d u z ¥ d ] h o m g ] n d e b ] h ] f o m busers.uoi.gr/giannoul/iciii/vak.pdfd u _ z ¥ d ] h o m...
TRANSCRIPT
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 ,
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