Giáo trình giải tích 2 part 6 ppt

f : U −→ R
m
U ⊂ R
n
f
U
f C
1
∂f
∂x
i
i =1, ··· ,n U
Df : U −→ L(R
n
, R
m
)
g :[a, b] −→ R g
(a, b)
g(b) − g(a)=g

(c)(b − a), c a<c<b.
n>1,m=1
f : U → R U ⊂ R
n
f U
[x, x + h]={x + th, t ∈ [0, 1]}⊂U
f(x + h) − f (x)=Df(x + θh)h, 0 <θ<1.
g(t)=f (x + th)
m>1
f : R → R

2
,f(x)=(x
2
,x
3
)
f(1) −f(0) = Df(c)(1 −0) ⇔ (1, 1) − (0, 0) = (2c, 2c
2
)
f(x, y)=(e
x
cos y, e
y
sin y)
f : U → R
m
U ⊂ R
n
[x, x + h] ⊂ U
f(x + h) − f(x)≤ sup
t∈[0,1]
Df(x + th)h.
T
T  =sup
h=1
Th Th≤T h.
g(t)=f(x + th) g

(t)=Df(x + th)h.
g(1) −g(0) =


1
0
g

(t)dt =

1
0
Df(x + th)hdt,

1
0
(φ
1
(t), ··· ,φ
m
(t))dt =(

1
0
φ
1
, ··· ,

1
0
φ
m
)


f : U → R
m
U Df(x)=0, ∀x ∈ U
f ≡
f : U → R
n
U ⊂ R
n
C
1
K
U L>0 f
f(x) − f(y)≤Lx − y, ∀x, y ∈ K.
0 <L<1 f : K → K f K
f : U −→ R
m
U ⊂ R
n
Df : U → L(R
n
, R
m
), L(R
n
, R
m
)
R
n

→ R
m
m × n
R
mn
Df a ∈ U a
R
n
−→ L(R
n
, R
m
) ≡ R
mn
L(R
n
,L(R
n
, R
m
)),L(R
n
,L(R
n
,L(R
n
, R
m
))), ···
∂f

∂x
i
U
∂
∂x
j

∂f
∂x
i

(a) f (i, j)
a
D
j
D
i
f(a)
∂
2
f
∂x
j
∂x
i
(a).
∂
k
f
∂x

i
k
···∂x
i
1
(a).
f
k U f C
k
U f
≤ k U
f(x, y)=yx
2
cos y
2
∂
2
f
∂y∂x
=
∂
2
f
∂x∂y
=
∂
2
f
∂x∂y
=

∂
2
f
∂y∂x
f(x, y)=xy
x
2
− y
2
x
2
+ y
2
(x, y) =(0, 0) f(0, 0) = 0
∂
2
f
∂y∂x
(0, 0) = 1
∂
2
f
∂x∂y
(0, 0) = −1
f 2
x f C
2
∂
2
f

∂x
i
∂x
j
(x)=
∂
2
f
∂x
j
∂x
i
(x), ∀i, j
f ∈ C
k
≤ k
2
S
h,k
= f(x + h, y + k) − f(x + h, y) −f(x, y + k)+f(x, y)
g
k
(u)=f(u, y + k) − f (u, y)
S
h,k
= g
k
(x + h) − g
k
(x)=g


k
(c)h, c ∈ (x, x + h)
=(
∂f
∂x
(c, y + k) −
∂f
∂x
(c, y))h
=
∂
2
f
∂y∂x
(c, d)hk, d ∈ (y,y + k).
S
h,k
g
h
(v)=f(x + h, v) −f(x, v)
S
h,k
=
∂
2
f
∂x∂y
(c


,d

)kh, c

∈ (x, x + h),d

∈ (y, y + k).
h, k → 0 S
h,k

g :(a, b) → R ∈ C
k
x, x + h ∈ (a, b) θ ∈ (0, 1)
g(x+h)=g(x)+
1
1!
g

(x)h+
1
2!
g

(x)h
2
+···+
1
(k −1)!
g
(k−1)

(x)h
k−1
+
1
k!
g
k
(x+θh)h
k
.
f : R
n
→ R
g(t)=f (x + th),t∈ [0, 1].
∇ =(D
1
, ··· ,D
n
)=(
∂
∂x
1
, ··· ,
∂
∂x
n
).
h =(h
1
, ··· ,h

n
)
h∇ = h
1
∂
∂x
1
+ ···+ h
n
∂
∂x
n
,
(h∇)
k
=
n

i
1
,···,i
k
=1
h
i
1
···h
i
k
∂

k
∂x
i
1
···∂x
i
k
.
f
(h∇)f = h
1
∂f
∂x
1
+ ···+ h
n
∂f
∂x
n
.
(h∇)
k
f =

i
1
,···,i
k
∂
k

f
∂x
i
1
···
∂
k
f
∂x
i
k
h
i
1
···h
i
k
k h
1
, ··· ,h
n
.
f : U → R C
k
U ⊂ R
n
[x, x + h] ⊂ U θ ∈ (0, 1)
f(x + h)=f(x)+h∇f(x)+···+
1
(k −1)!

(h∇)
k−1
f(x)+
1
k!
(h∇)
k
f(x + θh)
g(t)=f(x + th)
g
(k)
(t)=(h∇ )
k
f(x + th). 
C
k
f
x
k x
T
k
x
(h)=f(x)+h∇f (x)+···+
1
k!
(h∇)
k
f(x),
R
k

(x, h)=
1
k!

(h∇)
k
f(x + θh) −(h∇)
k
f(x)

, 0 <θ<1.
|f(x + h) − T
k
x
(h)| = |R
k
(x, h)| = o(h
k
) f ∈ C
k
f ∈ C
∞
f x
0
Tf(x)=
∞

k=0
1
k!

((x − x
0
)∇)
k
f(x
0
).
Tf(x) f(x)=
∞

k=0
e
−k
cos k
2
x
Tf(x) Tf(x)=f(x)
f(x)=e
−
1
x
2
.
f f f
x
0
f |f
(k)
(x)|≤M
k

, ∀x ∈ (a, b), ∀k ∈ N (a, b)
f : U → R,U⊂ R
n
f
a ∈ U f(a) ≥ f(x) x a
f
a ∈ U f(a) ≤ f(x) x a
f
a f
f a
f Df(a)=0
max, min
f
f U f a ∈ U Df(a)=0
i g
i
(t)=f(a + te
i
) t =0
g

i
(0) =
∂f
∂x
i
(a)=0. Df(a)=0 
f(x)=x
3
f(x, y)=x

2
− y
2
f f

f
f

(a) > 0 f a f

(a) < 0
f
x
2
+ y
2
, −x
2
− y
2
,x
2
− y
2
,x
2
, −x
2
, 0.
(0, 0) (0, 0)

f C
2
f a Hf(a)
R
n
 h → Hf(a)(h)=(h∇)
2
f(a)=
n

i,j=1
∂
2
f(a)
∂x
i
∂x
j
h
i
h
j
∈ R.
f C
2
Df(a)=0
Hf(a) Hf(a)(h) > 0, ∀h ∈ R
n
\ 0 f a
Hf(a) Hf(a)(h) < 0, ∀h ∈ R

n
\ 0 f a
Hf(a) f a
f(a + h)=f(a)+Df(a)h +
1
2
Hf(a)(h)+o(h
2
)
Df(a)=0 Hf(a) > 0 m = min
h=1
Hf(a)(h) > 0
Hf(a)(h) ≥ mh
2
, ∀h ∈ R
n
f a

H(h)=
n

i,j=1
a
ij
h
i
h
j
,h∈ R
n

.
D
k
=det(a
ij
)
1≤i,j≤k
H D
1
> 0,D
2
> 0, ··· ,D
n
> 0.
H D
1
< 0,D
2
> 0, ··· , (−1)
n
D
n
> 0.
f(x, y)=x
3
+ y
3
− 3xy
f
∂f

∂x
=3x
2
− 3y =0,
∂f
∂y
=3y
2
− 3x =0.
(0, 0) (1, 1)
f
Hf =




∂
2
f
∂x
2
∂
2
f
∂x∂y
∂
2
f
∂y∂x
∂

2
f
∂y
2




=

6x −3
−36y

(0, 0) D
2
= −9 < 0 Hf(0, 0) (0, 0)
(1, 1) D
1
=6> 0,D
2
=27> 0 Hf(1, 1) > 0 f (1, 1)
f : U → R
m
f
f a
f a Df(a)
Df(a) f a
Df(a) f a f(a)
Df(a) f a
f(a)

Ax = y, A ∈ Mat(n, n).
det A =0 A x = A
−1
y.
x
1
, ··· ,x
n





f
1
(x
1
, ··· ,x
n
)=y
1
···
f
n
(x
1
, ··· ,x
n
)=y
n

,
y
1
, ··· ,y
n
f : R → R f
f

(a) =0 f
−1
a f
−1
C
1
f : U −→ R
n
U ⊂ R
n
f C
k
(k ≥ 1) a ∈ U det Jf(a) =0 V a W f(a)
f : V −→ W f
−1
: W −→ V f
−1
C
k
Df
−1
(y)=(Df(x))

−1
,y= f(x),x∈ V.
Df(a)
−1
a = f (a)=0 Df(0) = I
n
R
n
x y
y = f(x) 0 y ∈ R
n
g
y
(x)=y + x −f(x) g
y
0 x g
y
(x)=x
f(x)=y x y
g(x)=x − f(x) Dg(0) = 0 g ∈ C
1
r>0
g(x) − g(x

)≤
1
2
x − x

,x,x


∈ B(0,r).
y≤r/2,g
y
: B(0,r) −→ B(0,r)
g
y
(x) − g
y
(x

)≤
1
2
x − x

.
x ∈
B(0,r) g
y
f f
−1
: B(0,r/2) −→ B(0,r).
f
−1
y, y

∈ B(0,r/2) x = f
−1
(y),x


= f
−1
(y

) ∈ B(0,r)
g
x − x

≤f(x) − f(x

) + g(x) − g(x

)≤f(x) − f(x

) +
1
2
x − x

.
f
−1
(y) − f
−1
(y

)≤2y −y

 f

−1
r>0 f
−1
∈ C
k
det f ∈ C
k
det Df(a) =0 r>0
(Df(x))
−1
, ∀x ∈ B(0,r).
y = f(x),y

= f(x

),x,x

∈ B(0,r)
f
−1
(y) − f
−1
(y

) − (Df(x))
−1
(y −y

) = x −x


− (Df(x))
−1
(Df(x)(x − x

)+
+o(x − x

) = (Df(x))
−1
(ox − x

) = o(y −y

) .
Df
−1
(y)=(Df(x))
−1
, y = f(x).
Jf
−1
(y)=
1
det Jf(x)
(A
ij
(x))
n×n
A
ij

(x) Jf(x)=
f x Jf
−1
C
k−1
f
−1
C
k

f : U → V
C
k
C
k
f f, f
−1
C
k
f
a f a
f(a)
u(x, y)=e
x
cos y, v(x, y)=e
x
sin y
x, y u, v
det J(u, v)=






e
x
cos y −e
x
sin y
e
x
sin ye
x
cos y





= e
2x
=0.
f : R
2
→ R
2
,f(x, y)=(u(x, y),v(x, y)) (x, y)
det Jf(x, y) =0, ∀(x, y) ∈ R
2
f : R → R f(x)=x

3
f
−1
(y)=
3
√
y f

(0) = 0
f : U → R
m
,U⊂ R
n
f C
1
a ∈ U
n<m Df(a) rankDf(a)=n
g f(a) 0 R
m
g ◦f(x
1
, ··· ,x
n
)=(x
1
, ··· ,x
n
, 0, ··· , 0)
n>m Df(a) rankDf(a)=m
h 0 a R

n
f ◦ h(x
1
, ··· ,x
n
)=(x
1
, ··· ,x
m
)
Df(a)
Jf(a) n
Φ:U × R
m−n
→ R
m
, Φ(x, y
n+1
, ··· ,y
m
)=f(x)+(0, ··· , 0,y
n+1
, ··· ,y
m
)
Φ ∈ C
1
JΦ(a, 0) Φ
(a, 0) f (x)=Φ(x, 0) g =Φ
−1

Df(a) Jf(a) m
Ψ:U → R
m
× R
n−m
, Ψ(x)=(f(x) − f(a),x
m+1
− a
m+1
, ··· ,x
n
− a
n
).
Ψ ∈ C
1
JΨ(a) Ψ
a Ψ f(x)=pr ◦ (Ψ(x) − f(a)) pr
m h =(Ψ− f(a))
−1

F (x, y)=0
y x y = g(x) g
F (x, y)=x
2
+ y
2
− 1=0
y = g(x) x = a ∈ (−1, 1)
∂F

∂y
=0 F
0x
y x a = ±1





a
11
x
1
+ ··· +a
1n
x
n
+b
11
y
1
+ ··· +b
1m
y
m
=0
··· ··· ··· ··· ··· ···
a
m1
x

1
+ ··· +a
mn
x
n
+b
m1
y
1
+ ··· +b
mn
y
m
=0
A =(a
ij
)
m×n
,B=(b
ij
)
m×m
det B =0 y x
y = −B
−1
Ax






F
1
(x, y)=0
···
F
m
(x, y)=0
x ∈ R
n
,y∈ R
m
y x
F : A → R
m
,A⊂ R
n
× R
m
(a, b) ∈ A
F C
k
(k ≥ 1) F (a, b)=0 F y
D(F
1
, ··· ,F
m
)
D(y
1

, ··· ,y
m
)
(a, b) = det
∂F
∂y
(a, b)=










∂F
1
∂y
1
(a, b) ···
∂F
1
∂y
m
(a, b)
··· ··· ···
∂F
m

∂y
1
(a, b) ···
∂F
m
∂y
m
(a, b)










=0.
U ⊂ R
n
a V ⊂ R
m
b g : U → V
C
k
F (x, y)=0, (x, y) ∈ U × V ⇐⇒ y = g(x),x∈ U, y ∈ V.
Dg(x)=−

∂F

∂y

−1
∂F
∂x
(x, g(x)),x∈ U.
f(x, y)=(x, F (x, y))
f
−1
(x, z)=(x, G(x, z)) (x, z) (a, 0)
g(x)=G(x, 0)

F (x, g(x)) = 0,x∈ U
Dg
∂F
i
∂x
j
+
m

k=1
∂F
i
∂y
k
∂g
k
∂x
j

=0,i=1, ··· ,m; j =1, ··· ,k,
Dg m =1
∂F
∂y
(a, b) =0 y = g(x
1
, ··· ,x
n
) (a, b) y = g(x)
F (x
1
, ··· ,x
n
,y)=0,x ∈ U
dF =
∂F
∂x
1
dx
1
+ ···+
∂F
∂x
n
dx
n
+
∂F
∂y
dy =0.

∂F
∂y
dg = −

∂F
∂x
1
dx
1
+ ···+
∂F
∂x
n
dx
n

,
∂g
∂x
j
= −
∂F/∂x
j
∂F/∂y
,j=1, ··· ,n.

xu + yv
2
=0
xv

3
+ y
2
u
6
=0