TechTalk

2011/09/01

oono@preferred.jp

+

Hodge-de Rham

3

4

y=x2

Rn, Cn=R2n

x2+y2=1

-1

0

1

1

y2 = x2(x+1)

()

V=Rn

Vn,k = {(v1, , vk) | vjV, vj

vj}

{(v1, , vk) | vjV, |vj|=1, vivj}

=> Vn,knk-k(k+1)/2

k=1 Vn,1 = (1) = Sn (n)

RPn(n) = (Rn+1\{0}) /

(Rn+1)

xy (xy) 0x = y

(1, 2, 3) (3, 6, 9)

i:RPn \ {xi = 0} Rn

i([x0, , xn]) = ( x0/xi, x1/xi, , ^xi/xi, ,xn/xi )

i

[0, ,1 , , n] = i(0, 1, , ^i, , n)

RPn\{xi = 0}RnRPnnRn

MnMp

pUp

RnVp

p:Up Vp

MnMp

pUp

RnVp

p:Up Vp

p, qMUpUq

qp-1|p(UpUq): q(UpUq) q(UpUq)

f:Rn Rm, f = (f1, f2, , fn)

Df(x) full rank

x0Rn y = f(x0)

f-1(y) = {xRn|f(x)=y}

f : Rnk R(k(k+1)/2)

=> Vn,k = f-1(yij)1

Stokes

Md = M

y3dx+z2dy+(x+2y)dz1

y3dxdy+z2dydz+(x+2y)dzdx2

y3dxdydz3

GreenGaussStokes

variety

32

()

RPn,

f-1(y)

SPCM http://www.ms.u-tokyo.ac.jp/~kawazumi/spcm.html