 0

⎞

⎠

detDg(r, ϑ,ϕ) = r2 sin ϑ > 0 ∀ (r, ϑ,ϕ) ∈ Aσ

⎛

⎝rϑϕ

⎞

⎠ = g−1(x,y, z) =

⎛

⎜⎜⎝

√x2 + y2 + z2

arccosz√

x2 + y2 + z2

φ(x,y)

⎞

⎟⎟⎠ , φ(x,y) ,

arccos : (−1, 1) → (0,π), ∀ (x,y, z) ∈ g(Aσ) = R3 \ {(x, 0, z) : x ≥ 0, z ∈ R},

∫

g(T)f(x,y, z)d(x,y, z) =

∫

Tf(r sin ϑ cosϕ, r sin ϑ sinϕ, r cos ϑ) r2 sin ϑd(r, ϑ,ϕ),

T = [r1, r2]× [ϑ1, ϑ2]× [ϕ1,ϕ2] ⊂ Aσ

∫

g(T)f(x,y, z)d(x,y, z)

=

∫ϕ2

ϕ1

∫ϑ2

ϑ1

∫r2

r1

f(r sin ϑ cosϕ, r sin ϑ sinϕ, r cos ϑ) r2 sin ϑdrdϑdϕ

(r1, ϑ1, ϑ2,ϕ1,ϕ2) → (0, 0,π, 0, 2π)∞

Aσ

T = [r1, r2]× [ϑ1, ϑ2]× [ϕ1,ϕ2] ⊂ [0,∞)× [0,π]× [0, 2π]f : gE(T) → R gE(r, ϑ,ϕ) = (r sin ϑ cosϕ, r sin ϑ sinϕ, r cos ϑ)

∫

gE(T)f(x,y, z)d(x,y, z)

=

∫ϕ2

ϕ1

∫ϑ2

ϑ1

∫r2

r1

f(r sin ϑ cosϕ, r sin ϑ sinϕ, r cos ϑ) r2 sin ϑdrdϑdϕ

v(M) =

∫

M1d(x,y, z) =

∫

M−(x0,y0,z0)1d(ξ,η, ζ)

=

∫2π

0

∫π

0

∫r

0r2 sin ϑdrdϑdϕ = 2π 2

r3

3,

∫π0 sin ϑdϑ = − cos ϑ|πϑ=0 = − cosπ+ cos 0 = 2

g : A → Rn A ⊂ Rn

1− 1 detDg(x) = 0 ∀ x ∈ A g(T)T ⊂ A

∂g(T) = g(∂T)

g(T)∂T = T \ T ◦ ∂g(T) = g(T) \ g(T)◦

g 1− 1 g(T \ T ◦) = g(T) \ g(T ◦)

y ∈ g(T ◦) ε > 0 U ⊂ T ◦ B(y, ε) = g(U) ⊂ g(T ◦) ⊂ g(T)g(T ◦) ⊂ g(T)◦

g−1 : g(A) → Rn g(A) g−1

y ∈ g(A)g−1(g(T)◦) ⊂ g−1(g(T))◦ = T ◦ g(T)◦ ⊂ g(T ◦)

g(∂T) = g(T \ T ◦) = g(T) \ g(T ◦) = g(T) \ g(T)◦ = ∂g(T).

T ∂Tg(∂T) = ∂g(T)

g(T)

A ⊂ Rn g : A → Rm m ≥ nN ⊂ A g(N)

U ⊂ Rn f : U → Rm S := {x+ tη : t ∈[0, 1]} ⊂ U

f(x+ η)− f(x) =

∫1

0Df(x+ tη)dt η,

∥f(x+ η)− f(x)∥ ≤ M∥η∥, M := maxy∈S

∥Df(y)∥,

f = (f1, . . . , fm)T ϕj(t) := fj(x+ tη), t ∈ [0, 1], j = 1, . . . ,m,

fj(x+ η)− fj(x) = ϕj(1)−ϕj(0)

y ∈ g(A) ε > 0 U ⊂ A B(y, ε) = g(U) ⊂ g(A) g(A) ⊂ g(A)◦

=

∫1

0ϕ ′

j(t)dt

=

∫1

0Dfj(x+ tη) · ηdt

=

∫1

0Dfj(x+ tη)dt · η.

j = 1, . . . ,m

✷

nε > 0 n

n< ε

N ⊂ Rn

J ⊂ Rn N ⊂ J

∫

JχN = 0.

ε > 0 δ > 0 P J∥P∥ < δ

∑

S∈SP

supχN|S · v(S) = U(χN,P) = U(χN,P)− L(χN,P) < ε.

J m< δ J

N S ′P S ∈ SP S ∩N = ∅

N ⊂⋃

S∈S ′P

S∑

S∈S ′P

v(S) < ε.

N ⊂ Rn g : N → Rm

m ≥ n L > 0

∥g(x)− g(y)∥ ≤ L∥x− y∥ ∀ x, y ∈ N,

g(N) ⊂ Rm

Rn Rm ∥ · ∥∞∥ · ∥

ε > 0 Nk ∈ N 2r r ≤ 1

Ui := {x ∈ Rn : ∥x− xi∥∞ ≤ r}, i = 1, . . . , k,

N ⊂ Rn < ε

N ⊂k⋃

i=1

Ui

k∑

i=1

v(Ui) = k(2r)n < ε.

N =k⋃

i=1

N∩Ui

g(N) =k⋃

i=1

g(N∩Ui).

i = 1, . . . , k yi ∈ N∩Ui

∥x− yi∥∞ ≤ ∥x− xi∥∞ + ∥xi − yi∥∞ ≤ 2r ∀ x ∈ N∩Ui,

∥ · ∥∞ L∞ > 0 L

∥g(x)− g(yi)∥∞ ≤ 2L∞r ∀ x ∈ N∩Ui,

g(N∩Ui) mg(yi) 4L∞r g(N)

k

k(4L∞r)m = (22m−nLm∞)rm−nk(2r)n ≤ (22m−nLm∞)k(2r)n < (22m−nLm∞)ε,

r ≤ 1 m ≥ nε > 0 g(N) ⊂ Rm

✷

∥ · ∥∞N A ⊂ Rn

k ∈ N xi ∈ N 2ri > 0N A

N ⊂k⋃

i=1

Ui, Ui := {x ∈ Rn : ∥x− xi∥∞ ≤ ri} ⊂ A, i = 1, . . . , k.

A N

B(x,ε(x)

2

)⊂ B(x, ε(x)) ⊂ A, x ∈ N,

N

B(xi, ri) ⊂ B(xi, ri) = Ui ⊂ A, ri =ε(xi)

2, i = 1, . . . , k.

Ui

∥g(x)− g(y)∥ ≤ Mi∥x− y∥ ∀ x, y ∈ Ui, Mi := maxz∈Ui

∥Dg(z)∥,

g N ∩Ui

g(N∩Ui)

g(N) = g

( k⋃

i=1

N∩Ui

)=

k⋃

i=1

g(N∩Ui).

✷

g 1− 1A

B ⊂ Rn T ⊂ B ⊂ ADg|B

T AT

A BT Dg A

A DgB

TT B

B(xi, ε(xi)) ⊂ B ∥ · ∥∞ i = 1, . . . , k xi ∈ T

T ⊂k⋃

i=1

B

(xi,ε(xi)

3

)⊂

k⋃

i=1

B(xi, ε(xi)) ⊂ B.

r := min{ε(xi) : i = 1, . . . , k} W≤ r/3 T W ⊂ B

y ∈ W ∩ T i ∈ {1, . . . , k} y ∈ B(xi,

ε(xi)3

)

x0 W x ∈ W

∥x− xi∥∞ ≤ ∥x− x0∥∞ + ∥x0 − y∥∞ + ∥y− xi∥∞ ≤ r

6+

r

6+

r

3=

2r

3< ε(xi),

x ∈ B(xi, ε(xi)) ⊂ BJ ⊂ Rn T ⊂ J

≤ r/3T W B

U ⊂ T

∫

g(U)f(x)dx =

∫

Uf(g(y))| detDg(y)|dy.

J ⊂ Rn T P JR ∂T

S Tε > 0

P ∂TB v(R) < ε

T ⊂ R∪ S T \ S ⊂ R

v(T \ S) < ε.

Dg Bg ∂T

L := supy∈B ∥Dg(y)∥P

Ui i = 1, . . . ,k ∂T xi2r > 0 ∥ · ∥ ∥ · ∥∞

∥g(x)− g(xi)∥∞ ≤√nLr ∀ x ∈ Ui ∀ i = 1, . . . , k,

v(g(R)) = v

(g( k⋃

i=1

Ui

))= v

( k⋃

i=1

g(Ui)

)=

k∑

i=1

v(g(Ui))

≤ k(2√nLr)n = (

√nL)nv(R) < (

√nL)nε,

g(T) ⊂ g(R ∪ S) = g(R) ∪ g(S) g(T) \ g(S) ⊂ g(R)

v(g(T) \ g(S)

)< (

√nL)nε.

∫

g(S)f(x)dx =

∫

Sϕ(y)dy, ϕ(y) := f(g(y))| detDg(y)|, y ∈ T ,

M := max{max |f|, max |ϕ|}

∣∣∣∣∫

g(T)f(x)dx−

∫

Tϕ(y)dy

∣∣∣∣

=

∣∣∣∣∫

g(T)\g(S)f(x)dx+

∫

g(S)f(x)dx−

∫

T\Sϕ(y)dy−

∫

Sϕ(y)dy

∣∣∣∣

≤∣∣∣∣∫

g(T)\g(S)f(x)dx

∣∣∣∣+∣∣∣∣∫

T\Sϕ(y)dy

∣∣∣∣

≤ M((√nL)n + 1)ε.

ε > 0g : A → Rn

g(y) =(y ′,g(y)

)T, y = (y ′,yn) ∈ A, y ′ := (y1, . . . ,yn−1),

g : A → R

detDg(y) =∂g(y)

∂yn, y ∈ A,

A g 1− 1

U = U ′ × [an,bn] ⊂ T , U ′ := [a1,b1]× · · ·× [an−1,bn−1].

f : [a,b] → R g : [α,β] → [a,b] 1− 1g ′(y) = 0 y ∈ [α,β]

∫g(β)

g(α)f(x)dx =

∫β

αf(g(y))g ′(y)dy,

g(α) = a g(β) = b g ′ > 0 g(α) = b g(β) = a g ′ < 0I = [α,β] g(I) = [a,b]

∫

g(I)f(x)dx =

∫

If(g(y))|g ′(y)|dy,

∫ab f(x)dx = −

∫ba f(x)dx = −

∫J f(x)dx a < b J = [a,b]

A y ′ ∈ U ′

[an,bn] ∋ yn 4→ g(y ′,yn)

g(y ′,an) < g(y ′,bn) g(U)

g(U) = {(y ′, xn) ∈ Rn : y ′ ∈ U ′, g(y ′,an) ≤ xn ≤ g(y ′,bn)}.

U ′

U ′ ∋ y ′ 4→ g(y ′,an) U ′ ∋ y ′ 4→ g(y ′,bn)

g(U) 0y1 · · · yn−1

f : g(U) → R

∫

g(U)f(x)dx =

∫

U ′

∫g(y ′,bn)

g(y ′,an)f(y ′, xn)dxn dy ′.

y ′ ∈ U ′

[g(y ′,an),g(y

′,bn)]∋ xn → f(y ′, xn)

∫

g(U)f(x)dx =

∫

U ′

∫bn

an

f(y ′,g(y ′,yn)

)∂g(y ′,yn)

∂yndyn dy ′

∫

g(U)f(x)dx =

∫

Uf(g(y))| detDg(y)|dy,

−1

g 1− 1 A

U ⊂ Rn n ≥ 2 g = (g1, . . . ,gn) : U → Rn

detDg(x) = 0 x ∈ U x0 ∈ UW ⊂ U x0

ψ : W → Rn ω : ψ(W) → Rn

ψ(W) ⊂ Rn

ψ ω 1− 1

x = (x1, . . . , xn) ψxn ω n− 1 x1, . . . , xn−1

g|W = ω ◦ ψ

U0 ⊂ U x0 g|U01− 1

Dg(x0) n− 10

ψ0(x) := (g1(x), . . . ,gn−1(x), xn) , x = (x1, . . . , xn) ∈ U.

ψ0

Dψ0(x0)W ⊂ U0 x0

ψ = (ψ1, . . . ,ψn) := ψ0|W : W → ψ(W)

1− 1 ψ(W) ⊂ Rn

χ = (χ1, . . . ,χn) := ψ−1 : ψ(W) → W.

ω(x) := (x1, . . . , xn−1,gn (χ1(x), . . . ,χn−1(x), xn)) , x = (x1, . . . , xn) ∈ ψ(W),

χ(ψ(x)

)= x ⇔ χi

(ψ(x)

)= xi ∀ i = 1, . . . ,n ∀ x = (x1, . . . , xn) ∈ W,

x = (x1, . . . , xn) ∈ W

ω(ψ(x)

)=(ψ1(x), . . . ,ψn−1(x),gn

(χ1(ψ(x)

), . . . ,χn−1

(ψ(x)

),ψn(x)

))

= (g1(x), . . . ,gn−1(x),gn(x1, . . . , xn−1, xn))

= g(x),

ω 1− 1 g|W 1− 1 ✷

1− 1 g : A → Rn

A ⊂ Rn

U ⊂ T

n n = 1n− 1 n ≥ 2

nx0 ∈ U ⊂ T ⊂ B

W ⊂ B x0 ∈ W g = ω ◦ ψω ψ

x0 ∈ U U WU U Wj

j = 1, . . . ,m U Ui

i = 1, . . . , k Ui Wj

Ui g|Ui= (ωj ◦ ψj)|Ui

ωj : ψj(Wj) → Rn ψj : Wj → Rn

Ui i = 1, . . . , kU Ui g(U)

g(Ui)

i j Ui Wj ωj ψj

i jn− 1

U = U ′ × [an,bn] ⊂ W = W ′ × (an − ε,bn + ε), ε > 0,

U ′ W ′ Rn−1 U ′ ⊂ W ′

ω(η) = (η ′,ωn(η)), η ′ = (η1, . . . ,ηn−1), η = (η ′,ηn) ∈ ψ(W),

ψ(y) = (g1(y), . . . ,gn−1(y),yn), y ′ = (y1, . . . ,yn−1), y = (y ′,yn) ∈ W

g|W = ω ◦ ψyn ∈ [an,bn]

hyn(y′) = (g1(y

′,yn), . . . ,gn−1(y′,yn)) ∈ Rn−1, y ′ ∈ W ′,

hyn 1− 1 ψ 1− 1

detDhyn(y′) = detDψ(y ′,yn) = 0 ∀ y ′ ∈ W ′,

detDg(y) = detDω(ψ(y)) detDψ(y) = 0 ∀ y ∈ W,

U Wj

Wj U UU

U

W ′ detDhyn : W ′ → Rhyn

g n− 1

F(η ′,yn) := f(ω(η ′,yn))| detDω(η ′,yn)|, η ′ ∈ hyn(U′),

hyn(U′)× {yn} = ψ(U

′ × {yn}) ⊂ ψ(U) ⊂ ψ(W), (ω ◦ ψ)(U) = g(U) ⊂ g(T),

n− 1

∫

hyn(U ′)F(η ′,yn)dη

′ =∫

U ′F(hyn(y

′),yn)|detDhyn(y′)|dy ′,

(hyn(y′),yn) = ψ(y ′,yn)

∫

hyn(U ′)F(η ′,yn)dη

′ =∫

U ′F(ψ(y ′,yn))| detDψ(y

′,yn)|dy′.

hyn(U′) ⊂ Rn−1

hyn(U′)× [an,bn] = ψ(U) ⊂ Rn.

F : ψ(U) → R (F ◦ ψ)| detDψ| = (f ◦ g)|detDg| : U → R

∫

ψ(U)F(η)dη =

∫

Uf(g(y))|detDg(y)|dy,

∫

ψ(U)F(η)dη =

∫

ψ(U)f(ω(η))|detDω(η)|dη =

∫

g(U)f(x)dx,

ψ(W)ψ(U)

ω : ψ(W) → Rn 1− 1detDω : ψ(W) → R

W ω(ψ(U)) = g(U)

F g|U = (ω ◦ ψ)|U

g1−1 detDg N ⊂ Tε > 0 N Ui

i = 1, . . . , k B < εR S = T \ R

S TT \ S ⊂ R v(T \ S) < ε

v(g(T) \ g(S)) ≤ v(g(R)) ≤k∑

i=1

v(g(Ui)) ≤ (√nL)n

k∑

i=1

v(Ui) < (√nL)nε,

✷

K ={(x,y, z) ∈ R3 : (x− x0)

2 + (y− y0)2 ≤ r20, z1 ≤ z ≤ z2

}

K K0

K 0xy

x2

a2+

y2

b2≤ 1 a,b > 0

x2

a2+

y2

b2+

z2

c2≤ 1 a,b, c >

0

T < cB N < δ

< ε/2n T< min{c/2, δ} N

∂T Ui i = 1, . . . ,k

B ⊂ R3

R ≥ 0

∫

Bz d(x,y, z)

Σ x2 + y2 ≤ R2 x2 + y2 +z2 ≤ 4R2 Σ

Hα x2+y2 ≤ Rx z ≥ 0

x2 + y2 + z2 ≤ R2 Hα13

(π− 4

3

)R3

x2

a2 + y2

b2 ≤ 1 a,b > 0

x2

a2+

y2

b2+

z2

c2≤ 1 a,b, c > 0 Σε

Σε

B ⊂ R3

z = x2 + y2 z = 1∫

B

√x2 + y2 d(x,y, z).

B ⊂ R3 x = 0y = 0 z = 1 z = x2 + y2

x2 +y2 ≤ 1 x ≥ 0

y ≥ 0

∫

Bxyzd(x,y, z)

B ⊂ R3

R ≥ 0

∫

B∥(x,y, z)∥d(x,y, z)

e−(x2+y2) (x,y) ∈ R2

[−R,R]× [−R,R] B((0, 0),R) R > 0∫∞

0e−x2

dx =

√π

2.

f(R) =

∫

B((0,0),R)e−(x2+y2)d(x,y), g(R) =

∫

[−R,R]2e−(x2+y2)d(x,y)

f(R) = 2π

∫R

0e−r2rdr = π

(1− e−R2

)

f(R) ≤ g(R) = 4

(∫R

0e−x2

dx

)2

≤ f(√2R),

R → ∞

γ : [α,β] → Rn α,β ∈ R α < β C1

f : γ([α,β]) → Rn

∫

γf · dx :=

∫β

αf(γ(t)) · γ ′(t)dt

f γ

f γ∫

γf(x) · dx.

f = (f1, . . . , fn)∫

γf1dx1 + · · ·+ fndxn

∫

γf1(x)dx1 + · · ·+ fn(x)dxn.

γ = (γ1, . . . ,γn) : [α,β] → Rn

γi : [α,β] → R i = 1, . . . ,nγ

g : [α,β] → R[α,β] α,β g ′ : [α,β] → R

g(α,β) α,β

C1

γ f

R

γ(t) = (r cos t, r sin t), t ∈ [α,β], r > 0,

f(x,y) = (−y, x), g(x,y) = (x,y), (x,y) ∈ R2.

∫

γf · d(x,y) =

∫

γ(−y, x) · d(x,y)

=

∫β

α(−r sin t, r cos t) · (−r sin t, r cos t)dt

=

∫β

αr2dt

= r2(β−α)

∫

γg · d(x,y) =

∫

γ(x,y) · d(x,y)

=

∫β

α(r cos t, r sin t) · (−r sin t, r cos t)dt

=

∫β

α0dt

= 0.

γ(t) = tv ∈ Rn, t ∈ [α,β],

f(x) = x, x ∈ Rn.

∫

γf · dx =

∫

γx · dx =

∫β

αtv · vdt = ∥v∥2β

2 −α2

2.

γ = γ1 ⊕ γ2 : [α,β] → Rn C1 f, g : γ([α,β]) → Rn

λ,µ ∈ R∫

γ(λf+ µg) · dx = λ

∫

γf · dx+ µ

∫

γg · dx

∫

γ1⊕γ2

f · dx =

∫

γ1

f · dx+∫

γ2

f · dx

∣∣∣∣∫

γf · dx

∣∣∣∣ ≤ ∥f∥∞ L(γ) ∥f∥∞ := max {∥f(x)∥ : x ∈ γ([α,β])}

✷

C1 γ : [α,β] → Rn f : γ([α,β]) → Rn

f γ−

γ ∫

γ−f · dx = −

∫

γf · dx.

∫

γ−f · dx =

∫β

αf(γ−(t)) · (γ−) ′(t)dt

= −

∫β

αf(γ(α+β− t)) · γ ′(α+β− t)dt

= −

∫β

αf(γ(τ)) · γ ′(τ)dτ

= −

∫

γf · dx

✷

γ : [α,β] → Rn C1 f : γ([α,β]) → Rn

ϕ : [A,B] → [α,β] C1

γ ◦ϕ : [A,B] → Rn

C1

∫

γ◦ϕf · dx =

∫

γf · dx.

γ ◦ ϕγ ϕ

f γ ◦ϕ

∫

γ◦ϕf · dx =

∫B

Af((γ ◦ϕ)(τ)

)· (γ ◦ϕ) ′(τ)dτ

=

∫ϕ(β)

ϕ(α)(f ◦ γ)(ϕ(τ)) · γ ′(ϕ(τ))ϕ ′(τ)dτ

=

∫β

αf(γ(t)) · γ ′(t)dt

=

∫

γf · dx.

✷

C1 C ⊂ Rn C ⊂ Rn

C1

γ : [α,β] → Rn γ([α,β]) = Cγ C

f C ⊂ Rn

f γ∫

Cf · dx :=

∫

γf · dx.

C3

γ

γ−(t) = γ(α+β− t) =(r cos(α+β− t), r sin(α+β− t)

), t ∈ [α,β].

α > 0 C = γ([α,β]) ⊂ R2

ζ(t) = γ(t2) =(r cos(t2), r sin(t2)

), t ∈ [

√α,√β], ζ

([√α,√β])

= C.

∫

γ−(−y, x) · d(x,y)

=

∫β

α

(− r sin(α+β−t), r cos(α+β−t)

)·(r sin(α+β−t),−r cos(α+β−t)

)dt

= −

∫β

αr2dt

= −r2(β−α)

∫

γ−(x,y) · d(x,y)

=

∫β

α

(r cos(α+β−t), r sin(α+β−t)

)·(r sin(α+β−t),−r cos(α+β−t)

)dt

= 0.

∫

ζ(−y, x) · d(x,y)

=

∫√β√α

(− r sin(t2), r cos(t2)

)·(− r sin(t2)2t, r cos(t2)2t

)dt

= r2∫√β√α2tdt

= r2(β−α)

∫

ζ(x,y) · d(x,y)

=

∫√β√α

(r cos(t2), r sin(t2)

)·(− r sin(t2)2t, r cos(t2)2t

)dt

= 0,

C1

ϕ

ϕ(t) = t2, t ∈[√α,√β], ϕ

([√α,√β])

= [α,β] ζ = γ ◦ϕ,

ϕ ′(t) = 2t > 0 ∀ t ∈[√α,√β].

γ = γ1 ⊕ · · · ⊕ γk : [α,β] → Rn C1

f : γ([α,β]) → Rn

f γi i = 1, . . . , k

∫

γf · dx :=

k∑

i=1

∫

γi

f · dx

f γ

C1

C1

C1,C2

∫

Ci

(y, x− y) · d(x,y)∫

Ci

(y,y− x) · d(x,y), i = 1, 2, 3,

Ci ⊂ R2

C1 (0, 0) (0, 1) (1, 1)

C2 (0, 0) (1, 0) (1, 1)

C3 y = x2 (0, 0) (1, 1)

f = (f1, f2) : R2 → R2

∫

C1

f · d(x,y) =∫

γ1

f · d(x,y) +∫

γ2

f · d(x,y),∫

C2

f · d(x,y) =∫

γ3

f · d(x,y) +∫

γ4

f · d(x,y),∫

C3

f · d(x,y) =∫

γ5

f · d(x,y),

γ1(t) = (t, 0), t ∈ [0, 1], γ2(t) = (1, t), t ∈ [0, 1],

γ3(t) = (0, t), t ∈ [0, 1], γ4(t) = (t, 1), t ∈ [0, 1],

γ5(t) = (t, t2), t ∈ [0, 1].

∫

C1

f · d(x,y) =∫1

0f1(t, 0)dt+

∫1

0f2(1, t)dt,

∫

C2

f · d(x,y) =∫1

0f2(0, t)dt+

∫1

0f1(t, 1)dt,

∫

C3

f · d(x,y) =∫1

0f1(t, t

2) + 2tf2(t, t2)dt

f(x,y) = (f1(x,y), f2(x,y)) = (y, x− y)

∫

C1

(y, x− y) · d(x,y) =∫1

0(1− t)dt =

1

2,

∫

C2

(y, x− y) · d(x,y) =∫1

0(−t)dt+

∫1

01dt = −

1

2+ 1 =

1

2,

∫

C3

(y, x− y) · d(x,y) =∫1

0

(t2 + 2t(t− t2)

)dt =

1

3+ 2

1

3− 2

1

4=

1

2,

f(x,y) = (f1(x,y), f2(x,y)) = (y,y− x)

∫

C1

(y,y− x) · d(x,y) =∫1

0(t− 1)dt = −

1

2,

∫

C2

(y,y− x) · d(x,y) =∫1

0tdt+

∫1

01dt =

3

2,

∫

C3

(y,y− x) · d(x,y) =∫1

0

(t2 + 2t(t2 − t)

)dt = −

1

3+

1

2=

1

6.

R2

(x,y) 4→ (y, x− y) (x,y) 4→ (y,y− x), (x,y) ∈ R2,

Ci ⊂ R2

Ci

∫

γ(y, x) · d(x,y) γ(t) = (t, t2) t ∈ [0, 1]

∫

γ(x2,y2) · d(x,y) γ(t) = (2t, 4t) t ∈ [0, 1]

∫

γ(ex, ey) · d(x,y) γ(t) = (

√t, t) t ∈ [0, 1]

∫

γ(xy,yex) · d(x,y) γ

(0, 0) (2, 0) (2, 1) (0, 1) (0, 0)∫

γ(y− x,−y, 1) · d(x,y, z) γ(t) = (− sin t, cos t, 0) t ∈ [0, 2π]

∫

γ(x2+ 5y+ 3yz, 5x+ 3xz− 2, 3xy− 4z) ·d(x,y, z) γ(t) = (sin t, cos t, t)

t ∈ [0, 2π]

U ⊂ Rn f : U → Rn

ϕ : U → R

f(x) = gradϕ(x) ∀ x ∈ U.

ϕ f

f(x) = −Gmx

∥x∥3 , x ∈ R3 \ {0}, G,m > 0,

1x ∈ R3 \ {0} m

0 Gf(x) x

0 x

∥f(x)∥ =Gm

∥x∥2 ∀ x ∈ Rn \ {0},

f(x) = gradϕ(x) ∀ x ∈ R3 \ {0}, ϕ : R3 \ {0} → R, ϕ(x) =Gm

∥x∥ ,

−ϕ

U ⊂ Rn

a, b ∈ U γ : [α,β] → Rn

γ([α,β]) ⊂ U γ(α) = a γ(β) = b

U ⊂ Rn f : U → Rn

ϕ : U → R f

{ϕ+ c : c ∈ R}.

gradϕ = f grad (ϕ+ c) = fψ : U → R gradψ = f g : U → R g = ψ−ϕ

gradg(x) = 0 ∀ x ∈ U.

−ϕ ϕf

Rn

x0, x ∈ U γ = γ1 ⊕ · · ·⊕ γkγi(t) = xi−1 + t(xi − xi−1) ∈ U ∀ t ∈ [0, 1], i = 1, . . . , k, xk = x,

x0 x

g(x)− g(x0) =k∑

i=1

g(xi)− g(xi−1) =k∑

i=1

g(γi(1))− g(γi(0))

=k∑

i=1

(g ◦ γi) ′(t) =k∑

i=1

gradg(γi(t)) · (γi) ′(t) = 0,

g(x) = g(x0) ∀ x ∈ U,

ψ = ϕ U

✷

U ⊂ Rn f : U → Rf

a, x ∈ U fC1 U

a x

U ⊂ Rn f : U → Rf

C1

U

⇒: γ : [α,β] → Rn C1 γ([α,β]) ⊂U γ(α) = γ(β) γ−

∫

γf · dx =

∫

γ−f · dx = −

∫

γf · dx,

∫

γf · dx = 0.

⇐: C1 γ1, γ2γ = γ1 ⊕ γ−2 C1

0 =

∫

γf · dx =

∫

γ1

f · dx+∫

γ−2

f · dx =

∫

γ1

f · dx−∫

γ2

f · dx.

✷

U ⊂ Rn

U ⊂ Rn f : U → Rn

ff

ϕ : U → R fa, x ∈ U C1

γ : [α,β] → Rn γ([α,β]) ⊂ U, γ(α) = a, γ(β) = x

∫

γf(y) · dy = ϕ(x)−ϕ(a).

⇒: ϕ : U → R f = gradϕ a, x ∈ U γ

P = {t0, . . . , tk}, α = t0 < · · · < tk = β,

[α,β] γi := γ|[ti−1,ti] i = 1, . . . , k

∫

γf · dy =

k∑

i=1

∫

γi

gradϕ · dy =k∑

i=1

∫ti

ti−1

gradϕ(γi(t)) · γ ′i(t)dt

=k∑

i=1

∫ti

ti−1

(ϕ ◦ γi) ′(t)dt =k∑

i=1

((ϕ ◦ γi)(ti)− (ϕ ◦ γi)(ti−1)

)

=k∑

i=1

((ϕ ◦ γ)(ti)− (ϕ ◦ γ)(ti−1)

)= (ϕ ◦ γ)(β)− (ϕ ◦ γ)(α)

= ϕ(x)−ϕ(a),

f

a x⇐: a ∈ U ϕ : U → R

ϕ(x) :=

∫

γf · dy ∀ x ∈ U,

γ : [α,β] → Rn C1 γ([α,β]) ⊂ Uγ(α) = a γ(β) = x

x ∈ U U ⊂ Rn ϕR

gradϕ(x) = f(x) x ∈ U

limh→0

ϕ(x+ h)−ϕ(x)− f(x) · h∥h∥ = 0 ∀ x ∈ U.

x ∈ U U ε > 0B(x, ε) ⊂ U

γh(t) := x+ th ∈ B(x, ε) ⊂ U, t ∈ [0, 1], h ∈ B(0, ε),

ϕ(x+ h) =

∫

γ⊕γh

f · dy =

∫

γf · dy+

∫

γh

f · dy

= ϕ(x) +

∫

γh

f · dy

= ϕ(x) + f(x) · h+

∫

γh

(f(y)− f(x)

)· dy

= ϕ(x) + f(x) · h+

∫1

0

(f(x+ th)− f(x)

)· hdt,

γ⊕ γh C1 Ua x+ h

∣∣∣∫1

0

(f(x+ th)− f(x)

)· hdt

∣∣∣ ≤ ∥h∥ maxt∈[0,1]

∥∥f(x+ th)− f(x)∥∥,

h → 0 f⇐

ϕ(a) = 0 ✷

U ⊂ Rn f : U → Rn

Df(x) ∈ Rn×n

x ∈ U

f ϕ : U → R f = gradϕf

ϕ

Df(x) = Hϕ(x) ∀ x ∈ U,

✷

f : U → Rn U ⊂ Rn

U ⊂ Rn f

U ⊂ Rn

x0 ∈ U x ∈ U x0 xU

∃ x0 ∈ U ∀ x ∈ U : {x0 + t(x− x0) : t ∈ [0, 1]} ⊂ U.

Rn

Rn

R2 \ (−∞, 0]× {0}

U ⊂ Rn

U ⊂ Rn f : U → Rn

Df(x) ∈ Rn×n x ∈ U f

x0 ∈ U γx([0, 1]) ⊂ U x ∈ Uγx(t) := x0 + t(x− x0) t ∈ [0, 1]

ϕ(x) :=

∫

γx

f · dy =

∫1

0f(x0 + t(x− x0))dt · (x− x0), x ∈ U,

′

x Df(x) ∈ Rn×n

t

gradϕ(x)−

∫1

0f(x0 + t(x− x0))dt

U ⊂ Rn x1, x2 ∈ Ux1 x2 U ∀ x1, x2 ∈ U : {x1 + t(x2 − x1) : t ∈ [0, 1]} ⊂ U

= (x− x0)D

∫1

0f(x0 + t(x− x0))dt

= (x− x0)

∫1

0D(f(x0 + t(x− x0))

)dt

= (x− x0)

∫1

0t(Df)(x0 + t(x− x0))dt

=

∫1

0t(Df)(x0 + t(x− x0))(x− x0)dt

=

∫1

0td

dt

(f(x0 + t(x− x0))

)dt

=[tf(x0 + t(x− x0))

]1t=0

−

∫1

0f(x0 + t(x− x0))dt

= f(x)−

∫1

0f(x0 + t(x− x0))dt,

tt

f ✷

B ⊂ Rn A = [a,b]×Bf : A → R

F : [a,b] → R, F(x) =

∫

Bf(x, y)dy,

∫b

aF(x)dx =

∫

Af(x, y)d(x, y) =

∫

B

( ∫b

af(x, y)dx

)dy.

∂∂xf : A → R F

F ′(x) =∫

B

∂

∂xf(x, y)dy ∀ x ∈ [a,b].

f [a,b]×B f(x, ·) : B → Rx ∈ [a,b], B

F(x) x ∈ [a,b] f(·, y) : [a,b] → R∫ba f(x, y)dx y ∈ B

A (xν, yν) ⊂ A[a,b] (xkν) x0 ∈ [a,b] xkν → x0

(ykν) ⊂ B B(ykℓν ) y0 ∈ B ykℓν → y0 (xkℓν , ykℓν ) → (x0, y0) ∈ A

A f : A → R fε > 0 δ > 0

(x, y), (x0, y) ∈ A ∥(x, y)− (x0, y)∥ = |x− x0| < δ|f(x, y)− f(x0, y)| < ε/v(B)

|F(x)− F(x0)| =

∣∣∣∣∫

Bf(x, y)dy−

∫

Bf(x0, y)dy

∣∣∣∣ =∣∣∣∣∫

B(f(x, y)− f(x0, y))dy

∣∣∣∣

≤∫

B|f(x, y)− f(x0, y)|dy ≤ εv(B) ∀ x, x0 ∈ [a,b], |x− x0| < δ,

FA = {(x, y) : y ∈ B,a ≤ x ≤ b}

f : A → R

x ∈ [a,b] y ∈ B

Fx ∈ [a,b]

x x, x+h ∈ [a,b] h > 0

F(x+ h)− F(x)

h=

∫

B

f(x+ h, y)− f(x, y)

hdy =

∫

B

∂f

∂x(x+ ϑ(x, y)h, y)dy

=

∫

B

∂f

∂x(x, y)dy+

∫

B

(∂f

∂x(x+ ϑ(x, y)h, y)−

∂f

∂x(x, y)

)dy

ϑ(x, y) ∈ (0, 1) ∂f∂x : A → Rε > 0 δ > 0

yν → y (x, yν) → (x, y) f(x, yν) → f(x, y)

y ∈ B x, x0 ∈ [a,b] |x− x0| < δ

y ∈ B |h| < δ∣∣∣∣∂f

∂x(x+ ϑ(x, y)h, y)−

∂f

∂x(x, y)

∣∣∣∣ < ε/v(B),

∣∣∣∣∫

B

(∂f

∂x(x+ ϑ(x, y)h, y)−

∂f

∂x(x, y)

)dy

∣∣∣∣ < ε ∀ |h| < δ,

0 h → 0

✷

U ⊂ Rn fU f

f : Rn \ {0} → Rn, f(x) =x

∥x∥k , k ∈ N,

C1

f : R2 \ {(0, 0)}, f(x,y) =( −y

x2 + y2,

x

x2 + y2

)

Df(x) =1

(x2 + y2)2

(2xy y2 − x2

y2 − x2 −2xy

)∀ (x,y) ∈ R2 \ {(0, 0)},

γ(t) = (cos t, sin t) t ∈ [0, 2π]∫

γf(x,y) · d(x,y) =

∫2π

0f(γ(t)) · γ ′(t)dt

=

∫2π

0(− sin t, cos t) · (− sin t, cos t)dt

= 2π

f U1 = R2 \ {(x, 0) : x ≤ 0} U2 = R2 \ {(x, 0) : 0 ≤ x}

ϕ1 ϕ2 f U1 U2

ϕi i = 1, 2 f|Ui

ai ∈ Ui ϕi(x) ϕi

x ∈ Ui fC1 γi ai x

Ui

ϕi(x) =

∫

γi

f(y) · dy, x ∈ Ui, i = 1, 2.

ϕi : Ui →R f|Ui

(1, 0) ∈ U1 (−1, 0) ∈ U2

(0,y) ∈ U1 ∩U2 y > 0 γi

γi(t) = ((−1)i+1, 0) + t((−1)i,y), t ∈ [0, 1], i = 1, 2,

ϕi(0,y) =

∫1

0f((−1)i(t− 1),yt) · ((−1)i,y)dt

=

∫1

0

1

(t− 1)2 + y2t2(−yt, (−1)i(t− 1)) · ((−1)i,y)dt

= (−1)i+1y

∫1

0

1

(t− 1)2 + y2t2dt ∀ y > 0,

U = R2 \ {(0, 0)}U1 U2 f

f U

⇐

f : U → Rn U ⊂ Rn

a ∈ Uϕ : U → R x ∈ U

ϕ(x) =

∫

γf(y) · dy

C1 γ U ax

aγ U f

U = (x1, x2)× (y1,y2) ⊂ R2 ϕ :U → R f = (f1, f2) : U → R2 (x,y) ∈ U

(x0,y0) ∈ Uf γ = γ1 ⊕ γ2 γ1 (x0,y0)(x,y0) γ2 (x,y0) (x,y)

γ1(t) = (x0,y0) + t(x− x0, 0), γ2(t) = (x,y0) + t(0,y− y0), t ∈ [0, 1].

ϕ(x,y) =

∫

γf(ξ,η) · d(ξ,η)

=

∫1

0f1(x0 + t(x− x0),y0)(x− x0)dt

+

∫1

0f2(x,y0 + t(y− y0))(y− y0)dt

=

∫x

x0

f1(τ,y0)dτ+

∫y

y0

f2(x, τ)dτ,

gradϕ = f U

f = (f1, . . . , fn) : U → R U ⊂ Rn

ϕ : U → R gradϕ = f

∂

∂xiϕ(x1, . . . , xn) = fi(x1, . . . , xn) ∀ x = (x1, . . . , xn) ∈ U, ∀ i = 1, . . . ,n.

x1i = 1

ϕ(x1, . . . , xn) = F1(x1, . . . , xn) + g1(x2, . . . , xn),

F1(x1, . . . , xn) :=

∫f1(x1, . . . , xn)dx1

g1 x1 x2, . . . , xnx2

i = 2

∂

∂x2ϕ(x1, . . . , xn) =

∂

∂x2

(F1(x1, . . . , xn) + g1(x2, . . . , xn)

)= f2(x1, . . . , xn),

g1 x2

g1(x2, . . . , xn) = F2(x1, . . . , xn) + g2(x3, . . . , xn),

F2(x1, . . . , xn) :=

∫ (f2(x1, . . . , xn)−

∂

∂x2F1(x1, . . . , xn)

)dx2

g2 x3, . . . , xn

ϕ(x1, . . . , xn) = F1(x1, . . . , xn) + F2(x1, . . . , xn) + g2(x3, . . . , xn).

n− 1

ϕ(x1, . . . , xn) =n−1∑

i=1

Fi(x1, . . . , xn) + gn−1(xn),

xn i = n

∂

∂xnϕ(x1, . . . , xn) =

∂

∂xn

(n−1∑

i=1

Fi(x1, . . . , xn) + gn−1(xn))= fn(x1, . . . , xn),

xn

gn−1(xn) = Fn(x1, . . . , xn) + gn,

Fn(x1, . . . , xn) :=

∫ (fn(x1, . . . , xn)−

n−1∑

i=1

∂

∂xnFi(x1, . . . , xn)

)dxn

gn ∈ Rϕ f

ϕ(x1, . . . , xn) =n∑

i=1

Fi(x1, . . . , xn).

gradϕ = f U

f(x,y) = (y, x− y) ∈ R2, (x,y) ∈ R2,

R2 f

Df(x,y) =

(0 11 −1

)∀ (x,y) ∈ R2.

fC1

ϕ : R2 → R f : R2 → R2

(x0,y0) = (0, 0)γ = γ1 ⊕ γ2γ1(t) = (tx, 0), γ2(t) = (x, ty), t ∈ [0, 1], (x,y) ∈ R2,

ϕ(x,y) =

∫

γ(η, ξ− η) · d(ξ,η)

=

∫1

0(0, tx− 0) · (x, 0)dt+

∫1

0(ty, x− ty) · (0,y)dt

= xy−1

2y2

(x,y) ∈ R2

gradϕ(x,y) = (y, x− y) = f(x,y) ∀ (x,y) ∈ R2.

ϕ : R2 → R gradϕ = f

∂

∂xϕ(x,y) = y,

∂

∂yϕ(x,y) = x− y.

x

ϕ(x,y) = xy+ g(y).

y

∂

∂yϕ(x,y) = x+ g ′(y) = x− y,

g ′(y) = −y g(y) = −12y

2 + c c ∈ R g(y)

f

ϕ(x,y) = xy−1

2y2, (x,y) ∈ R2,

f

f : U → R U ⊂ R2

f(x,y) =(f1(x,y), f2(x,y)

)=(−

tany

x2+ 2xy+ x2,

1

x cos2 y+ x2 + y2

),

x = 0 cosy = 0

U ={(x,y) ∈ R2 : x = 0 y = kπ+

π

2∀ k ∈ Z

}

=⋃

k∈Z

{(x,y) ∈ R2 : kπ+

π

2< y < (k+ 1)π+

π

2

}\ {0}× R.

U

Uf

U

ff

∂

∂yf1(x,y) = −

1

x2 cos2 y+ 2x =

∂

∂xf2(x,y) ∀ (x,y) ∈ U.

fϕ : U → R f

gradϕ = f

∂

∂xϕ(x,y) = −

tany

x2+ 2xy+ x2,

∂

∂yϕ(x,y) =

1

x cos2 y+ x2 + y2 ∀ (x,y) ∈ U.

x

ϕ(x,y) =tany

x+ x2y+

1

3x3 + g(y),

yg ′(y)

∂

∂yϕ(x,y) =

1

x cos2 y+ x2 + g ′(y) =

1

x cos2 y+ x2 + y2,

g(y) = 13y

3 + c c ∈ R g

ϕ(x,y) =tany

x+ x2y+

1

3x3 +

1

3y3, (x,y) ∈ U.

ϕ fU

f1(x,y) = (12xy+ 3, 6x2),

f2(x,y) = (xy,y),

f3(x,y) = (3x2y, x3),

g1(x,y, z) = (x,y, z),

g2(x,y, z) = (x2y, zex, xy ln z),

g3(x,y, z) = (x+ z,−y− z, x− y).

a,b ∈ R a < b ϕ : [a,b] → RC1

P = {t0, . . . , tk} k ∈ N [a,b] a = t0 < . . . < tk = bϕ|[ti−1,ti] i = 1, . . . ,k

0x

B = {(x,y) ∈ R2 : a ≤ x ≤ b, ϕ1(x) ≤ y ≤ ϕ2(x)},

a,b ∈ R a < b ϕ1,ϕ2 : [a,b] → R ϕ1 ≤ ϕ2

C1 0x ϕ1,ϕ2

0y

C = {(x,y) ∈ R2 : c ≤ y ≤ d, ψ1(y) ≤ x ≤ ψ2(y)},

c,d ∈ R c < d ψ1,ψ2 : [c,d] → R ψ1 ≤ ψ2

C1 0y ψ1,ψ2

D ⊂ R2 C1 C1

0x C1

0y

D ⊂ R2 C1 ∂D

C1

C1

U ⊂ R2 D ⊂ U (f1, f2) : U → R2

∫

∂D(f1, f2) · d(x,y) =

∫

D

(∂f2∂x

−∂f1∂y

).

D∂f2∂x , ∂f1∂y D

∫

D

(∂f2∂x

−∂f1∂y

)=

∫d

c

( ∫ψ2(y)

ψ1(y)

∂f2∂x

(x,y)dx)dy−

∫b

a

( ∫ϕ2(x)

ϕ1(x)

∂f1∂y

(x,y)dy)dx

=

∫d

c

(f2(ψ2(y),y)− f2(ψ1(y),y)

)dy

−

∫b

a

(f1(x,ϕ2(x))− f1(x,ϕ1(x))

)dx.

D0x

∂D = γ([t1, t2])

γ = γ1 ⊕ γ2 ⊕ γ−3 ⊕ γ−4 ,

γ1(t) = (t,ϕ1(t)), t ∈ [a,b],

γ2(t) =(b,ϕ1(b) + t(ϕ2(b)−ϕ1(b))

), t ∈ [0, 1],

γ3(t) = (t,ϕ2(t)), t ∈ [a,b],

γ4(t) =(a,ϕ1(a) + t(ϕ2(a)−ϕ1(a))

), t ∈ [0, 1],

0y ∂D = δ([t3, t4])

δ = δ−1 ⊕ δ2 ⊕ δ3 ⊕ δ−4 ,

δ1(t) = (ψ1(t), t), t ∈ [c,d],

δ2(t) =(ψ1(c) + t(ψ2(c)−ψ1(c)), c

), t ∈ [0, 1],

δ3(t) = (ψ2(t), t), t ∈ [c,d],

δ4(t) =(ψ1(d) + t(ψ2(d)−ψ1(d)),d

), t ∈ [0, 1],

a b x

c

D

d

y

γ−4

γ−3

γ2

γ1

a b x

c

D

d

y

δ−1

δ−4

δ3

δ2

∂D C1 D 0x0y

ϕ1,ϕ2 ψ1,ψ2

(f1, f2) ∂D∂D

∫

∂D(f1, f2) · d(x,y) =

∫

∂D(f1, 0) · d(x,y) +

∫

∂D(0, f2) · d(x,y)

∫

∂D(f1, 0) · d(x,y) =

∫

γ1

(f1, 0) · d(x,y) +∫

γ2

(f1, 0) · d(x,y)

−

∫

γ3

(f1, 0) · d(x,y)−∫

γ4

(f1, 0) · d(x,y)

=

∫b

af1(t,ϕ1(t))dt−

∫b

af1(t,ϕ2(t))dt

∫

∂D(0, f2) · d(x,y) = −

∫

δ1

(0, f2) · d(x,y) +∫

δ2

(0, f2) · d(x,y)

+

∫

δ3

(0, f2) · d(x,y)−∫

δ4

(0, f2) · d(x,y)

= −

∫f2(ψ1(t), t)dt+

∫d

cf2(ψ2(t), t)dt.

γ1, γ3 δ1, δ3C1

✷

f1, f2,∂f2∂x , ∂f1∂y : U → R

D ⊂ R2 C1 ∂Dv(D) D

v(D) =1

2

∫

∂D(−y, x) · d(x,y).

✷

C1

D ⊂ R2

D ⊂ R2

C1

D =⋃k

i=1Di ⊂ R2 Di C1 i = jDi ∩Dj = ∅ Di ∩Dj = ∂Di ∩ ∂Dj U ⊂ R2

D ⊂ U (f1, f2) : U → R2

∫

∂D(f1, f2) · d(x,y) =

∫

D

(∂f2∂x

−∂f1∂y

),

∂D

C1

k∑

i=1

∫

∂Di

(f1, f2) · d(x,y) =k∑

i=1

∫

Di

(∂f2∂x

−∂f1∂y

)=

∫

D

(∂f2∂x

−∂f1∂y

),

∂Di Di

∂Di C1

∂DDi

✷

∂Dj ∩ ∂Di = ∅

∂D∫

∂D(2y, 6x) · d(x,y) D = [0, 1]× [0, 1]

∫

∂D(y2, 2x) · d(x,y) D = [0, 1]× [0, 1]

∫

∂D(xy, x− y) · d(x,y) D = [0, 1]× [1, 3]

∫

∂D(x− y3, x3 − y2) · d(x,y) D

∫

∂Dex(siny, cosy) · d(x,y) D (0, 0)

3

R2

(0, 0) r > 0

x2

a2 + y2

b2 = 1 a,b > 0

r(t− sin t, 1− cos t) t ∈ [0, 2π]r > 0 0x

D = B((0, 0),R) \B((0, 0), r), R > r > 0,

f(x,y) = (2x3 − y3, x3 + y3), (x,y) ∈ R2.

γ : [α,β] → Rn α,β ∈ R α < β C1

f : γ([α,β]) → R

∫

γfds :=

∫β

αf(γ(t))∥γ ′(t)∥dt

f γ

f : γ([α,β]) → R f = 1

∫

γ1ds = L(γ),

L(γ) γ

C1

γ = γ1 ⊕ γ2 : [α,β] → Rn C1 f,g : γ([α,β]) → Rϕ : [A,B] → [α,β] C1 λ,µ ∈ R

∫

γ◦ϕfds =

∫

γfds

∫

γ(λf+ µg)ds = λ

∫

γfds+ µ

∫

γgds

∫

γ1⊕γ2

fds =

∫

γ1

fds+

∫

γ2

fds

∣∣∣∣∫

γfds

∣∣∣∣ ≤ ∥f∥∞ L(γ) ∥f∥∞ := max {|f(x)| : x ∈ γ([α,β])}

✷

γ = γ1 ⊕ · · · ⊕ γk : [α,β] → Rn C1

f : γ([α,β]) → Rn

f γi i = 1, . . . , k

∫

γfds :=

k∑

i=1

∫

γi

fds

f γ

C = γ([α,β]) ⊂ Rn

C1

∫

Cfds :=

∫

γfds.

∫

γ(x+ y)ds γ(t) = (cos t, sin t) t ∈ [0,π]

∫

γ

x

x2 + y2ds γ(t) = (cos t, sin t) t ∈

[−π

2,π

2

]

∫

γ(x2 + y)ds γ(t) = (t, cosh t) t ∈ [0, 1]

∫

γ

√x2 + y2 + z2ds γ(t) = (t cos t, t sin t, t) t ∈ [0, 2π]

R3

R3

K ⊂ R2 U ⊂ R2

K ⊂ U Φ : U → R3

Φ|K : K → R3 Φ

K Φ(K) ⊂ R3

Φ|K

S = Φ(K) ⊂ R3

Φ|K : K → R3

S

Φ|K

ΦK ⊂ U Φ ∈ C1(U;R3)

S = Φ(K) ⊂ R3 ΦΦ R3

U ⊂ R2

K ⊂ R2 ΦΦ

S ⊂ R3

U ⊂ R2

S ⊂ R3

ΦΦ(u, v) = (x0,y0, z0) ∈ R3 (u, v) ∈ R2

S = {(x0,y0, z0)} ⊂ R3

R3

Φ : U → R3

U ⊂ R2

Φ(u, v) =

⎛

⎝x(u, v)y(u, v)z(u, v)

⎞

⎠ ∈ R3, (u, v) ∈ U,

DΦ(u, v) =

⎛

⎜⎜⎜⎜⎜⎜⎝

∂x

∂u(u, v)

∂x

∂v(u, v)

∂y

∂u(u, v)

∂y

∂v(u, v)

∂z

∂u(u, v)

∂z

∂v(u, v)

⎞

⎟⎟⎟⎟⎟⎟⎠∈ R3×2, (u, v) ∈ U,

(u, v) ∈ U(u, v) ∈ U

DΦ(u, v) ∈ R3×2

R3 Φ uv

∂Φ

∂u(u, v) =

( ∂x∂u

(u, v),∂y

∂u(u, v),

∂z

∂u(u, v)

)T∈ R3, (u, v) ∈ U,

∂Φ

∂v(u, v) =

(∂x∂v

(u, v),∂y

∂v(u, v),

∂z

∂v(u, v)

)T∈ R3, (u, v) ∈ U,

DΦ(u, v) =

(∂Φ

∂u(u, v)

∂Φ

∂v(u, v)

)∈ R3×2, (u, v) ∈ U.

U ⊂ R2 f : U → RK ⊂ U

Φ(x,y) = (x,y, f(x,y)), (x,y) ∈ K,

K f|K

S = {(x,y, f(x,y)) : (x,y) ∈ K} = Γf|K ,

ΦΦ U

Φ

DΦ(x,y) =

(∂Φ

∂x(x,y)

∂Φ

∂y(x,y)

)=

⎛

⎜⎝

1 00 1

∂f

∂x(x,y)

∂f

∂y(x,y)

⎞

⎟⎠ , (x,y) ∈ U,

f ∈ C1(U)K

R3

f(x,y) = x2 + y2, (x,y) ∈ R2,

K ⊂ R2

K = B((0, 0), r) = {(x,y) ∈ R2 : x2 + y2 ≤ r2} (r > 0)

R3

S = {(x,y, x2 + y2) ∈ R3 : x2 + y2 ≤ r2},

Φ(x,y) = (x,y, x2 + y2), x2 + y2 ≤ r2,

DΦ(x,y) =

(∂Φ

∂x(x,y)

∂Φ

∂y(x,y)

)=

⎛

⎝1 00 12x 2y

⎞

⎠ , x2 + y2 ≤ r2.

Φ(λ,µ) =

⎛

⎝x(λ,µ)y(λ,µ)z(λ,µ)

⎞

⎠ =

⎛

⎝x0y0z0

⎞

⎠+ λ

⎛

⎝a1

a2

a3

⎞

⎠+ µ

⎛

⎝b1b2b3

⎞

⎠ , λ,µ ∈ R,

DΦ(λ,µ) =

(∂Φ

∂λ(λ,µ)

∂Φ

∂µ(λ,µ)

)=

⎛

⎝a1 b1a2 b2a3 b3

⎞

⎠

R3 a = (a1,a2,a3), b = (b1,b2,b3) ∈ R3

R3

= 0R3 a = b = 0

K ⊂ R2 Φ′

DΦ(λ,µ) 2 a, b

a = 0 c ∈ R b = ca λa+ µb = (λ+ µc)a λ,µ ∈ R2

Φ(R2) ⊂ R3

det

(a1 b1a2 b2

)= 0,

(x,y, z) ∈ Φ(K) (λ,µ) ∈ K (x,y, z) = Φ(λ,µ)

(λµ

)=

(λ(x,y)µ(x,y)

):=

(a1 b1a2 b2

)−1 (x− x0y− y0

),

(xy

)=

(a1 b1a2 b2

)(λµ

)+

(x0y0

)∈(a1 b1a2 b2

)K+

(x0y0

):= D ⊂ R2,

(x,y)

z = f(x,y) := z0 + λ(x,y)a3 + µ(x,y)b3.

(x,y, z) ∈ Φ(K) (x,y, z) ∈ Γf(D)(x,y, z) ∈ Γf(D) (λ,µ) ∈ K (x,y, z) ∈ Φ(K)

R3

det

(a1 b1a3 b3

)= 0 det

(a2 b2a3 b3

)= 0.

R3

f : U → R U ⊂ R2

(x0,y0, z0) ∈ R3 r > 0

∂B((x0,y0, z0), r) = {(x,y, z) ∈ R3 : ∥(x,y, z)− (x0,y0, z0)∥ = r}

= {(x,y, z) ∈ R3 : (x− x0)2 + (y− y0)

2 + (z− z0)2 = r2},

Φ(ϑ,ϕ) =

⎛

⎝x(ϑ,ϕ)y(ϑ,ϕ)z(ϑ,ϕ)

⎞

⎠ =

⎛

⎝x0y0z0

⎞

⎠+ r

⎛

⎝sin ϑ cosϕsin ϑ sinϕ

cos ϑ

⎞

⎠ , (ϑ,ϕ) ∈ R2,

DΦ(ϑ,ϕ) =

(∂Φ

∂ϑ(ϑ,ϕ)

∂Φ

∂ϕ(ϑ,ϕ)

)= r

⎛

⎝cos ϑ cosϕ − sin ϑ sinϕcos ϑ sinϕ sin ϑ cosϕ− sin ϑ 0

⎞

⎠

K = [0,π]× [0, 2π],

Φ(K) = ∂B((x0,y0, z0), r).

(x,y, z) ∈ Φ(K) ∥(x− x0,y− y0, z− z0)∥ = r(x,y, z) ∈ R3

ϑ = arccosz− z0

r, ϕ =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[0, 2π] x = x0, y = y0,π2 , x = x0, y > y0,3π2 , x = x0, y < y0,

0 2π x > x0, y = y0,

arctan y−y0x−x0

, x > x0, y > y0,

2π+ arctan y−y0x−x0

, x > x0, y < y0,

π+ arctan y−y0x−x0

, x < x0

arccos : [−1, 1] → [0,π] arctan : R → (−π2 ,π2 ) Φ(ϑ,ϕ) =

(x,y, z)DΦ(ϑ,ϕ) 2 ϑ ∈ (0,π)

R3

R3

R

a = (a1,a2,a3), b = (b1,b2,b3) ∈ R3

a× b =(a2b3 − b2a3,a3b1 − b3a1,a1b2 − b1a2

)

=( ∣∣∣∣

a2 a3

b2 b3

∣∣∣∣ ,−∣∣∣∣a1 a3

b1 b3

∣∣∣∣ ,∣∣∣∣a1 a2

b1 b2

∣∣∣∣)

=

∣∣∣∣∣∣

e1 e2 e3a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣

a, b

a, b, c ∈ R3 λ ∈ R

a∧ bR3 R3

ei i = 1, 2, 3 R3

a× b = −b× a

a× a = 0

(λa)× b = a× (λb) = λ(a× b)

a× (b+ c) = (a× b) + (a× c)

(a+ b)× c = (a× c) + (b× c)

a · (a× b) = b · (a× b) = 0

′

a · (a× b) =

∣∣∣∣∣∣

a1 a2 a3

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣, b · (a× b) =

∣∣∣∣∣∣

b1 b2 b3a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣

✷

a× ba b

a× b

e1 × e2 = e3,

′

a× ba b

∥a× b∥ = ∥a∥∥b∥ sin ϑ,

ϑ ∈ [0,π] a b

a× b a ba bϑ ∈ [0,π] a, b ∈ Rn \ {0}

cos ϑ =a · b

∥a∥∥b∥ .

b

∥a∥ sin ϑ a

ϑ

a b ∥a× b∥

a× ba b

R3

Φ K(u, v) ∈ K

N(u, v) :=∂Φ

∂u(u, v)× ∂Φ

∂v(u, v)

Φ Φ(u, v)

Φ(u, v) N(u, v) = 0 N(u, v) = 0

Φ(u, v)

n(u, v) :=N(u, v)

∥N(u, v)∥

Φ Φ(u, v)

{Φ(u, v) + λ

∂Φ

∂u(u, v) + µ

∂Φ

∂v(u, v) : λ,µ ∈ R

}⊂ R3

Φ Φ(u, v)

Φ

ΦK

f : U → R U ⊂ R2 K ⊂ U

N(x,y) =

⎛

⎜⎝

10

∂f(x,y)

∂x

⎞

⎟⎠×

⎛

⎜⎜⎝

01

∂f(x,y)

∂y

⎞

⎟⎟⎠ =

⎛

⎜⎜⎜⎜⎜⎝

−∂f(x,y)

∂x

−∂f(x,y)

∂y

1

⎞

⎟⎟⎟⎟⎟⎠= (−∇f(x,y), 1),

0z

n(x,y) =(−∇f(x,y), 1)√∥∇f(x,y)∥2 + 1

Φ(x,y) = (x,y, f(x,y)) (x,y) ∈ K

⎧⎪⎪⎨

⎪⎪⎩

⎛

⎝xy

f(x,y)

⎞

⎠+ λ

⎛

⎜⎝

10

∂f(x,y)

∂x

⎞

⎟⎠+ µ

⎛

⎜⎜⎝

01

∂f(x,y)

∂y

⎞

⎟⎟⎠ : λ,µ ∈ R

⎫⎪⎪⎬

⎪⎪⎭

2

a, b ∈ R3

N(x,y) = a× b,

x y

n(x,y) =a× b

∥a× b∥ ,

R3

N(ϑ,ϕ) = r2

⎛

⎝cos ϑ cosϕcos ϑ sinϕ− sin ϑ

⎞

⎠×

⎛

⎝− sin ϑ sinϕsin ϑ cosϕ

0

⎞

⎠ .

= r2

∣∣∣∣∣∣

e1 e2 e3cos ϑ cosϕ cos ϑ sinϕ − sin ϑ

− sin ϑ sinϕ sin ϑ cosϕ 0

∣∣∣∣∣∣= r2 sin ϑ

⎛

⎝sin ϑ cosϕsin ϑ sinϕ

cos ϑ

⎞

⎠

∥N(ϑ,ϕ)∥ = r2 sin ϑ,

n(ϑ,ϕ) =

⎛

⎝sin ϑ cosϕsin ϑ sinϕ

cos ϑ

⎞

⎠ ∀ (ϑ,ϕ) ∈ (0,π)× [0, 2π],

2

N(u, v) Φ Φ(u, v)α ′(0)

α = Φ ◦ γ : (−ε, ε) → R3 α((−ε, ε)) ⊂ Φ(U) α(0) = Φ(u, v)

ε > 0 U

γ = (γ1,γ2) : (−ε, ε) → R2 γ((−ε, ε)) ⊂ U γ(0) = (u, v)

t = 0α

α(0)

α ′(0) = DΦ(u, v)γ ′(0) =∂Φ

∂u(u, v)γ ′

1(0) +∂Φ

∂v(u, v)γ ′

2(0).

N(u, v)′

N(u, v) · α ′(0) = 0.

Φ(u, v)

R3

λ∂Φ

∂u(u, v) + µ

∂Φ

∂v(u, v) ∈ R3, λ,µ ∈ R.

γ(t) = (u, v) + t(λ,µ), t ∈ (−ε, ε),

ε > 0 ΦΦ(u, v)

Φ(u, v)

Φ(u, v)Φ(u0, v0) U0 ⊂ U

(u0, v0) ∈ U0 U

α : (−ε, ε) → R3 α((−ε, ε)) ⊂ Φ(U0) α(0) = Φ(u0, v0)

ε > 0 t = 0 U0 UΦ(u0, v0)

∂(x,y)

∂(u, v):=

⎛

⎜⎜⎝

∂x

∂u

∂x

∂v

∂y

∂u

∂y

∂v

⎞

⎟⎟⎠ ,∂(x, z)

∂(u, v),

∂(y, z)

∂(u, v)

(u0, v0) ∈ U

f(u, v,w) := Φ(u, v) + (0, 0,w)T ∈ R3, (u, v,w) ∈ U× R,

Df(u, v,w) =

⎛

⎜⎜⎜⎜⎜⎜⎝

∂x

∂u(u, v)

∂x

∂v(u, v) 0

∂y

∂u(u, v)

∂y

∂v(u, v) 0

∂z

∂u(u, v)

∂z

∂v(u, v) 1

⎞

⎟⎟⎟⎟⎟⎟⎠

(u0, v0, 0)

U0 ⊂ U (u0, v0) ∈ U0 δ, ε0 > 0 f

f : U0 × (−δ, δ) → B(Φ(u0, v0), ε0)

1− 1

f−1(B(Φ(u0, v0), ε0)∩ Φ(U0)) = U0 × {0}.

π : R3 → R2 π(x,y, z) = (x,y) R3 R2

ε > 0

γ := π ◦ f−1 ◦ α : (−ε, ε) → R2

t = 0 γ(0) = (u0, v0) γ((−ε, ε)) ⊂ U0 Φ ◦ γ = αt ∈ (−ε, ε)

ε > 0

f−1(α(t)) = (u(t), v(t), 0) (u(t), v(t)) ∈ U0

π f

(Φ ◦ γ)(t) = Φ(π(u(t), v(t), 0)) = Φ(u(t), v(t))

= f(u(t), v(t), 0) = f(f−1(α(t))) = α(t).

Φ(u, v) ΦU0 ⊂ U (u, v) ∈ U0

Φ(u, v) Φ(U0) ⊂ R3

Φ : U0 → R3 1− 1

Φ(U0) ⊂ R3 ΦΦ

1 − 1

R3

R3

1− 1

ΦR3

K

Φ(u, v) = c+ ua+ vb a, b, c ∈ R3 a, b(u, v) ∈ K = [0, 1]× [0, 1] Φ

S = Φ(K) = {c+ ua+ vb : u, v ∈ [0, 1]} ⊂ R3

{c+ λa+ µb : λ,µ ∈ R} ⊂ R3,

Φ S Φ(u, v) ∈ SΦ N(u, v) = a× b

ΦS ⊂ R3 ∥a× b∥

Φ : K → R3

Φ K R2

K kΦ [ui,ui +∆ui]× [vi, vi +

∆vi] i = 1, . . . , k K

Φi(u, v) = Φ(ui, vi) + u∂Φ

∂u(ui, vi) + v

∂Φ

∂v(ui, vi), (u, v) ∈ [0,∆ui]× [0,∆vi],

ΦΦ(ui, vi)

∥N(ui, vi)∥∆ui∆vi.

Φ : K → R3

k∑

i=1

∥N(ui, vi)∥∆ui∆vi.

K

Φ(K) ⊂ R3

N : K → R3

Φ

Φ K

A(Φ) :=

∫

K∥N(u, v)∥d(u, v) =

∫

K

∥∥∥∂Φ

∂u(u, v)× ∂Φ

∂v(u, v)

∥∥∥d(u, v)

Φ

Φ K

Φ

Φ S = Φ(K)

R3

Φ A(Φ)S = Φ(K) ⊂ R3 Φ

S ⊂ R3

Ψ|Tψ(T) = S ψ : V → R3 V ⊂ R2 T ⊂ V

Φ|K Ψ|T

U,V ⊂ R2 K ⊂ U T ⊂ Vg : V → U 1 − 1

Dg(s, t) ∈ R2×2 (s, t) ∈ V g(T) = K gK T

detDg(s, t) > 0 (s, t) ∈ T gdetDg(s, t) < 0 (s, t) ∈ T g

g KT g−1 : g(V) → U

T Kg

g : V → g(V) 1− 1

detDg(s, t) (s, t) ∈ VV V

U,V ⊂ R2 K ⊂ U T ⊂ Vg : V → U

K T ΦK Ψ = Φ ◦ g

T Ψ(T) = Φ(K) ⊂ R3 A(Ψ) = A(Φ)

Ψ : V → R3

V T Ψ : T → R3

T

g Ψ(T) = Φ(g(T)) = Φ(K)

A(Ψ) =

∫

T

∥∥∥∂Ψ

∂s(s, t)× ∂Ψ

∂t(s, t)

∥∥∥d(s, t).

DΨ(s, t) = DΦ(g(s, t))Dg(s, t)

(a b

)=(c d

) (α1 β1α2 β2

)=(α1c+α2d β1c+β2d

),

a× b = (α1c+α2d)× (β1c+β2d)

= α1c× (β1c+β2d) +α2d× (β1c+β2d)

= α1β2 c× d+α2β1 d× c

=

∣∣∣∣α1 β1α2 β2

∣∣∣∣ c× d

A(Ψ) =

∫

T

∥∥∥∂Φ

∂u(g(s, t))× ∂Φ

∂v(g(s, t))

∥∥∥ |detDg(s, t)|d(s, t),

A(Ψ) =

∫

K

∥∥∥∂Φ

∂u(u, v)× ∂Φ

∂v(u, v)

∥∥∥d(u, v).

A(Ψ) = A(Φ) ✷

S = Φ(K) ⊂ R3

S = Φ(K) ⊂ R3

Φ|K

A(S) := A(Φ).

R3

S := {(x,y, z) ∈ R3 : ∥(x,y, z)− (x0,y0, z0)∥ = r}, (x0,y0, z0) ∈ R3, r > 0,

Φ

A(S) = A(Φ) =

∫

K∥N(ϑ,ϕ)∥d(ϑ,ϕ) =

∫

[0,π]×[0,2π]r2 sin ϑd(ϑ,ϕ),

A(S) =

∫π

0

∫2π

0r2 sin ϑdϕdϑ = 4π r2.

r > 0x2 + y2 = (r − ε)2 ε ∈

(0, r)

ε→ 0

ε→ 0

Φ

A(Φ) =

∫

K

√

1+

(∂f

∂x

)2 ( ∂f∂y

)2

.

f : [a,b] → Rf 0x

Φ(u, v) = (u, f(u) cos v, f(u) sin v), (u, v) ∈ [a,b]× [0, 2π].

A(Φ) = 2π

∫b

af(u)

√1+ (f ′(u))2 du.

z = x2 + y2 z = 0 z = 4

2az = x2−y2 a > 0 0xy0 ≤ r ≤ a

√cosϕ 0 ≤ ϕ ≤ π

2

z = xy0xy

Aε 0 < ε < min{a,b}z =

√2xy [ε,a]× [ε,b] a,b > 0

A := limε→0Aε[0,a]× [0,b]

A

(0, 0, 0) R > 0 (x− R2 )

2 + y2 ≤R4

4

K h > 0y = ax a > 0 0xy 0x

Φ K

E :=∂Φ

∂u· ∂Φ∂u

, F :=∂Φ

∂u· ∂Φ∂v

, G :=∂Φ

∂v· ∂Φ∂v

,

Φ

∥N∥ =√

EG− F2 A(Φ) =

∫

K

√EG− F2.

1

Φ Kf : Φ(K) → R

∫

Φf dσ :=

∫

Kf(Φ(u, v))∥N(u, v)∥d(u, v)

=

∫

Kf(Φ(u, v))

∥∥∥∂Φ

∂u(u, v)× ∂Φ

∂v(u, v)

∥∥∥d(u, v)

f Φ

f ΦK

Φ : K → R3 f : Φ(K) → R N : K → R3

∥N∥ : K → R

Φ Kf(x,y, z) = 1 (x,y, z) ∈ Φ(K) Φ

A(Φ) =

∫

Φdσ :=

∫

Φ1dσ.

f Φ

f S = Φ(K) ⊂ R3

∫

Sf dσ :=

∫

Φf dσ,

Φ|K S ⊂ R3

f,g : Φ(K) → RΦ α,β ∈ R

∫

Φ(αf+βg)dσ = α

∫

Φf dσ+β

∫

Φgdσ.

f : R3 → R, f(x,y, z) = (x2 + y2)z,

S = {(x,y, z) ∈ R3 : x2 + y2 + z2 = r2, z ≥ 0}, r > 0,

R3

Φ(ϑ,ϕ) = r

⎛

⎝sin ϑ cosϕsin ϑ sinϕ

cos ϑ

⎞

⎠ , (ϑ,ϕ) ∈[0,π

2

]× [0, 2π],

Φ([0, π2 ]× [0, 2π]) = S ∥N(ϑ,ϕ)∥ = r2 sin ϑ

′

∫

Sf dσ =

∫

Φ(x2 + y2)z dσ

=

∫

[0,π2 ]×[0,2π](r2 sin2 ϑ cos2ϕ+ r2 sin2 ϑ sin2ϕ)r cos ϑ r2 sin ϑd(ϑ,ϕ)

= r5∫ π

2

0

∫2π

0sin3 ϑ cos ϑdϕdϑ = 2π r5

∫1

0s3 ds =

1

2π r5.

Rn R3

Φ Kf : Φ(K) → R3

∫

Φf · n dσ :=

∫

Kf(Φ(u, v)) · N(u, v)d(u, v)

=

∫

Kf(Φ(u, v)) · ∂Φ

∂u(u, v)× ∂Φ

∂v(u, v)d(u, v)

f Φ

K(f ◦ Φ) · N : K → R f

ΦΦ

∫

Kf(Φ(u, v)) · N(u, v)d(u, v) =

∫

Kf(Φ(u, v)) · n(u, v)∥N(u, v)∥d(u, v),

n(u, v) Φ Φ(u, v)

Φ

α,β ∈ Rf, g : Φ(K) → R3

∫

Φ(αf+βg) · n dσ = α

∫

Φf · n dσ+β

∫

Φg · n dσ.

U,V ⊂ R2 K ⊂ U T ⊂ VΦ Ψ = Φ ◦ g

K = g(T) T g : V → UK T S = Φ(K) = Ψ(T) ⊂ R3

f : S → R3

∫

Ψf · n dσ =

⎧⎪⎪⎨

⎪⎪⎩

∫

Φf · n dσ, g

−

∫

Φf · n dσ, g

Φ g Ψ = Φ ◦ gT Ψ(T) = Φ(K)

∫

Ψf · n dσ =

∫

Tf(Ψ(s, t)) · ∂Ψ

∂s(s, t)× ∂Ψ

∂t(s, t)d(s, t),

∫

Ψf · n dσ =

∫

Tf(Φ(g(s, t))) · ∂Φ

∂u(g(s, t))× ∂Φ

∂v(g(s, t)) detDg(s, t)d(s, t)

⎧⎨

⎩

detDg(s, t) > 0 ∀ (s, t) ∈ T g

detDg(s, t) < 0 ∀ (s, t) ∈ T g

✷

R3 S = Φ(K) ⊂ R3

n(u, v)f S = Φ(K) ⊂ R3

∫

Sf · n dσ :=

∫

Φf · n dσ,

Φ|K S ⊂ R3

n

N(u, v) = 0

Φ(u, v) = (u, v,uv)K f R3

f Φ

I =

∫

Φf · n dσ =

∫

KΦ(u, v) · N(u, v)d(u, v), N(u, v) =

∂Φ(u, v)

∂u× ∂Φ(u, v)

∂v,

Φ f K

N(u, v) =

⎛

⎝10v

⎞

⎠×

⎛

⎝01u

⎞

⎠ =

⎛

⎝−v−u1

⎞

⎠

I =

∫

K(u, v,uv) · (−v,−u, 1)d(u, v) = −

∫

Kuvd(u, v)

= −

∫1

0

∫2π

0r cosϕ r sinϕ r dϕdr = −

1

4

1

2

∫2π

0sin(2ϕ)dϕ = 0.

S R3

a > 0 f R3

f SS

S

f Sz ≥ 0 z ≤ 0

z

S x2 + y2 + z2 = R2 z ≥ 0 R > 0r(q) q ∈ S p = (0, 0, ζ) ∈ S

J(ζ) :=

∫

S

1

rdσ lim

ζ→∞ζ J(ζ).

S

∫

S(x2 + y2)dσ.

x2

a2+

y2

b2+

z2

c2= 1 a,b, c > 0

∫

S

√x2

a4+

y2

b4+

z2

c4dσ.

Φ(u, v) = (u cos v,u sin v, v)[0, 1]× [0, 2π] f(x,y, z) = (y,−x, 0)

f Φ

R3

U ⊂ R2 Φ : U → R3

C1 K ⊂ U∂K = γ([α,β]) γ : [α,β] → R2

C1 V ⊂ R3 Φ(K) ⊂ V f : V → R3

∫

Φcurl f · n dσ =

∫

Φ◦γf · d(x,y, z).

∫

Φ◦γf · d(x,y, z) =

∫β

αf((Φ ◦ γ)(t)) · (Φ ◦ γ) ′(t)dt

=

∫β

α(f ◦ Φ)(γ(t)) ·DΦ(γ(t))γ ′(t)dt

=

∫β

α

(f ◦ Φ · ∂Φ

∂u, f ◦ Φ · ∂Φ

∂v

)(γ(t)) · γ ′(t)dt,

v ·(a b

) (c1c2

)= v · (c1a+ c2b) = c1v · a+ c2v · b = (v · a, v · b) ·

(c1c2

)

curl f = rot f = ∇× f

v, a, b ∈ R3 c1, c2 ∈ R2

∫

Φ◦γf · d(x,y, z) =

∫

γ

(f ◦ Φ · ∂Φ

∂u, f ◦ Φ · ∂Φ

∂v

)· d(u, v).

[α,β] γ

∫

Φ◦γf · d(x,y, z) =

∫

K

(∂

∂u

(f ◦ Φ · ∂Φ

∂v

)−∂

∂v

(f ◦ Φ · ∂Φ

∂u

)).

∫

Φcurl f · n dσ =

∫

K(∇× f) ◦ Φ · ∂Φ

∂u× ∂Φ

∂v,

′

D = ∂∂u D = ∂

∂v

∂

∂u

(f ◦ Φ · ∂Φ

∂v

)=∂Φ

∂v· ∂∂u

(f ◦ Φ

)+ f ◦ Φ · ∂

∂u

∂Φ

∂v,

∂

∂v

(f ◦ Φ · ∂Φ

∂u

)=∂Φ

∂u· ∂∂v

(f ◦ Φ

)+ f ◦ Φ · ∂

∂v

∂Φ

∂u,

∂

∂u

(f ◦ Φ · ∂Φ

∂v

)−∂

∂v

(f ◦ Φ · ∂Φ

∂u

)=∂Φ

∂v· (Df) ◦ Φ∂Φ

∂u−∂Φ

∂u· (Df) ◦ Φ∂Φ

∂v.

a =∂Φ

∂u, b =

∂Φ

∂v, (Df) ◦ Φ =

⎛

⎝cx cy czdx dy dz

ex ey ez

⎞

⎠

︸ ︷︷ ︸= A

, (∇× f) ◦ Φ =

⎛

⎝ey − dz

cz − exdx − cy

⎞

⎠

︸ ︷︷ ︸= v

,

b ·Aa− a ·Ab = v · a× b,

b ·Aa− a ·Ab = b · (A−AT )a = b ·

⎛

⎝0 cy − dx cz − ex

dx − cy 0 dz − eyex − cz ey − dz 0

⎞

⎠ a

= b ·

⎛

⎝0 −v3 v2v3 0 −v1−v2 v1 0

⎞

⎠ a = v ·

⎛

⎝a2b3 − a3b2a3b1 − a1b3a1b2 − a2b1

⎞

⎠ = v · a× b.

✷

R3

S = {(x,y, z) ∈ R3 : x2 + y2 + z2 = 4, z ≥ 0}

f(x,y, z) = (2y, 3x,−z2), (x,y, z) ∈ R3.

S

Φ(ϑ,ϕ) = 2(sin ϑ cosϕ, sin ϑ sinϕ, cos ϑ), (ϑ,ϕ) ∈ K :=[0,π

2

]× [0, 2π].

Φ R2

K ⊂ R2 C1

∂K Kγ = γ1 ⊕ γ2 ⊕ γ−3 ⊕ γ−4

γ1(t) = (t, 0), t ∈[0,π

2

], γ2(t) =

(π2, t), t ∈ [0, 2π],

γ3(t) = (t, 2π), t ∈[0,π

2

], γ4(t) = (0, t), t ∈ [0, 2π],

f : R3 → R3

∇× f(x,y, z) =

∣∣∣∣∣∣

e1 e2 e3∂∂x

∂∂y

∂∂z

2y 3x −z2

∣∣∣∣∣∣=

⎛

⎝001

⎞

⎠ .

Φ

N(ϑ,ϕ) = 4 sin ϑ

⎛

⎝sin ϑ cosϕsin ϑ sinϕ

cos ϑ

⎞

⎠

∫

Φ∇× f · n dσ =

∫

K(0, 0, 1) · N(ϑ,ϕ)d(ϑ,ϕ) = 4

∫

Ksin ϑ cos ϑd(ϑ,ϕ)

= 2

∫2π

0

∫π/2

0sin(2ϑ)dϑdϕ = 2π

∫π

0sin ϑdϑ = 4π.

∂K γ = γ1 ⊕γ2 ⊕ γ−3 ⊕ γ−4

∫

Φ◦γf · d(x,y, z) =

∫

Φ◦γ1

f · d(x,y, z) +∫

Φ◦γ2

f · d(x,y, z)

−

∫

Φ◦γ3

f · d(x,y, z)−∫

Φ◦γ4

f · d(x,y, z)

=

∫

Φ◦γ2

f · d(x,y, z).

γ : [α,β] → Rn γ− : [α,β] → Rn

t ∈ [α,β]

(Φ ◦ γ−)(t) = Φ(γ−(t)) = Φ(γ(α+β− t)) = (Φ ◦ γ)(α+β− t) = (Φ ◦ γ)−(t).

Φ γ1, γ3γ4

(Φ ◦ γ1)(t) = (Φ ◦ γ3)(t) ∀ t ∈[0,π

2

], (Φ ◦ γ4) ′(t) = (0, 0) ∀ t ∈ [0, 2π].

(Φ ◦ γ2)(t) = 2(cos t, sin t, 0) ∀ t ∈ [0, 2π],

∫

Φ◦γf · d(x,y, z) = 4

∫2π

0(2 sin t, 3 cos t, 0) · (− sin t, cos t, 0)dt

= 4

∫2π

0(−2 sin2 t+ 3 cos2 t)dt = 4π,

cos2 t+ sin2 t = 1 ∀ t ∈ R

∫2π

0cos2 t dt = cos t sin t

∣∣∣2π

t=0+

∫2π

0sin2 t dt =

∫2π

0sin2 t dt = π.

I =

∫

C(y, z, x) · d(x,y, z),

C x+ y+ z = 0

R3

∇× (y, z, x) =

∣∣∣∣∣∣

e1 e2 e3∂∂x

∂∂y

∂∂z

y z x

∣∣∣∣∣∣= −

⎛

⎝111

⎞

⎠ .

C ⊂ R3 x+ y+ z = 0

C = {(x,y,−x− y) ∈ R3 : x2 + y2 + (x+ y)2 = 1}.

D = {(x,y,−x− y) ∈ R3 : x2 + y2 + (x+ y)2 ≤ 1}.

D Φ(x,y) = (x,y,−x− y)

K = {(x,y) ∈ R2 : x2 + y2 + (x+ y)2 ≤ 1},

R2

K =

{(x,y) ∈ R2 : −

√23 ≤ x ≤

√23 , −x

2 −√

12 − 3x2

4 ≤ y ≤ −x2 +

√12 − 3x2

4

}

x yK I

C = Φ(∂K)

N(x,y) =∂Φ

∂x(x,y)× ∂Φ

∂y(x,y) =

⎛

⎝10

−1

⎞

⎠×

⎛

⎝01

−1

⎞

⎠ =

⎛

⎝111

⎞

⎠ ,

I =

∫

Φ∇× (y, z, x) · n dσ = −

∫

K(1, 1, 1) · (1, 1, 1)d(x,y) = −3v(K),

v(K) KK

R2

ΦΦ D = Φ(K)

A(D) = A(Φ) =

∫

K∥N(x,y)∥d(x,y) =

√3v(K),

D KD

DA(D) = π

v(K) = π√3

I = −√3π.

IC

I∂K

Φ(x,y) = (x,y, x2 − y2), (x,y) ∈ B((0, 0),a) (a > 0)

f(x,y, z) = (z, x,y), (x,y, z) ∈ R3.

Φ

f(x,y, z) = (1, xz, xy), (x,y, z) ∈ R3.

R3

R3

U ⊂ R3 f = (f1, f2, f3) : U → R3

f div f : U → R div f = ∇ · f = ∂∂x f1 + ∂

∂y f2 + ∂∂z f3

0xy

V = {(x,y, z) ∈ R3 : (x,y) ∈ K, ϕ1(x,y) ≤ z ≤ ϕ2(x,y)},

K ⊂ R2 ϕ1,ϕ2 : K → Rϕ1 ≤ ϕ2 C1 0xy

∂K = γ([α,β]) γ : [α,β] → R2

i = 1, 2 Φi Ki

Φi(u, v) = (gi(u, v), (ϕi ◦ gi)(u, v)) ∀ (u, v) ∈ Ki,

gi : Ui → R2 Ui ⊂ R2 Ki ⊂ Ui

gi(Ki) = K

detDg1 < 0, detDg2 > 0 K1 K2

f : K → R

∫

Kf(x,y)d(x,y) =

∫

Ki

f(gi(u, v)) |detDgi(u, v)|d(u, v).

V ⊂ R3 C1 C1

0xy 0xz 0yz

S2 := Φ2(K2) = Γϕ2 S1 := Φ1(K1) = Γϕ1

V ⊂ R3

K ⊂ R2

V

C1 R3

gigi

1− 1

gi

z

gi = (xi,yi)T Ni Φi

∂Φi

∂u× ∂Φi

∂v=

⎛

⎜⎝

∂xi∂u∂yi∂u

∂(ϕi◦gi)∂u

⎞

⎟⎠×

⎛

⎜⎝

∂xi∂v∂yi∂v

∂(ϕi◦gi)∂v

⎞

⎟⎠ =

⎛

⎜⎝

∂yi∂u

∂(ϕi◦gi)∂v − ∂yi

∂v∂(ϕi◦gi)∂u

∂xi∂v

∂(ϕi◦gi)∂u − ∂xi

∂u∂(ϕi◦gi)∂v

detDgi

⎞

⎟⎠ ,

Φi

VK

V R3

R3 C1

V = [a1,a2]× [b1,b2]× [c1, c2] ⊂ R3 VC1 0xy

ϕi(x,y) = ci, i = 1, 2, (x,y) ∈ K = [a1,a2]× [b1,b2],

g1(u, v) = (v,u), Φ(u, v) = (v,u, c1), (u, v) ∈ K1 = [b1,b2]× [a1,a2],

g2(u, v) = (u, v), Φ(u, v) = (u, v, c1), (u, v) ∈ K2 = [a1,a2]× [b1,b2].

g2 Φ2

ϕ2 : K → R ϕ2(x,y) = c2

n2(u, v) =N2(u, v)

∥N2(u, v)∥=

⎛

⎝001

⎞

⎠ , N2(u, v) =

⎛

⎝100

⎞

⎠×

⎛

⎝010

⎞

⎠ =

⎛

⎝001

⎞

⎠ ,

V

n2 Vg1 u v

Φ1

n1(u, v) =N1(u, v)

∥N1(u, v)∥=

⎛

⎝00−1

⎞

⎠ , N1(u, v) =

⎛

⎝010

⎞

⎠×

⎛

⎝100

⎞

⎠ =

⎛

⎝00−1

⎞

⎠ ,

Vg1

Vgi i = 1, 2

V ⊂ R3 C1 0xz 0yzC1 R3

′

ϕ1 ϕ2

g1

B((x0,y0, z0), r) (x0,y0, z0) ∈ R3

r > 0 C1 R3

0xy 0xz 0yz

ϕ1(x,y) = z0 −√

r2 − (x− x0)2 − (y− y0)2, (x,y) ∈ K,

ϕ2(x,y) = z0 +√

r2 − (x− x0)2 − (y− y0)2, (x,y) ∈ K,

gi(ϑ,ϕ) =

(x0y0

)+ r

(sin ϑ cosϕsin ϑ sinϕ

), (ϑ,ϕ) ∈ Ki, i = 1, 2,

Φi(ϑ,ϕ) =

⎛

⎝x0y0z0

⎞

⎠+ r

⎛

⎝sin ϑ cosϕsin ϑ sinϕ

cos ϑ

⎞

⎠ , (ϑ,ϕ) ∈ Ki, i = 1, 2,

K = B((x0,y0), r), K1 =[π2,π]× [0, 2π], K2 =

[0,π

2

]× [0, 2π].

gi

detDg1(ϑ,ϕ) = r2 cos ϑ sin ϑ < 0, (ϑ,ϕ) ∈(π2,π)× [0, 2π] ⊂ K1,

detDg2(ϑ,ϕ) = r2 cos ϑ sin ϑ > 0, (ϑ,ϕ) ∈(0,π

2

)× [0, 2π] ⊂ K2.

gi R2

1− 1 detDgi = 0 intKi

(0,π

2

)∋ ϑ 4→ sin ϑ ∈ (0, 1)

(π2,π)∋ ϑ 4→ sin ϑ ∈ (0, 1)

1 − 1intKi

giUi ⊂ R2 Ki ⊂ Ui gi : Ui → R2

1−1 Ui

2π ϕ gi ϑ ∈ (π2 − ε, π2 + ε)sin ϑ 1− 1

g2(−ε,ϕ) = g2(ε,ϕ± π), g1(π+ ε,ϕ) = g1(π− ε,ϕ± π)

ϕ± π ∈ [0, 2π] 0 < ε ≪ 1 Ki

Ui ⊂ R2 Ki ⊂ Ui

K2

K2,ε :=[ε,π

2− ε]×[ε, 2π− ε

]⊂ intK2, 0 < ε <

π

4,

f : K → R

∫

g2(K2,ε)f(x,y)d(x,y) =

∫

K2,ε

f(g2(ϑ,ϕ)) |detDg2(ϑ,ϕ)|d(ϑ,ϕ)

ε → 0

∫

g2(K2)f(x,y)d(x,y) =

∫

K2

f(g2(ϑ,ϕ)) | detDg2(ϑ,ϕ)|d(ϑ,ϕ).

K1 ε→ 0

K1,ε :=[π2+ ε,π− ε

]×[ε, 2π− ε

]⊂ intK1, 0 < ε <

π

4.

Φi i = 1, 2 Ki

ni(ϑ,ϕ) =

⎛

⎝sin ϑ cosϕsin ϑ sinϕ

cos ϑ

⎞

⎠ , (ϑ,ϕ) ∈ (0,π)× [0, 2π]∩ Ki,

cos ϑ ≥ 0 cos ϑ ≤ 0

0xy

ϕ1(x,y) = z0 = ϕ2(x,y) ∀ (x,y) ∈ ∂K = ∂B((x0,y0), r).

C1 0xz 0yzC1 R3

C1

0xy

ϕi : K → Ri = 1, 2

U ⊂ R2 K ⊂ UK

∂KΦi

ϕi

C1 0xy

Vε = {(x,y, z) ∈ R3 : (x,y) ∈ Kε, ϕ1(x,y) ≤ z ≤ ϕ2(x,y)},

Kε = B((x0,y0), r− ε), 0 < ε < r,

Φi,ε ϕi|Kε VεV C1

0xz 0yz ε → 0V

V C1 R3

VV

VεV

V ⊂ R3 C1

∂V nV f : W → R3

W ⊂ R3 V ⊂ W∫

Vdiv f =

∫

∂Vf · n dσ.

f = (f1, f2, f3) V ⊂ R3 C1

0xy V0xy

∫

V

∂

∂zf3 =

∫

K

( ∫ϕ2(x,y)

ϕ1(x,y)

∂

∂zf3(x,y, z)dz

)d(x,y)

∫

V

∂

∂zf3 =

∫

K

(f3(x,y,ϕ2(x,y)

)− f3

(x,y,ϕ1(x,y)

))d(x,y).

gi∫

V

∂

∂zf3 =

∫

K2

f3(g2(u, v),ϕ2(g2(u, v))

)detDg2(u, v)d(u, v)

+

∫

K1

f3(g1(u, v),ϕ1(g1(u, v))

)detDg1(u, v)d(u, v)

=

∫

K2

f3(Φ2(u, v)

)detDg2(u, v)d(u, v)

+

∫

K1

f3(Φ1(u, v)

)detDg1(u, v)d(u, v),

∫

V

∂

∂zf3 =

∫

Φ2

(0, 0, f3) · n dσ+

∫

Φ1

(0, 0, f3) · n dσ,

Φi

Vn

V

∫

V

∂

∂zf3 =

∫

S1

(0, 0, f3) · n dσ+

∫

S2

(0, 0, f3) · n dσ,

Si = Φi(Ki)γ = γ1 ⊕ · · ·⊕ γk : [α,β] → R2

γ([α,β]) = ∂K γj : [αj,βj] → R2 j = 1, . . . , k

Φj(u, v) := (γj(u), v), (u, v) ∈ Kj, j = 1, . . . , k,

Kj := {(u, v) ∈ R2 : αj ≤ u ≤ βj, ϕ1(γj(u)) ≤ v ≤ ϕ2(γj(u))},

Φj Kj Sj := Φj(Kj)V C1

0xy S1 = Φ1(K1) S2 = Φ2(K2)

∂V = S1 ∪ S2 ∪k⋃

j=1

Sj.

Φj

γj

γj,ε(u) :=

⎧⎪⎨

⎪⎩

γj(αj) + (u−αj)γ′j(αj), u ∈ (αj − ε,αj)

γj(u), u ∈ [αj,βj]

γj(βj) + (u−βj)γ′j(βj), u ∈ (βj,βj + ε)

, ε > 0,

Φj,ε(u, v) := (γj,ε(u), v), (u, v) ∈ Uj := (αj − ε,βj + ε)× R ⊃ Kj,

Φj γj =(γ(1)j ,γ

(2)j

)

Nj =∂Φj

∂u× ∂Φj

∂v=

⎛

⎜⎝

dduγ

(1)j

dduγ

(2)j

0

⎞

⎟⎠×

⎛

⎜⎝0

0

1

⎞

⎟⎠ =

⎛

⎜⎝

dduγ

(2)j

− dduγ

(1)j

0

⎞

⎟⎠ ,

γjNj Φj

V VV γj∂Φj

∂u∂Φj

∂v Nj

Nj

∫

Φj(0, 0, f3) · n dσ = 0, j = 1, . . . ,k,

Sj = Φj(Kj)V

∫

Sj(0, 0, f3) · n dσ = 0, j = 1, . . . , k.

∫

V

∂

∂zf3 =

∫

S1

(0, 0, f3) · n dσ+

∫

S2

(0, 0, f3) · n dσ+k∑

j=1

∫

Sj(0, 0, f3) · n dσ,

=:

∫

∂V(0, 0, f3) · n dσ,

n ∂VV

S1 S2

γ S1 S2V

∂VV

V C1 0xz 0yz

∫

V

∂

∂yf2 =

∫

∂V(0, f2, 0) · n dσ

∫

V

∂

∂xf1 =

∫

∂V(f1, 0, 0) · n dσ,

n ∂V

∂VV C1 0xy 0xz 0yz

∂VR3

∂V

✷

V ⊂ R3

C1

V =⋃k

i=1 Vi Vi ⊂ R3 C1 ∂Vi

Vi

i = j Vi ∩ Vj = ∅ Vi ∩ Vj = ∂Vi ∩ ∂Vj

f : W → R3 W ⊂ R3

V ⊂ W∫

Vdiv f =

∫

∂Vf · n dσ,

n ∂V

Vi

∫

Vdiv f =

k∑

i=1

∫

Vi

div f =k∑

i=1

∫

∂Vi

f · n dσ.

∂Vi R3

∂V Sij := ∂Vi ∩ ∂Vj = ∅ Vi Vj

j = i Sij = ∅ ∂V

k∑

i=1

∫

∂Vi

f · n dσ =

∫

∂Vf · n dσ+

∑

Sij =∅

( ∫

Sij∩∂Vi

f · n dσ+

∫

Sij∩∂Vj

f · n dσ

).

Sij = ∅ Kij

f Sij Kij

Sij ∩ ∂Vi

Sij ∩ ∂Vj

∫

Sij∩∂Vi

f · n dσ = −

∫

Sij∩∂Vj

f · n dσ.

✷

S ⊂ R3

R > 0 n S

I =

∫

S(x3,y3, z3) · n dσ.

B = B((0, 0, 0),R) C1 R3

f : R3 → R3, f(x,y, z) = (x3,y3, z3)

∂B BS S

I =

∫

B∇ · (x3,y3, z3)d(x,y, z) = 3

∫

B(x2 + y2 + z2)d(x,y, z)

= 3

∫

B∥(x,y, z)∥2 d(x,y, z).

I = 3

∫2π

0

∫π

0

∫R

0r4 sin ϑdrdϑdϕ = 6π

R5

5[− cos ϑ]πϑ=0 = 12π

R5

5.

I =

∫

S(4xz,−y2,yz) · n dσ,

S [0, 1]× [0, 1]× [0, 1] n

K := [0, 1]× [0, 1]× [0, 1] R3 C1

f : R3 → R3, f(x,y, z) = (4xz,−y2,yz)

nS

I =

∫

K(4z− 2y+ y)d(x,y, z) =

∫1

0

∫1

0

∫1

0(4z− y)dxdydz = 4

1

2−

1

2=

3

2.

f(x,y, z) = (2xy+ z,y2,−x− 3y), (x,y, z) ∈ R3,

R3

x = 0, y = 0, z = 0 2x+ 2y+ z = 6.

f(x,y, z)

div f(x,y, z) = 2y+ 2y+ 0 = 4y (x,y, z) ∈ R3.

z = 6− 2x− 2y 0xy 0 = 6− 2x− 2yy = 3− x 0x 0 = 3− x

V ⊂ R3

V = {(x,y, z) ∈ R3 : (x,y) ∈ K3, 0 ≤ z ≤ 6− 2x− 2y},

K3 = {(x,y) ∈ R2 : 0 ≤ x ≤ 3, 0 ≤ y ≤ 3− x},

V 0xy

ϕ1(x,y) = 0, ϕ2(x,y) = 6− 2x− 2y, (x,y) ∈ K3,

K3

V C1 0xyϕi i = 1, 2 R2

K3

K3

(x,y, 0) (x,y) ∈ K3

n = (0, 0, 1)V

R3

n = (0, 0,−1)V ⊂ R3

C1 0xz 0yz

V = {(x,y, z) ∈ R3 : (x, z) ∈ K2, 0 ≤ y ≤ 3− x− 12z},

K2 = {(x, z) ∈ R2 : 0 ≤ x ≤ 3, 0 ≤ z ≤ 6− 2x},

V = {(x,y, z) ∈ R3 : (y, z) ∈ K1, 0 ≤ x ≤ 3− y− 12z},

K1 = {(y, z) ∈ R2 : 0 ≤ y ≤ 3, 0 ≤ z ≤ 6− 2y}.

V C1 R3 f : R3 →R3

I1 :=

∫

Vdiv f(x,y, z)d(x,y, z) =

∫

∂Vf(x,y, z) · n dσ =: I2,

I1I2

I1 = I2

I1 =

∫

V4yd(x,y, z) =

∫

K3

∫6−2x−2y

04ydzd(x,y)

=

∫3

0

∫3−x

04y(6− 2x− 2y)dydx =

∫3

0

(4(6− 2x)

(3− x)2

2− 8

(3− x)3

3

)dx

=

∫3

0

(82−

8

3

)(3− x)3 dx =

8

6

34

4= 27.

∂V0xy 0xz 0yz

S3 = {(x,y, 0) ∈ R3 : (x,y) ∈ K3}, n = (0, 0,−1),

S2 = {(x, 0, z) ∈ R3 : (x, z) ∈ K2}, n = (0,−1, 0),

S1 = {(0,y, z) ∈ R3 : (y, z) ∈ K1}, n = (−1, 0, 0),

R3 ∂Vϕ2 : K3 → R

S4 = {(x,y, 6− 2x− 2y) ∈ R3 : (x,y) ∈ K3}, n =N

∥N∥ =1√5(2, 2, 1),

N

N =

⎛

⎝10

−2

⎞

⎠×

⎛

⎝01

−2

⎞

⎠ =

∣∣∣∣∣∣

⎛

⎝e1 e2 e31 0 −20 1 −2

⎞

⎠

∣∣∣∣∣∣=

⎛

⎝221

⎞

⎠ ,

I2 =4∑

i=1

∫

Si

f · n dσ

=

∫

S1

(−2xy− z)dσ+

∫

S2

(−y2)dσ+

∫

S3

(x+ 3y)dσ

+

∫

S4

(2xy+ z,y2,−x− 3y) · (2, 2, 1) 1√5dσ

=

∫

K1

(−z)d(y, z) + 0+

∫

K3

(x+ 3y)d(x,y)

+

∫

K3

(2xy+ 6− 2x− 2y,y2,−x− 3y) · (2, 2, 1)d(x,y)

= −

∫3

0

∫6−2y

0z dzdy+ 2

∫3

0

∫3−x

0

(6− 2x+ 2(x− 1)y+ y2

)dydx

= − 2

∫3

0(3− y)2 dy+ 2

∫3

0

((x+ 1)(3− x)2 +

(3− x)3

3

)dx

= 2

∫3

0

(x(3− x)2 +

(3− x)3

3

)dx = 4

∫3

0

(3− x)3

3dx = 27.

V ⊂ R3 C1

∂V nn V u, v : W → R

W ⊂ R3 V ⊂ W∫

V∇u ·∇v+ u∆v =

∫

∂Vu∂v

∂ndσ.

∆v = ∇ ·∇v∂v

∂n= ∇v · n

∫

∂Vu∂v

∂ndσ =

∫

∂Vu∇v · n dσ =

∫

V∇ · (u∇v).

∇ · (u∇v) = ∇u ·∇v+ u∇ ·∇v,

∫

V(u∆v− v∆u) =

∫

∂V

(u∂v

∂n− v

∂u

∂n

)dσ.

V ⊂ R3 C1 ∂Vn n

V v : W → R W ⊂ R3 V ⊂ W

∫

∂V

∂v

∂ndσ = 0.

∫

∂Vv∂v

∂ndσ =

∫

V∥∇v∥2.

V ⊂ R3

x2 + y2 ≤ 4 z = 4− x2 − y2

∂V V

f : R3 → R3, f(x,y, z) = (x+ y,y+ z, x+ z).

V C1 f∂V

V ⊂ R3

x2 + y2 ≤ 9 0 ≤ z ≤ 5

R3

R3

C1

C1

C1

0x0y

0x0y

R3

C1

C1

0xy0xy

k+ 1

1

p

∞

C1

n

C1

k+ 1

k+ 1