← attire ✓ Jon : Rp ' = Real projective plane . Rp '=R%→v → Coordinate charts : R' = # ii. xD L fi , u rt : auto si . . - Ea . sina.tt a - - E . p.es , Mr : art . ⇐ I 5,3=1 a' = nat p 's .az W W Us : asf . E- an 's 5 ! ,=÷ a " - I ; p " - E Minus : a' =L P' =L ⇒ F- I ga÷l . ) ! ' ' Ts ' / ' II ¥ =µp . ÷ If = to → Riots not orientable . I Definition : let M be an orientable manifold of diner . . We can define an n form W I W # no one all M . Uv , ah ' ¥me form . ^h⇒dnhdn : Ian has > o : o - defined on VI
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←attire✓ Jon : Rp'
= Real projective plane .
Rp'=R%→v → Coordinate charts : R'= # ii. xD
L fi, u
rt : auto si.
.- Ea. sina.tt a -- E. p.es,
Mr : art . ⇐ I 5,3=1 a'= nat p 's.azW W
Us : asf . E- an's 5!,=÷ a"
- I; p"
-- E
Minus : a'=L P'=L ⇒ F- I ga÷l . ) !''
Ts'
/'II ¥
=µp.
÷ If = to → Riots not orientable .
I
Definition: let M be an orientable manifold of diner. .
We can define an n - form W I W # no one all M .
Uv
, ah' ¥me form
.^h⇒dnhdn: - Ian has > o:o-
defined on VI
I 1
on Ninh w = hoax 'D df÷, gn÷. - fg÷.
.
dat 's - din-
dnihdniin drain= E" " in dnlidnin di
'
←
on Vin Uz W -
- hcxcx 's) §n÷. . . Finn, E" " "
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 .