Série 13 Fichier

O=R2\ {(0, x)|x≤0}x
(r, θ)∈(0,∞)×(−π, π)
(x=rcos(θ),
y=rsin(θ),
O∂
∂r ,∂ ∂θ
∂
∂x ,∂ ∂y
f
∆(f) = ∂2
∂x2+∂2 ∂y2f .
∆f=h∂2
∂r2+1
r
∂
∂r +1
r2
∂2
∂θ2if
ω∈Ω2(R3)
ω=xdy ∧dz +ydz ∧dx +zdx ∧dy .
R3\ {(0,0, z)∀z∈R}ω
(x, y, z)=(rsin(θ) cos(φ), r sin(θ) sin(φ), r cos(θ)) .
dω
˜ω ω 1
(1, θ, φ)
U=S2\ {(0,0,1),(0,0,−1)}
˜ω
ω Sn
˜ω≡i?(ω),
i:S2→R3
c: [a, b]→M M α
M
Zc
α=Z[a,b] c∗α=Zb
a
αc(t)(c0(t))dt
α c
α f ∈C∞(M)α=df Rcα
c(a) = c(b)

α=xdy −ydx
x2+y2
R2\ {0}
R4x, y, z, t x, y, z
t E B
R3
(E) = −∂B
∂t (B)=0
(B) = ∂E
∂t (E)=0.
∗: Ω2(R4)→Ω2(R4)
∗(dx ∧dt) = dy ∧dz, ∗(dy ∧dt) = dz ∧dx, ∗(dz ∧dt) = dx ∧dy,
∗(dy ∧dz) = −dx ∧dt, ∗(dz ∧dx) = −dy ∧dt, ∗(dx ∧dy) = −dz ∧dt,
2F∈Ω2(R4)
F=Exdx ∧dt +Eydy ∧dt +Ezdz ∧dt +Bxdy ∧dz +Bydz ∧dx +Bzdx ∧dy.
dF = 0, d ? F = 0
Rn(x1, . . . , xn−1, xn=t)
Rnω∈Ωk(Rn)ω ω =α+dt ∧β
α=
n−1
X
i1,...,ik=1
ai1,...,ik(x, t)dxi1∧. . . ∧dxik=: aI(x, t)dxI∈Ωk(Rn)
β=
n−1
X
j1,...,jk−1
bj1,...,jk−1(x, t)dxj1∧. . . ∧dxjk−1=: bJ(x, t)dxJ∈Ωk−1(Rn),
I={i1, . . . , ik}⊂{1, . . . , n −1}J={j1, . . . , jk−1}⊂{1, . . . , n −1}α β
dt
dω =d0α+d00α−dt ∧d0β d0α d0β dα dβ d00β
dβ
ω dω = 0
d0β=∂aI
∂t (x, t)dxI.
Rnω∈Ωk(Rn)k ω ∈Ωk(Rn)

BRn
RnRn
U⊂Rn
Hp
DR(U)=0 p≥1Hp DR(U)p
U k
Rn