Giáo trình giải tich 3 part 3 ppt

R
n
rank (DF
1
, ··· ,DF
m
)(x)=m, ∀x ∈ M M n − m
C
p
k = n − m x =(x

,y) ∈ R
k
× R
m
= R
n
F =(F
1
, ··· ,F
m
)
a ∈ M det
∂F
∂y
(a) =0.
V

a =(a

,b)

M ∩ V

= {(x

,y) ∈ V

: F(x

,y)=0} = {(x

,y) ∈ V

: y = g(x

)},
g C
p
U a

ϕ : U → R
n
ϕ(x

)=(x

,g(x

))
M a 
R

3
S
2
F (x, y, z)=x
2
+ y
2
+ z
2
− 1=0
F

(x, y, z)=(2x, 2y, 2z) =(0, 0, 0) S
2
S
2
2
C 1

F
1
(x, y, z)=x
2
+ y
2
+ z
2
− 1=0
F
2

(x, y, z)=x + y + z =0
(ψ, W) M x
W

,U

ψ
−1
(x),ϕ
−1
(x) W

ψ = ϕ ◦ h,
h = ϕ
−1
◦ ψ : W

→ U

h
−1
h = ϕ
−1
◦ψ ψ
−1
(ψ(W )∩ϕ(U)) ϕ
−1
(ψ(W )∩
ϕ(U)) h C
p

rank Dϕ = k k Dϕ(u)
u U

D(ϕ
1
, ··· ,ϕ
k
)
D(u
1
, ··· ,u
k
)
=0
U

x =(x

,y) ∈ R
k
× R
n−k
i : R
k
→ R
k
× R
n−k
i(u)=(u, 0) p = R
k

× R
n−k
→ R
k
p(x

,y)=x

Φ(u, y)=(ϕ(u),y) det DΦ=
D(ϕ
1
, ··· ,ϕ
k
)
D(u
1
, ··· ,u
k
)
=0
Φ
−1
∈ C
p
h = ϕ
−1
◦ ψ =(Φ◦ i)
−1
◦ ψ = p ◦Φ
−1

◦ ψ.
C
p
h C
p

M ⊂ R
n
k x
0
∈ M
γ :(−, ) → M C
1
M γ(0) = x
0
γ

(0)
M x
0
M x
0
M x
0
T
x
0
M
(ϕ, U) M x
0

= ϕ(u
0
)
T
x
0
M = {v ∈ R
n
: v = t
1
D
1
ϕ(u
0
)+···+ t
k
D
k
ϕ(u
0
),t
1
, ··· ,t
k
∈ R} = Dϕ(u
0
).
R
n
M F

1
= ···= F
m
=0 x
0
T
x
0
M = {v ∈ R
n
: v ⊥ grad F
i
(x
0
),i=1, ···,m}.
T
x
0
M
v ∈ R
n
: < grad F
1
(x
0
),v >= ···=< grad F
m
(x
0
),v >=0

S
2
C
H
k
= {x =(x
1
, ··· ,x
k
) ∈ R
k
: x
k
≥ 0} R
k
∂H
k
= {x ∈ H
k
: x
k
=0} = R
k−1
× 0 H
k
H
k
+
= {x ∈ H
k

: x
k
> 0} H
k
M ⊂ R
n
k C
p
x ∈ M
V ⊂ R
n
x U ⊂ R
k
ϕ : U → R
n
C
p
ϕ : U ∩H
k
→ M ∩V
rank ϕ

(u)=k u ∈ U
x = ϕ(u),u∈ U
M u ∈ H
k
+
M u ∈ ∂H
k
∂M = {x ∈ M : x M}

M
s
✲
R
1
✻x
U

H
✲
ϕ
s
x
✲

✒
M

V








V ⊂ R
n
C

p
F
1
, ··· ,F
m
,F
m+1
: V → R
M = {x ∈ V : F
1
(x)=···= F
m
(x)=0,F
m+1
(x) ≥ 0}
∂M = {x ∈ V : F
1
(x)=···= F
m
(x)=F
m+1
(x)=0}
rank (DF
1
, ··· ,DF
m
)(x)=m, ∀x ∈ M rank (DF
1
, ··· ,DF
m+1

)(x)=
m +1, ∀x ∈ ∂M M n − m C
p
∂M

R
3
B x
2
+ y
2
+ z
2
≤ 1
R
n
∂B x
2
+ y
2
+ z
2
=1
M k
∂M k − 1 ∂(∂M)=∅
x ∈ ∂M T
x
∂M k − 1 T
x
M

i : R
k−1
→ R
k
,i(u
1
, ··· ,u
k−1
)=(u
1
, ··· ,u
k−1
, 0)
(ϕ, U) M x x ∈ ∂M (ϕ ◦i, i
−1
(U))
∂M x x ∂M ∂(∂M)=∅
T
x
∂M D
1
ϕ(u), ··· ,D
k−1
ϕ(u)
k − 1 T
x
M 
F =(F
1
, ··· ,F

m
):V → R
m
C
1
V ⊂ R
n
M = {x ∈ V : F
1
(x)=···= F
m
(x)=0} rank F

(x)=m, ∀x ∈ M
f : V → R C
1
f|
M
f
F
1
= ···= F
m
=0
M a ∈ M (ϕ, U) M
a a = ϕ(b)
f F
1
= ··· = F
m

=0 a
grad f(a) ⊥ T
a
M λ
1
, ··· ,λ
m
∈ R
grad f(a)= λ
1
grad F
1
(a)+···+ λ
m
grad F
m
(a)
f|
M
a f ◦ϕ
b
(f ◦ ϕ)

(b)=f

(a)ϕ

(b)=0 < grad f (a),v >=0, ∀v ∈ Imϕ

(b)=T

a
M
grad f(a) ⊥ T
a
M rank (grad F
1
(a), ··· , grad F
m
(a)) = m =codimT
a
M
grad f(a) grad F
1
(a), ··· , grad F
m
(a) 
f F
1
= ···= F
m
=0
L(x, λ)=f(x) − λ
1
F
1
(x) −···−λ
m
F
m
(x),x∈ V,λ =(λ

1
, ··· ,λ
m
) ∈ R
m
a λ ∈ R
m
(a, λ)













∂L
∂x
(x, λ)=0
F
1
(x)=0
F
m
(x)=0

f(x, y, z)=x + y + z x
2
+ y
2
=1,x+ z =1
L(x, y, z, λ
1
,λ
2
)=x + y + z −λ
1
(x
2
+ y
2
− 1) −λ
2
(x + z − 1)
























∂L
∂x
=1− 2λ
1
x −λ
2
=0
∂L
∂y
=1− 2λ
1
y =0
∂L
∂z
=1 −λ
2
=0
x
2

+ y
2
− 1=0
x + z − 1=0
(0, ±1, 1) f
max f|
E
=max{f(0, 1, 1) = 1,f(0, −1, 1) = 0} = f(0, 1, 1) = 1
min f|
E
= min{f(0, 1, 1) = 1,f(0, −1, 1) = 0} = f(0, −1, 1) = 0
f, F
1
, ··· ,F
m
C
2
grad f(a)= λ
1
grad F
1
(a)+···+ λ
m
grad F
m
(a)
∂L
∂x
(a, λ)=0
H

x
L(x, a) L x
H
x
L(a, λ)|
T
a
M
f|
M
a
H
x
L(a, λ)|
T
a
M
f|
M
a
H
x
L(a, λ)|
T
a
M
f|
M
a
f|

M
f◦ϕ f

(a)ϕ

(b)=0 H(f◦ϕ)(a)(h)=
Hf(a)(ϕ

(b)h)
F
i
◦ ϕ =0 H(F
i
◦ ϕ)=0 H(F
i
◦ ϕ)(b)(h)=
HF
i
(a)(ϕ

(b)(h)
H
x
L(a, λ)|
T
a
M
= H(f ◦ ϕ)(b)|
T
a

M

k ∈ N a ∈ R f(x
1
, ··· ,x
n
)=x
k
1
+ ···+ x
k
n
x
1
+ ···+ x
n
= an
R
3
R
3
< ·, · >
T =(x
t
,y
t
,z
t
) T  =


x
2
t
+ y
2
t
+ z
2
t
u =(x
u
,y
u
,z
u
),v=(x
v
,y
v
,z
v
)
(u, v)=uv
⊥
 = u × v
=






u
2
<u,v>
<v,u> v
2





1
2
=

u
2
v
2
−|<u,v>|
2
.
v = v

+ v
⊥
v

v u v
⊥

⊥ u
v

= αu, < v
⊥
,u>=0





<u,u> <u,v>
<v,u> <v,v>





=





<u,u> <u,v

> + <u,v
⊥
>
<v,u> <v,v


> + <v,v
⊥
>





=





<u,u> α<u,u

>
<v,u> α<v,u

>





+






<u,u> 0
<v,u> v
⊥

2





= u
2
v
⊥

2

u, v, w ∈ R
3
(u, v, w)= (u, v) w
⊥

= | <u×v, w > | = |det(u, v, w)|
=








<u,u> <u,v> <u,w>
<v,u> <v,v> <v,w>
<w,u> <w,v> <w,w>







1
2
w = w

+ w
⊥
w

w
u, v
✂
✂
✂
✂
✂✍
w
✲

u
✟
✟
✟✯
v
✻w
✟
✟
✟
✂
✂
✂
✂
✂
✟
✟
✟
✂
✂
✂
✂
✂
✂
✂
✂
✂
✂
✟
✟
✟


k R
n
R
n
k
v
1
, ··· ,v
k
∈ R
n
k
V
1
(v
1
)=v
1
,V
k
(v
1
, ··· ,v
k
)=V
k−1
(v
1
, ··· ,v

k−1
)v
⊥
k

v
k
= v

k
+ v
⊥
k
v

k
v
k
v
1
, ··· ,v
k−1
G(v
1
, ··· ,v
k
)=(<v
i
,v
j

>)
1≤i,j≤k
V
k
(v
1
, ··· ,v
k
)=

det G(v
1
, ··· ,v
k
)

C ⊂ R
3
ϕ : I → R
3
,ϕ(t)=(x(t),y(t),z(t))
l(C)
I I
i
=[t
i
,t
i
+∆t
i

] l(C)=

i
l(ϕ(I
i
))
∆t
i
l(ϕ(I
i
)) ∼ l(ϕ

(t
i
)∆t
i
)=ϕ

(t
i
)∆t
i
dl = ϕ

(t)dt =

x

2
t

+ y

2
t
+ z

2
t
dt
C
l(C)=

C
dl =

I

x

2
t
+ y

2
t
+ z

2
t
dt

S ⊂ R
3
ϕ : U → R
3
,ϕ(u, v)=(x(u, v),y(u, v),z(u, v))
S
U U
i
=[u
i
,u
i
+∆u
i
]×[v
i
,v
i
+∆v
i
]
(S)=

i
(ϕ(U
i
))
∆u
i
, ∆v

i
(ϕ(U
i
)) ∼ (D
1
ϕ(u
i
,v
i
)∆u
i
,D
2
ϕ(u
i
,v
i
)∆v
i
)
dS = (D
1
ϕ, D
2
ϕ)dudv =

EG − F
2
dudv,
E = D

1
ϕ
2
= x

u
2
+ y

u
2
+ z

u
2
G = D
2
ϕ
2
= x

v
2
+ y

v
2
+ z

v

2
F = <D
1
ϕ, D
2
ϕ> = x

u
x

v
+ y

u
y

v
+ z

u
z

v
S
(S)=

S
dS =

U


EG − F
2
dudv
H
ϕ : A → R
3
,ϕ(u, v, w)=(x(u, v, w),y(u, v, w),z(u, v, w))
H
dV = (D
1
ϕ, D
2
ϕ, D
3
ϕ)dudvdw = |det Jϕ|dudvdw
H V (H)=

H
dV =

A
|det Jϕ|dudvdw
M ⊂ R
n
k
dV : M  x → dV (x)= k T
x
M.
(ϕ, U) M x = ϕ(u

1
, ··· ,u
k
)
dV (x)(D
1
ϕ(x)∆u
1
, ··· ,D
k
ϕ(x)∆u
k
)=V
k
(D
1
ϕ(x), ··· ,D
k
ϕ(x))∆u
1
···∆u
k
G
ϕ
=(<D
i
ϕ, D
j
ϕ>)
1≤i,j≤k

dV =

det G
ϕ
du
1
···du
k
f : M → R k
f M

M
fdV
M = ϕ(U) (ϕ, U )

M
fdV =

U
f ◦ ϕ

det G
ϕ
, G
ϕ
=(<D
i
ϕ, D
j
ϕ>)

1≤i,j≤k
.
k =1

M
fdl
k =2

M
fdS
M
O = {(ϕ
i
,U
i
):i ∈ I} M Θ={θ
i
: i ∈ I}
M O i ∈ I
θ
i
: M → [0, 1]
θ
i
= {x ∈ M : θ(x) =0}
θ
i
⊂ ϕ
i
(U

i
)
x ∈ M V x i ∈ I
θ
i
=0 V

i∈I
θ
i
(x)=1, ∀x ∈ M
{supp θ
i
,i ∈ I}
x
O M
O
M k x ∈ M (ϕ
x
,U
x
) ∈O
x B
x
⊃ U
x
ϕ
−1
x
(x) B

x
= B(a, r)
g
x
: R
k
→ R
g
x
(u)=





e
−
1
r
2
−u−a
2
, u − a≤r
0 ,  u −a >r.
g
x
∈ C
∞
˜g
x

(y)=g
x
(ϕ
−1
x
(y)) y ∈ ϕ
x
(U
x
) ˜g
x
(y)=0
y ∈ ϕ
x
(U
x
) ˜g
x
M M
x
1
, ··· ,x
N
∈ M ϕ
x
1
(B
x
1
), ···ϕ

x
N
(B
x
N
) M θ
i
=
˜g
x
i
˜g
x
1
+ ···+˜g
x
N
{θ
i
: i =1, ···N }
M ϕ
x
(B
x
)
M

M O = {(ϕ
i
,U

i
):i ∈ I}
Θ={θ
i
: i ∈ I} M O

M
fdV =

i∈I

ϕ
i
(U
i
)
θ
i
fdV (=

i∈I

U
i
θ
i
f ◦ ϕ
i

det G

ϕ
i
).
M f
M ϕ(U)=ψ(W ) ψ = ϕ ◦ h
h
G
ψ
(w)=
t
Jh(w)G
ϕ
(h(w))Jh(w)

U
f ◦ ϕ

det G
ϕ
=

W
f ◦ ϕ ◦h|det Jh|

det G
ϕ
◦ h
=

W

f ◦ ψ

det
t
JhG
ϕ
◦ h det Jh =

W
f ◦ ψ

det G
ψ
.
Θ

= {θ

j
: j ∈ J} M

j

M
θ

j
f =

j


M
(

i
θ
i
)θ

j
f =

i,j

M
θ
i
θ

j
f =

i,j

M
θ

j
θ
i

f =

i

M
(

j
θ

j
)θ
i
f.

ϕ : I → R
n
,ϕ(t)=(x
1
(t), ··· ,x
n
(t)) C

C
fdl =

I
f ◦ ϕ ϕ

 =


I
f(ϕ(t))

(x

1
)
2
(t)+···+(x

n
)
2
(t)dt.
ϕ : U → R
3
,ϕ(u, v)=(x(u, v),y(u, v),z(u, v)) S

S
fdS =

U
f ◦ ϕ

EG − F
2
,
E = D
1

ϕ
2
= x

u
2
+ y

u
2
+ z

u
2
G = D
2
ϕ
2
= x

v
2
+ y

v
2
+ z

v
2

F = <D
1
ϕ, D
2
ϕ> = x

u
x

v
+ y

u
y

v
+ z

u
z

v
C x = a cos t, y = a sin t, z = bt, t ∈ [0,h]

C
dl =

h
0


a
2
sin
2
t + a
2
cos
2
t + b
2
dt = h

a
2
+ b
2
R
ϕ(φ, θ)=(R cos φ sin θ,R sin φ sin θ, R cos θ), (φ, θ) ∈ U =(0, 2π) × (0,π)
D
1
ϕ(φ, θ)=(−R sin φ sin θ,R cos φ sin θ, 0)
D
2
ϕ(φ, θ)=(R cos φ cos θ, R sin φ cos θ, −R sin θ)
E = R
2
sin
2
θ, F =0,G= R
2


S
dS =

U

EG − F
2
dφdθ =

2π
0

π
0
R
2
sin θdφdθ =4πR
2
R
ϕ(r, φ, θ)=(r cos φ sin θ, r sinφ sin θ, r cos θ), (r, φ, θ) ∈ U =(0,R) × (0, 2π) × (0,π)
D
1
ϕ(r, φ, θ)=(cosφ sin θ, sinφ sin θ, cos θ)
D
2
ϕ(r, φ, θ)=(−r sin φ sin θ, r cos φ sin θ, 0)
D
3
ϕ(r, φ, θ)=(r cos φ cos θ, r sinφ cos θ, −r sin θ)


B(0,R)
dV =

U
det(<D
i
ϕ, D
j
ϕ>)drdφdθ
=

R
0

2π
0

π
0







10 0
0 r
2

sin
2
θ 0
00r
2







drdφdθ =
4
3
πR
3