Giáo trình giải tich 3 part 5 pdf

M ⊂ R
n
M
F : M → R
n
,F(x)=(F
1
(x), ··· ,F
n
(x))
F (x) x
C ⊂ R
3
τ
τ : C → R
3
C τ(x) C
x x ∈ C
✿
✛
❍
❍
❍
❍
❍❨
t
τ (x)
x
C
ϕ(t)=(cost, sin t),t ∈ (0, 2π)
ϕ


(t)=(−sin t, cos t)
S ⊂ R
3
S
S N : S → R
3
N(x) ⊥ T
x
S, ∀x ∈ S
S
N
s
x
N (x)
❇
❇
❇
❇
❇▼
✲✒




S
ϕ(φ, θ)=(cosφ sin θ, sin φ sin θ, cos θ), (φ, θ) ∈ (0, 2π) × (0,π).
∂ϕ
∂φ
=(−sin φ sin θ,cos φ sin θ, 0),

∂ϕ
∂θ
=(−cos φ cos θ, sin φ cos θ, −sin θ)
N =
∂ϕ
∂φ
×
∂ϕ
∂θ
R
R
2
V k R
(v
1
, ··· ,v
k
) (w
1
, ··· ,w
k
) V
P =(p
ij
)
k×k
w
j
=


i
p
ij
v
i
(v
1
, ··· ,v
k
) (w
1
, ··· ,w
k
) det P>0
(v
1
, ··· ,v
k
) (w
1
, ··· ,w
k
) det P<0
V
(v
1
, ··· ,v
k
)
[v

1
, ··· ,v
k
] −[v
1
, ··· ,v
k
]
V
µ µ =[v
1
, ··· ,v
k
]
R
k
R R
2
R
3
✲
e
1
✲
e
1
✻
e
2
✩

✛
✲
e
1
✻
e
3


✒
e
2
✛
R
1
, R
2
, R
3
M ⊂ R
n
k
µ = {µ
x
: µ
x
T
x
M,x ∈ M}
a ∈ M

(ϕ, U) a [D
1
ϕ(u), ··· ,D
k
ϕ(u)] = µ
ϕ(u)
u ∈ U
M
M
M
µ M µ
µ
R
3
N = D
1
ϕ×D
2
ϕ
M ∂M M ∂M
O M µ
(ϕ, U) ∈O i : R
k−1
→ R
k
,i(u
1
, ··· ,u
k−1
)=(u

1
, ··· ,u
k−1
, 0)
{(ϕ ◦ i, i
−1
(U)) : (ϕ, U) ∈O,U

H
k
= ∅} ∂M
x ∈ ∂M (ϕ, U) ∈O x

x
=[D
1
ϕ(u), ··· ,D
k−1
ϕ(u)],x= ϕ(u).

x
(ϕ, U) ∈O ∂M
 = {
x
: x = ϕ(u) ∈ ∂M,(ϕ, U) ∈O} ∂M
(ϕ, U), (ψ, W) ∈O x ψ = ϕ ◦h det h

> 0
k h
h

k
(w
1
, ··· ,w
k−1
, 0) = 0, va h
k
(w
1
, ··· ,w
k−1
,w
k
) > 0khiw
k
> 0.
w =(w
1
, ··· ,w
k−1
, 0) h

(w)
(D
1
h
k
(w)=0 ··· D
k−1
h

k
(w)=0 D
k
h
k
(w) > 0).
det h

(w)=det(h ◦ i)

(w
1
, ··· ,w
k−1
)D
k
h
k
(w) > 0
det(h ◦ i)

(w
1
, ··· ,w
k−1
) > 0 (h ◦ i)

(w)
D
1

ϕ(u), ··· ,D
k−1
ϕ(u) D
1
ψ(w), ··· ,D
k−1
ψ(w) T
x
∂M
(x = ψ(w)=ϕ(u))
[D
1
ψ(w), ··· ,D
k−1
ψ(w)] = [D
1
ϕ(u), ··· ,D
k−1
ϕ(u)].

x
µ
x
M µ ∂M
∂µ
x ∈ ∂M (ϕ, U) x M µ
µ
x
=[D
1

ϕ(u), ··· ,D
k
ϕ(u)]
∂µ
x
=(−1)
k
[D
1
ϕ(u), ··· ,D
k−1
ϕ(u)].
(−1)
k
ϕ µ x = ϕ(u) T
x
∂M
T
x
M v ∈ T
x
M \T
x
∂M
v
M v ∈ ϕ

(u)(H
k
+

)
v
M
∂M
v
1
, ··· ,v
k−1
T
x
∂M v ∈ T
x
M M
µ =[v
1
, ··· ,v
k−1
,v]
∂µ
x
=(−1)
k
[v
1
, ··· ,v
k−1
]
s
x
✲



✒
v
✒
✛
✠








H
k
∂H
k
= R
k−1
× 0
R
k−1
k
k
M R
2
R
3

N ∂M
M R
3
∂M
F =(F
1
,F
2
,F
3
) R
3
• v ∈ R
3
x W
F
(x)(v)=<F(x),v >
F (x) v
W
F
= F
1
dx
1
+ F
2
dx
2
+ F
3

dx
3
C R
3
F C W
F
C

C
W
F
=

C
F
1
dx
1
+ F
2
dx
2
+ F
3
dx
3
.
• v
1
,v

2
∈ R
3
x ω
F
(x)(v
1
,v
2
)=<F(x),v
1
× v
2
>
F (x) ∆S v
1
,v
2
ω
F
= F
1
dx
2
∧ dx
3
+ F
2
dx
3

∧ dx
1
+ F
3
dx
1
∧ dx
2
.
S R
3
F
S ω
F
S

S
ω
F
=

S
F
1
dx
2
∧ dx
3
+ F
2

dx
3
∧ dx
1
+ F
3
dx
1
∧ dx
2
U R
k
ω ∈ Ω
k
(U)
ω = f(u)du
1
∧···∧du
k

U
ω =

U
f(u)du
1
∧···∧du
k
=


U
f(u)du
1
···du
k
.
M k µ
R
n
ω ∈ Ω
k
(V ) V M
ω M

M
ω
M = ϕ(U) (ϕ, U) µ

M
ω =

U
ϕ
∗
ω.
M O = {(ϕ
i
,U
i
):i ∈ I}

µ Θ={θ
i
: i ∈ I}
M O

M
ω =

i∈I

ϕ
i
(U
i
)
θ
i
ω

=

i∈I

U
i
ϕ
∗
i
(θ
i

ω)

,
M ω
k =1

M

i
F
i
dx
i
k =2

M

i<j
F
ij
dx
i
∧ dx
j
µ
(ϕ, U) (ψ, W ) µ
ψ = ϕ ◦ h h det Jh > 0 ϕ
∗
ω = f(u)du
1

∧···∧du
k
h
∗
(f(u)du
1
∧···∧du
k
)=h
∗
ϕ
∗
ω =(ϕ ◦ h)
∗
ω = ψ
∗
ω.

U
ϕ
∗
ω =

U
f =

W
f ◦◦h det Jh =

W

h
∗
(f(u)du
1
∧···∧du
k
)=

W
ψ
∗
ω.
Θ

= {θ

j
: j ∈ J} M

j

M
θ

j
ω =

j

M

(

i
θ
i
)θ

j
ω =

i,j

M
θ
i
θ

j
ω =

i,j

M
θ

j
θ
i
ω =


i

M
(

j
θ

j
)θ
i
ω

i

M
θ
i
ω.
M k µ V

M
:Ω
k
(V ) → R

M
ω = −

−M

ω −M M −µ

U
i
ϕ
∗
i
h(u
1
, ··· ,u
k
)=(−u
1
, ··· ,u
k
) det h

= −1
(ϕ, U) µ (ϕ ◦h, h
−1
(U))
−µ Θ

−M
ω =

θ∈Θ

h
−1

(U)
(ϕ ◦ h)
∗
θω =

θ∈Θ
(−

U
ϕ
∗
θω)=−

M
ω.
C ϕ : I → R
n

C

i
F
i
dx
i
=

I

i

F
i
◦ ϕdϕ
i
=

I
(

i
F
i
◦ ϕ(t)ϕ

i
(t))dt.

x
2
+y
2
=1
ydx − xdy
x
2
+ y
2
=

2π

0
sin td(cos t) − cos td(sin t)
cos
2
t +sin
2
t
= −

2π
0
dt = −2π.
S

S
xdy ∧dz =

[0,2π]×[0,π]
cos φ sin θd(sin φ sin θ) ∧ d(cos θ)
=

[0,2π]×[0,π]
cos φ sin θ(cos φ sin θdφ +sinφ cos θdθ) ∧ d(−sin θdθ)
=

[0,2π]×[0,π]
−cos
2
φ sin
3

θdφ ∧ dθ =?
F =(P, Q,R) C
1
V ⊂ R
3
C ⊂ V
T =(cosα, cos β,cos γ)

C
Pdx+ Qdy + Rdz =

C
<F,T >dl=

C
(P cos α + Q cos β + R cos γ)dl.
S ⊂ V N =
(cos α, cos β,cos γ)

S
Pdy∧dz+Qdz∧dx+Rdx∧dy =

S
<F,N>dS=

S
(P cos α+Q cos β+R cos γ)dS.
v ∈ R
3
T v

W
F
(v)=<F,v>
W
F
= Pdx+ Qdy + Rdz.
W
F
(v)=<F,T >v =<F,T>dl(v).
C R
3
T

C
W
F
=

C
<F,T >dl.
v
1
,v
2
∈ R
3
N v
1
× v
2

ω
F
(v
1
,v
2
)=<F,v
1
× v
2
>
ω
F
= Pdy ∧ dz + Qdz ∧dx + Rdx ∧ dy.
ω
F
(v
1
,v
2
)=<F,N>v
1
× v
2
 =<F,N>dS(v
1
,v
2
)
S N


S
ω
F
=

<F,N>dS.
M k
V ⊂ R
n
∂M

M
dω =

∂M
ω, ∀ω ∈ Ω
k−1
(V ).
M µ ∂µ ∂M
{(ϕ
i
,U
i
):i ∈ I} µ M
U
i
A
i
: R

k−1
→ R
k
, (u
1
, ··· ,u
k−1
)=(u
1
, ··· ,u
k−1
, 0) {(ϕ
i
◦ ,
−1
(U
i
)) :
i ∈ I

} I

= {i ∈ I : U
i
∩∂H
k
= ∅} ∂M (−1)
k
∂µ
{θ

i
: i ∈ I}

M
dω =

M
d(

i∈I
θ
i
ω)=

i∈I

ϕ
i
(U
i
∩H
k
)
dθ
i
ω.

∂M
ω =


∂M
(

i∈I

θ
i
ω)=

i∈I


ϕ
i
(U
i
∩∂H
k
)
θ
i
ω.
ϕ = ϕ
i
,U = U
i
,A = A
i
=[α
1

,β
1
] ×···×[α
k
,β
k
]
U ∩ ∂H
k
= ∅ i ∈ I \ I


ϕ(U)
dω =0
U ∩ ∂H
k
= ∅ i ∈ I


ϕ(U∩H
k
)
dω =(−1)
k

ϕ(U∩∂H
k
)
ω.
ϕ

∗
ω =
k

j=1
a
j
(u
1
, ··· ,u
k
)du
1
∧···∧

du
j
∧···∧du
k
∈ Ω
k−1
(U)
ϕ
∗
ω ∈ Ω
k−1
(A) a
j
(u)=0 u ∈ U
(ϕ ◦ )

∗
ω = a
k
(u
1
, ··· ,u
k−1
, 0)du
1
∧···∧du
k−1
.
ϕ
∗
(dω)=
k

j=1
da
j
∧ du
1
∧···

du
j
···∧du
k
=
k


j=1
(−1)
j−1
∂a
j
∂u
j
du
1
∧···∧du
k
.

ϕ(U)
dω =

U
ϕ
∗
(dω)=

A
k

j=1
(−1)
j−1
∂a
j

∂u
j
du
1
∧···∧du
k
=

j


l=j
[α
l
,β
l
]
(a
j
(··· ,β
j
, ···) − a
j
(··· ,α
j
, ···))du
1
···

du

j
···du
k
=0.
(u
1
, ··· ,β
j
, ··· ,u
k
) (u
1
, ··· ,α
j
, ··· ,u
k
) ∈ U a
j

ϕ(U∩H
k
)
dω =

U∩H
k
k

j=1
(−1)

j−1
∂a
j
∂u
j
du
1
∧···∧du
k
=

A∩H
k
k

j=1
(−1)
j−1
∂a
j
∂u
j
du
1
∧···∧du
k
=

j
(−1)

j−1
(

[α
1
,β
1
]×···×[0,β
k
]
∂a
j
∂u
j
du
1
∧···∧du
k
).
j = k,

[α
j
,β
j
]
∂a
j
∂u
j

du
j
= a
j
(u
1
, ··· ,β
j
, ··· ,u
k
) − a
j
(u
1
, ··· ,α
j
, ··· ,u
k
)=0
j = k,

[0,β
k
]
∂a
k
∂u
k
du
k

= a
k
(u
1
, ··· ,β
k
) − a
k
(u
1
, ··· , 0) = −a
k
(u
1
, ··· , 0)

ϕ(U∩H
k
)
dω =(−1)
k


j=k
[α
j
,β
j
]
a

k
(u
1
, ··· , 0)du
1
···du
k−1
.

ϕ(U∩∂H
k
)
ω =

A∩R
k−1
×0
a
k
(u
1
, ··· , 0)du
1
···du
k−1
.
M M
R ω(x)=xdx
V R
n

F : V → R C
1
ϕ :[a, b] → V

ϕ([a,b])
dF = F (ϕ(b)) − F (ϕ(a)).
D ⊂ R
2
C = ∂D
P, Q C
1
D

D
(
∂Q
∂x
−
∂P
∂y
)dxdy =

C
Pdx+ Qdy.
S ⊂ R
3
N
∂S = C P, Q,R
C
1

S

S
(
∂Q
∂x
−
∂P
∂y
)dx∧dy+(
∂R
∂y
−
∂Q
∂z
)dy∧dz+(
∂P
∂z
−
∂R
∂x
)dz∧dx =

C
Pdx+Qdy +Rdz.
V ⊂ R
3
∂V = S
P, Q,R C
1

V

V
(
∂P
∂x
+
∂Q
∂y
+
∂R
∂z
)dxdydz =

S
Pdy ∧ dz + Qdz ∧dx + Rdx ∧ dy.
D C R
2

D
dxdy =

C
xdy = −

C
ydx =
1
2


C
(xdy −ydx).
V S R
3

V
dxdydz =

S
xdy ∧dz =

S
ydz ∧ dx =

S
zdx ∧dy
=
1
3
(

S
xdy ∧dz +

S
ydz ∧ dx +

S
zdx ∧dy)
U R

n
ω =
n

i=1
a
i
dx
i
∈ Ω
1
(U)
ω f ∈ C
1
(U) df = ω
ω dω =0
∂a
i
∂x
i
=
∂a
i
∂x
j
i, j

C
ω =0 C ⊂ U
R

2
\{0}
xdy −ydx
x
2
+ y
2
2π =0
R
n
\{0}
n

i=1
(−1)
i
x
i
x
n/2
dx
1
∧···

dx
i
···∧dx
n
.


dx
i
dx
i
R
3
e
1
,e
2
,e
3
U
R
3
∇ =
∂
∂x
1
e
1
+
∂
∂x
2
e
2
+
∂
∂x

3
e
3
f : U → R
f
grad f = ∇f =
∂f
∂x
1
e
1
+
∂f
∂x
2
e
2
+
∂f
∂x
3
e
3
.
F = F
1
e
1
+ F
2

e
2
+ F
3
e
3
U F
rot F = ∇×F =








e
1
e
2
e
3
∂
∂x
1
∂
∂x
2
∂
∂x

3
F
1
F
2
F
3








F
div F =< ∇,F >=
∂F
1
∂x
1
+
∂F
2
∂x
2
+
∂F
3
∂x

3
.
h
1
: X(U) → Ω
1
(U),h
2
(F
1
e
1
+ F
2
e
2
+ F
3
e
3
)=F
1
dx
1
+ F
2
dx
2
+ F
3

dx
3
.
h
2
: X(U) → Ω
2
(U),h
2
(F
1
e
1
+F
2
e
2
+F
3
e
3
)=F
1
dx
2
∧dx
3
+F
2
dx

3
∧dx
1
+F
3
dx
1
∧dx
2
.
h
3
: C
∞
(U) → Ω
3
(U),h
3
(f)=fdx
1
∧ dx
2
∧ dx
3
.