positively oriented Vili open - James Pascaleffjpascale.pages.math.illinois.edu/481sp20/notes/lecture26.pdf · Vili open cover at Vic TitoD fili partition of unity subordinate to

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