f (x0)
f
x0
U ⊂ R
m
→
R
m
C 1
n
R
i = 1, 2, . . . , n
f : U ⊂ ∂f i ∂x j U
U
j = 1, 2 , . . . , m U
n
V
R
f
f : U → V f −1
f
C 1
f f −1
f
C 1
• • f (x) :
f : U → V R
n
→ Rn
x ∈ U
[f (x)]
−1
= f −1 (f (x)).
IdU (x) = f −1 ◦ f (x)
h = (f −1)(f (x)).f (x)h,
[f (x)] U ⊂
−1
n
= f −1 (f (x)).
R
f : U →
n
R
x ∈ U
f V x x
W f (x)
f : V x → W f (x) C 1
f : V x → W f (x)
c1
x ∈ U.
(M, d) f : M → N
(N, d1) 0≤k<1
d1(f (x), f (y )) ≤ kd (x, y ), ∀x, y ∈ M.
f : M →
(M, d) M
f
M.
x0 ∈ M x1 = f (x0), xn+1 = f (xn ), n = 1, 2, . . .
limn→∞ xn = a
f.
(xn ) a ∈ M,
f
f (a) = f (lim xn ) = lim f (xn ) = lim xn+1 = a. a f
f. a
b
d(a, b) = d(f (a), f (b)) ≤ kd (a, b), a = b.
a = b.
(xn ) d(x1, x2 ) ≤ kd (x0 , x1)
d(xn+1, xn ) ≤ k n d(x1, x0 ), ∀n ∈
n, p ∈
N.
N
d(xn , xn+ p ) ≤ d(xn , xn+1) + · · · + d(xn+ p−1, xn+ p )
≤ [k n + k n+1 + · · · + k n+ p−1 ]d(x0, x1 ) ≤
kn
1−k
d(x0, x1 ).
lim k n = 0 2
• f : [a, b] → [a, b]
supx∈[a,b] |f (x)| < 1.
f
|f (y ) − f (x)| ≤ |y − x| sup |f (c)| ≤ k |y − x|. c∈(a,b)
U n
R
f : U →
f (x) ≤ M, ∀x ∈ M
n
R
f (b) − f (a) ≤ M b − a, ∀a, b ∈ U. A
B − AA−1 < 1
B ∈ L(Rn ; Rn ) B
A → A−1
A W ⊂
W → Rn
C 1
n
f :
R
a ∈ W.
f (a)
b = f (a) U a
V b
R
n
f : U → V
C 1 g = f −1
g : V → U
C 1 .
f f (a) = A
λ
2λA−1 = 1. f
a
U
a
f (x) − A < λ, ∀x ∈ U. y ∈ Rn φ(x) = x + A−1 (y − f (x)), ∀x ∈ U.
f (x) = y ⇔ φ(x) = x.
φ (x) = I − A−1f (x) = A−1 [A − f (x)] .
φ(x) = A−1 · A − f (x) <
1 φ (x) < . 2
1 1 ·λ= , 2λ 2
x1
U h(t) = φ((1 − t)x1 + tx2).
x2
h (t) = φ((1 − t)x1 + tx2).(x2 − x1).
1 h (t) = φ((1 − t)x1 + tx2 ).x2 − x1 < x2 − x1. 2
1 φ(x2) − φ(x1 ) < x2 − x1. 2 φ
U φ(U ) ⊂ U
φ : U → U
φ
x ∈ U x ∈ U.
y = f (x) g = f −1.
f : U → f (U )
y0 = f (x0) ∈
V = f (U ) U.
B1
x0
B1 ⊂ U.
r>0
y
y − y0 < λr, y
V
φ
r
φ(x0 ) − x0 = A−1 (y − y0) < A−1λr = . 2 x ∈ B1
φ(x) − x0 ≤ φ(x) − φ(x0 ) + φ(x0) − x0 r 1 ≤ x − x0 + 2 2 r r < + = r. 2 2
φ(x) ∈ B1 ⊂ B1 . φ : B1 → B1
B1 x ∈ B1 .
φ y ∈ f (B1 ) ⊂ f (U ) = V
x f (x) = y y
V
C 1
g U = B x∈B
y ∈ V
V = f (B )
y + k ∈ V
x+h∈B f (x) = y
f (x + h) = y + k.
φ φ(x + h) − φ(x) = h + A−1 [f (x) − f (x + h)] = h − A−1 k.
1 h − A−1 k ≤ h. 2 1 h − A−1k ≤ h, 2 1 −A−1 k ≤ − h. 2 1 A−1 k ≥ h. 2
h ≤ 2A−1.k =
k λ
.
V
B ∈
A
B − AA−1 < 1
L(Rn ; Rn )
B
A → A−1
A
f (x) − AA−1 < f (x)
1 < 1, 2
T
g (y + k ) − g (y ) − T k = h − T k = −T [f (x + h) − f (x) − f (x)h]
1
k
≤
1 . λh
g (y + k ) − g (y ) − T k T f (x + h) − f (x) − f (x)h ≤ . . k λ h k→0 h→0
g (y ) = T = [f (x)] g
V 1
A → A−1
.
g
− g (y ) = T = [f (g(y ))]
f g
−1
2
g
•
C 1
f
f (x) =
x
2 0,
1 + x2 sin( ),
= 0, x
x x = 0.
f
f (x) =
f
1 1 1 + 2x sin( ) − cos( ), x x 2 1 , x = 0. 2
= 0, x
x0 = 0 f (0) =
1 2
f
> 0 f x0
f
C 1.
n
U ⊂
R
f (x) :
R
n
f : U →
→
n
R
n
x ∈ U
R
C 1 .
f
y = f (x) yi = f i (x1, x2, . . . , xn ), i = 1, 2, . . . , n x = x1 , . . . , x n y1 , y2 , . . . , yn b
•
x
y
a
C 1.
F (x, y ) = (exp(x)cos(y ), exp(x)sin(y ))
F (x, y ) =
exp(x)cos(y ) − exp(x)sin(y ) exp(x)sin(y )
exp(x)cos(y )
F (x, y )
exp(2x) =0
(x, y ) ∈ R2. (x0, y0) U (x0 , y0 )
V F (x0 , y0 )
F (0, 0) = F (0, 2π )
F
U = {(r, θ ); r > 0 R
F : U → V
2
0 < θ < 2π } (r, θ ) ∈ U.
F (r, θ) = (r cos(θ ), r sin(θ)), C 1
F
=0 |F (r, θ )|
F
C 1 ∂f (a, b) = 0. ∂y (x, y ) = (x, f (x, y )) C 1
U ⊂ R2 F : U →
f : U → R
2 R
(a, b). F
|F (x, y )| =
F : U 0 → U 1
1
0
∂f ∂f (a, b) (a, b) ∂x ∂y
=
∂f (a, b) = 0. ∂y U 0
C 1.
F (x, y ) = (exp(x) + exp(y ), exp(x) + exp(−y )).
U 1
(x, y ) u
v
x
y
x4 + y 4 = u, x
cos(x) + sin(y ) = v (x, y )? x y
z x(ρ,φ,θ ) = ρ sin(φ) cos(θ ) y (ρ,φ,θ ) = ρ sin(φ) sin(θ) z (ρ,φ,θ ) = ρ cos(φ),
ρ, θ
φ
x, y
z
F (x, y ) = 0 x
y I = (x0 − h, x0 + h) x ∈ I
y F (x, y ) = 0
x
y
I
F (x, y ) = 0 x
x
y I
F
y
F
P
F y
y
F, F x
x
F y
R2
A
P = (x0 , y0) F y (x0, y0) = 0.
F (x0, y0) = 0, h
k
R
A
x y
I = {x; |x − x0| < h}
J = {y ; |y − y0| < k }
F (x, y ) = 0
(x, y )
f
I
J
f
I x
R
T x y = y − y
J
R
F (x, y ) F y (x0, y0 )
1
h J T x y = y
y y
J
F (x, y ) = 0. T x T x y = y0 − c(x − x0) − ψ (x, y ),
c=
F x (x0, y0) F y (x0, y0 )
k
1
ψ (x, y ) =
F y (x0, y0)
[F (x, y ) − F x (x0 , y0 )(x − x0) − F y (x0, y0)(y − y0 )] .
F (x0, y0 ) = 0 ψ (x0, y0). ψx(x0 , y0 ) = 0, ψy (x0 , y0 ) = 0. ψx
ψy
k
1 1 |ψx (x, y )| < , |ψy (x, y )| < 2 2 (x, y ) S = {(x, y ); |x − x0| ≤ k
|y − y0 | ≤ k }.
ψ (x, y )
S
(x0 , y0 ) ψ (x, y ) = ψx (ζ, η )(x − x0) + ψy ζ, η )(y − y0 ),
(ζ, η ) ∈ S h≤k
R
|ψ (x, y )| ≤
h
k
+ . 2 2 h k
T x
J
J
h k h 1 |T x y − y0| ≤ |c(x − x0)| + |ψ (x, y )| ≤ c|h| + + = ( + c)|h| + . 2 2 2 2 h J
x
T x y I
T x
1 |T x y1 − T x y2 | = |ψ (x, y1) − ψ (x, y2 )| ≤ |y1 − y2|. 2
J
x ∈ I y x
F (x, y ) = 0
J
(x, y ) ∈ R.
2
F : Rn+1 → R
(x, z )
C 1 x∈
F (x0 , z0) = 0
n
R
z0
C 1
B
z∈
n+1
R R
∂F (x0 , y0 ) = 0. ∂z B ⊂
V
y
R
n
x0
z = g (x)
g
F (x, g (x)) = 0 ∂F ∂g ∂x = − i , i = 1, 2, . . . , n . ∂F ∂xi ∂z
• x3 + 2y 2 + 8xz 2 − 3z 3y = 1 z = k (x, y ) F (x,y,z ) = x3 + 2y 2 + 8xz 2 − 3z 3 y − 1 ∂F (x0 , y0 , z0 ) (x0 , y0 , z0) = 0. ∂z ∂F (x0 , y0 , z0) = z0 (16x0 − 9y0z0) = 0, ∂z z0 = 0
16x0 − 9y0z0 = 0
z = k (x, y ) ∂z F x 3x2 + 8z 2 =− =− ∂x F z 16xz − 9yz
∂z F y 6y − 3z 3 . =− =− 16xz − 9yz ∂y F z F : C 1
k
R
×
m
R
→
m
R
F (x0, y0 ) = 0
∂F det (x0 , y0 ) = 0. ∂y W ⊂
k
R
φ : W →
R
k
C 1
x0 ∈ W
φ(x0) = y0
F (x, φ(x)) = 0, ∀x ∈ W. g : Rk × Rm → Rk × Rm g (x, y ) = (x, f (x, y )). C 1
g
(x0 , y0 )
g (z0 ) =
I k
0
∂F (x0, y0 ) ∂x
∂F (x0, y0) ∂y
det g (x0, y0) = det
,
∂F (x0 , y0 ) = 0. ∂y U ⊂
R
m
(x0, y0)
C 1
g : U → V
(x, y ) = g (x, y )
V = g (U )
(x, y ) ∈ U (x, y ) = g−1 (x, y ).
R
k
×
x = x
y = F (x, y )
y =
g −1 (x, y ). F (x, y ) = 0 ⇔ y = g −1(x, 0) φ(x) = g −1 (x, 0) W = U ∩ Rk .
x∈
