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L fi, u
rt : auto si .
Mr : art . ⇐ I 5,3=1 a'= nat p 's.azW W
Us : asf . E- an's 5!,=÷ a "
- I; p "
Ts '
/'II ¥
=µp .
I
Definition: let M be an orientable manifold of diner. .
We can define an n - form W I W # no one all M .
Uv
defined on VI
.
'

dxi.ndxw.TW ⇐ h'ex 's 1197,11 dnindni . dm " so
- -
then we extend this . to Mr and then we confine ou IM .
W is positive over all d . It is ' called a volume form .
EXAMPLE : our 82 W = Dino dandy o IO En
w> o over all E- { North pole ]- { South pole } .
-
f. cus I
conned by a finite numb f dats I
- Def : partition f Unity : { Ei! : M- R are continuous faiths
on M .
I = 2-Ei-
e'is C- icp, -_ o 'f p f- Mi . . .%⇒-
iii) § Evp = I t pent
{ fw I fmfw⇐ ED = E f fue .
Hi
SinceGe
in Un ( Ot . ) Xi - ate ay XI - ate since
zti-X.tw/ht--cotezei9enUsCo--tn)Xi=tamEzbpXI--tanEzSinf 2- = Xitixi = tangent
in UµnUg : 2- + = z-
.
:* ÷÷, since
'
)
2- Joe E "I 1-6 In =
Za Xtc -Xa E = X, - i -Xu d 2- ad I = - zidxindxz .
ZE = xiexi - W+=£z¥dZ -
W ,
For simplicity . -2=5 5 -- ¥ z -- f. E --f .
W -
¥i÷÷⇒*.
I