positively oriented Vili open - James...
Transcript of positively oriented Vili open - James...
M oriented by And omw 7 and compact
L e h Mpositively oriented
Vili open cover at Vic Tito D
fili partition of unity subordinate to Vi
fi E f ricin
Independent it choice of Evil 43 and fi 3
If Mc Eo D then Vi3 MI Ef theopen
andf.mx foyafaadx'n add I traded di
Probled This definition is difficult to use in practice
ReasonI The fi M 112 are complicated to integrate
Ex A basic model of a funchei used to
construct partitions of unity is
f IR o l
1 Se tX E C1,17
Lo otherwise
II
Exercise check this is smooth at x It
We can avoid the hi if we can avoid overlaps among the
ritual
Fact suppose M is compact and oriented Suppose that
I 8µ Eo D M we I 1 on 0,17mi posihely oriented
and satisfyN
U ri l 6,17 Min
AND
pe Tito D n 416,151
tips t.it E2TLo lT7 fxeEoiI fIoflher
a 6,134717t
THEN
fu i ri3 Ino fi's orvis
since only the contributions of ti 240,17 are
redundant we just need to show they are all zero
ti'd fix dx In
The contribution of the part of the boundary
Xj 0 is
f Cx dad did tax
Icxi did o
Example M S
V i o I s c IR
t cos2ut shirt
I 1 on 0,1
positively oriented for one orientation
replace by F Ct cos2Mt shat if need be
8,16,13 s
o 8 Cto D h Vito 17 8,67
ti riot of altoD
We can now compute d for my rents
and choice of orientation
i S 1122 inclusion
i f x da t 2x dxZ
suppose s is oriented so that 8 is positively oriented
a 8 i Cx dx t 2x'dx2 It
i 8 x dx t 2x'd 5 It
s att dlashtt t 2usattdlsinatt dt
fo IT cos2ittsinatt t 41T cos att Lt
21T I t Cos4itt dt
21T
Example
V o 172 S c 1123
Sgt 1 7 cos2ft sin ITS Siri2kt hits cos ITS
Allows one to compute f d for any xc 17.52
Now we have derivative C F do 7 and
integrals ga Sma and so we need a
version of the Fundamental Thon of calculus
fab Itza dx fcb fca
pdetermined by f restricted
to 21 a b fa bl
Stokes'thin
f du f w
M 2M