Home work 4 :
Ä: ÷: :÷
-
4. 4 .
Differential fernes
Det.IM
mfd .
① Aldiffeeulid) k form
an M is a (I )Tensor
we TI (M )
s .
t . wk) e MIM Ux EM .
② We wrihe h
"
(M) for the vector
space of K fernes
an M
,
while is also modul over IM .IR) -
H
'
, asubspoce ofTE IM ) .
Convention A.
°
(M) : = LM.IR) .
Note for k > dimm ) one two R"
(M) = 0 .
Reina .
AFM : =
Gut
"
M ETM ① . . ① TM is
u
weder subbuudle euer M .
k
• R "
(m ) = T ( 1
"
TM)
By Prop . 4.10
,
we can beneide o
k form
we RKLM) also
as a k linear
, dteuoliug uwp
w : TITM) x . . . x TLTM) → CNN.IR)
that is linear in dem
euvy over
( *
IM .IR) .
Def.4.12-supp.ae f : MAN is a
C-
ueop
hekweeu uefds .
If we R
"
IN) ,
then ÄW
,
Called the pull-
back
of w Viet ,
is a k form
an M
giveu by
Fw HK . .
,
s ) : =
wlfk)) (Ifs,
k),
. .
T.fs.la) )
K Si E T (TM ) .
If Sa .
. .
ihre TLTM )
,
für Lsa,
. .
, s
. ) (w
. f) (Tt .
! ". _
, Tfos.
)
www.swowsthotfkrisiudeedosmootntluserfidcdon M .
Rennen In
general an conpultpbocn
( I ) - Tensors Vie mops
.
We love a naturel ueop
'
Alt : TILM) → R
"
LM )
Attlo) ) : = Alt ×
) V-x.CM .
whre WERK In) Altlw ) = w .
Def.4.tl If we R "
LM ) und
gehe( M) then the
n
Wedge product way
E Akte (M) is
gun by
(way ) k ) : =
4^4 = Alt Lwx 4) VXEM .
For f E ROLM) =
( re , R ) and we R"
LM ) : fnw = fw .
By live.by we can abend this to a li war ueop
^ :
ULM ) x DTM ) → ein )
dim IM )
whee SEIN ) = ⑦ •
"
IM ) .
K=D
By Prep . 4.7 we how .
Prop.tn The vector
space
R
'
( M ) : =
!
RKLM ) is
an lossocide ,
neutral) groelend ↳
uuuhdie
algebra euer tue
trug OLM , IR) ( in
purem
our IR) ,
i e
.
it sdistie ,
① ⑤
of Pvp.
4.7 , since
poiutwize .
Papi her f : M N
ke a Amap
hehweeuuehds .
Then f
'
: R
"
IN) → h
"
LM ) exkuahs to a
morphin
f
'
: h
.
( N ) → ATM ) of ( uuitid)
godaddy.ba s :
i. e .
t
'
is linear ,
f-
'
1--1 ,
f-
'
( ng ) = für
ffand t
'
( RUND c MIN) .
Moreau
,
if g : NT P is ouoku ueop
hekueeu ufels ,
Leon
(g. f)
*
=
f :
;
Profis First adoiu fdl.ws freu See . 4.2 Since für G) =
f)IHK))
[Txf : IM →
Tf iudnces #f)
'
: MT N → NIM
and
tue se cand Chain also freu tue t T (g. f) =
Tgotf .
O
It IV. n ) is wert ,
then dui
{
^^ . . ndnin : 1 Ein < izc . . <
in Ek}
forma basis of 1
"
M =
AKTIV Xx EU .
If we R "
(M ) ,
then
<
Local Gordie .
WIE „
§,!
in- -
in
dein . . nduin
*
rassig
www.t(Un ) .
for w
" .. ;
"
E IV. R ) . Wi.
.
i.
= w (% , . . .
!) Recall
that we have on
operator :
d : IM .IR ) → RUM )
dlf ) = dt
We war eeteud this to on
operator d :b
"
IM) - h
""
IM ) .
for any K 30 .
Def.4.IM mfdl .
,
WER
"
(M) .
Then we dehne
v
→
(MIR) .dw : TLTM ) × . . × TITM )
dwls.is. .
.
.
, su ) : =
ÄH) si .
wl? . . . .it . . . is
. )
←
+
¥,
l-njtiwltsi.si] ,
so . . .
Ä ,
_ .
II. , . -
es.),Wwe
Ä uueeu , we am it this
entry
.
By lruearrhy , we war exteud it to a hier
uuep
d: SELM) →
SELM ) ,
called the externer derivate loncdiff.
fernes )
"
lemma4.17-dw.ch
" "
(m )
Prof . dw is
alternativ : Suppe
sj-s.sn
dw
(so , . .
Iss ,
Sitz , .
.
.hu ) = (1) sj
W
(so , II.
sj .
. . . .
. ! ) )
-
was
_ ⇐
{II"
Hiss. ) I
•tun,
} o =/
+
¥! w iii. so -
isis;
%:)
and
g) =
t
?
'
wksi.si/s.Y..sj.j, ;)
\
+
jijt 1
• dw is (M.IR) linear
in eon
entry ; by King alhoud.mg
ist is
sufficieut to Check it for one
entry :
f- EGMR) fl;
-
also,
].
. ) t Kit ! wls. " s
.
)
ndwAso
, 4
,
. .
, su ) =
fs.
. wlsn .
. .
.nu/t!tnIsi.Pfso,
§
.. . "
)
fürs)
-4;
-
f)so
+ Lwiw Htt . I. .
.
. ! )
+
¥ ?:<!
Ä
"
wlkiijI.is.
.
.
,
_
% ,
. .
)
=
f- ohwls . . .
.
.su ) .
TKm.4.IM mfd ,
d : SELM) → R
"(m)
satiestie,
the follow :
mg pro partie,
① For ft hm ,
IR) dt 4) = 9. f Vs ETLTM ) .
② For WE SHIM ) ,
qe
Re (M) we home
d (warf = dw
nette)
"
wndy
③ dod = 0
④ d is Loud : If we VLN) von : des ident
why on
San
open sunset UEM ,
then
dwlo is als
icdhiudcgzro .
⑤ If (Weu) is oaorr for Mond WERKIM ) ,
then
dw
) ,
=
¥÷!
(
Wiii
) ^ dünn . . sdüu =
=
{
①
wie. .
in
ii..
ein
,
;
d"
"
^ du '
n .
.
sduie
for
WII
{ wie . .
iv. dünn .
. ndu"
.
ii. -
ein
⑥ d is naturel : If f :
N -1M is a map
hekuoenufds,
tuenden .
Prof.
① ✓ dfls.
) =
! .
f
④ Ihnen Suppen
WIE
0 .
Then for obikoyveaeotiehds
↳ .
. .
ihre we love wls. .
. .
.sn//u=OauddDS
e)/ °
for an
dsineyvf .
so .
dwlu -
O
( h
per
www.t/wIu=y1u,thalw-z)Iu=0aedso dlwa)
Iv =
cdwlu dnlu
- O -
⑤ We first prom a
Special Core
of ② : WE 24M),
fuhr,
.
dlfw) Ko . .
.
, a.) =
Äh)
"
? : fwlso . .
.
. ! . .
. "
.
)
-
-
+
÷!
-
Ä"
fwfsii.it.
.
.
I .
. .
.
.
)
dtki)
=
ÄH) " f) who , . .
is . . .
) t t si
.
wlso . . %
, . .
, se )
-
+
f
!!
Ä
"
wltsiij] . . .
I ,
%
) .
=)
dltw ) - tdw
=
df 1W ( since wo
alternative) -
India d ( dünn . .
ndüu ) = O
It follow , freu tue oeefrn.hu ,
sunce insertion of ↳ 1
Gordian
vf .
in d Ldu) . .
du in ) Im ts Zero .
this is hee con
Luce du "
s . . 1 du in is Cousteau upon iuzutieu
of
K Load vf
.
und tfi ,
ftp.I-0 →
dl
!!!
.
.
i.
dü ) ) -
¥4!
^ dünn .
.
nach
=
Edwin duion . . .
ndu "
.
in --
Liu , in
Thm.4.IM mfd .
,
d : h
"
IM ) → nun ( M) externer derivative .
Then the follow in
g
hdds ,
① df 4) =
g. f Vse TLTM ) .
② For we R
"
( M) ,
y
E NI M) we love
dlwnz) -
dwnqtG)
Indy
③ d. D= 0
④ d is a Local
Operator :
If WE R "
LM ) is
Vauishieg ideuwaedlgbu an
open subset WEM
,
Hau
dwluvaiinesidaetiudly ⑤
If ( Un) is a now for M and we SELM) ,
then
dwlu =
÷!!!ii. i ! ^ du "
s .
. rdu in
Wwe
Wto
=
{ Wir . .
iv.
du "
s . . 1 du in .
Inc -
rein
⑥ d is a natural operator : If f : N - M is more
between
mhds .
,
tun dfaw = frdw .
Prof
① v
, ④ V. ⑤ ✓
② By ④ we can
proud this in Loud coordiuotes :
"
Iüi!! -
i.
du "
s - ndui"
II !!!
. .
jedu) -
sdie
I Lin
.
. .
. in )
dut =
dünn . .
n du in
] -
Lisa , . .
, je )
wny1u_-fghE2gduIed_uI.dlwIRgI_-@wIIRstwIdwzs.o[ ¥;D) eidutzdu ] +
¥,
4de
du '
.
=
§ dw
,
ndu In
§ 4 ]
du ]
t l s
)
"
¥4du In
{dpa did
-
-
=
ldw-t-d.nl ) lo .
③ We war pure
tun in 6cal coardinedes :
We
ulreody know d ( dein . .
. duiu ) = O
Heine
② and ④ imply :
✓
WIE ¥. . ;!
?
wie -
-
in
^ Huhn .
.
cdu in = 0
It nennen , hosuowtuud auf - o
for FE ( Mdk) .
dleff) ( s . . ) -
s.
.
dfl) sidfls.
) -
dfkso.si)
=
So .
(4. f) -
%
.
(g.
.
f) -
Eso
.si/.f
⑥ GE (M
, IR) f : N → M
Fg =
got and Lance dfg) ( s ) =
s .
(got ) =
=
Hfg ) .
g
=
dgltfs )
=
# dg ) (s) .
H)
S up
non I Ku ) is n hort for M sie .
f IN) n U # ¢ .
For
wlü .?! u " we
µ
Pvp. 4.15
wow
(f)
|
=
{ Fw
.
.
.
..
"
fedn "
i .
. stduik Er
f-Yo )
ist -
in
(* ) dlfrw". .
;) fedwi.
.
and
dlfiiif.feduij-d-wff.yujkfedwi.irfeduinn . -
rlidui,
=
wto ) .
D
4. 5 lie derivat ives
Suppen
f : M → N Load diffeau . kehren ufds .
Tef . IM →
THIN is an isomorphen .
Tief f. f)
'
: T
;#
N - TIM
www.If
(Kf)
"
)
'
Hit
)"
:
Tintin Def.h.jo/eTpPlN)
,
f. M → N bud aiken .
The
punch of d with
respect
to f is the (f) -
tensor
¥9 ET ! (m) an M
giveu by
Folk) lw: . .
. wr.sn . . .
.se/:=dltHIlifTw;...Eit)Ip,
for Wi E TIM ,
s
,
c. IM .
It ! I - ,
If ! )
( Evident G ,
this is a (Y) - heuser an
M) .
Applied to the Glow of a neuer fields we
get :
Def.h.IS E TLTM) , § C- T
! ( M
) ,
M mtd .
Then
the hie
derivativefs.IO ETF IM) oft daegs is
gun by
2,44 ) :
=
! 1¥.IT/olx)V-xeMIM
'
④ TIM
⑦ 9
.
-
Sonnengott SETLTM) .
(f. FLI)k )
.
f- E Rolm) =
( re , R ) ,
tun Lsf 4) =
#¥ f k )
L
,
f--
dfls )
=
Txfsk) =
dfls) (e ).
•
RE TITM) :
42 =
Is
)
•
de TIM) , 4 c- TIM) :
Lg (01×04)=40×4+100×14 .
( This fellows freu f-
'
(100×4) =
fed # fey for any Local
außen .
f of M und freu
bilinaer.by Q :(01,4) QY .
)
•
For WE R "
( M) ,
µ E Re ( re)
Lslwn µ
) =
2µs µ t wahre
Moreau note that for MET!(M) und w ! .
.
,
wie e- TITAN)
und Ss ,
.
.
es
,
ETLTM) the full louhuaiae :
④ ① hi . .
QWP ⑦
↳ ① . . ⑦
gg 011W! . .
, w! ! ,
. .
, <
g) .
is linear und nannten wir
pullbocksbylocddiftean.fr
-4) Clf
'
aua .
.
§ ) ) CHI
① Iiia . . at;) )
→
= f ( WI.
. .
de )) 01W?. -e)of
-
L
,
① W
'
① . ①
Sg ) =
Led ⑤ wir _ .
①
se
+
%¥ -
④
Lywi ⑦ . .
QS
,
Q .
. ②
Se
(
"
¥.
.
QQ w
'
① .
. ① WPQ -
⑦
Lqsj ⑦ "
⑦
4 .
äüd
Kind) lw: . .
.
. ;) =
Lrddlw: .
.
.se )) -
{dtüieü.
.se)
-
-
-
-
-
_
{dlw? iw
! das;)
.
heute _
-
If 10 E R
"
(M ) tun
yieuds :
Film ist
-
zollen .is
.
) -
¥44.
.
. Ä?. )
Operator on differential focus :
•
d : h "
(M) → 1km ( M)
.
.
Lg : sülm ) → RKIM)
.
i
?
: SELM) → Rk -
^
(M) 3 ETLTM) .
W ,-7
igw = W
( s , _ . . .
)
insertion
operator .
Det Aigner (SE, n ) of degree r
is a linear
uuop D : SELM) >
"
(M) si . Dlwn g) =
=
DIW) n
qt
HI
"
wa
Dz Kw e- R "
IM) ,
gehe ( M ) .
-
Prophet
① d is a
grund derivation
of degree 1
② For s E TLTM )
, Lg is
grodud derivaten
of degree O .
③ For } E TITM) , is is a
groelend deine Hen
of degree -
1 .
Marlow , II Ds and Dz are
groduddeivdeus of degree r
,
resp .
rz
,
then ID . .
Da] : =
Di Dz -
L-
1)
" "
Du D
,
is o
groelend
derivation
of degree rntrz .
Prod ① und ② ✓ the rest will ↳ oliscusseed in hutoriel
W I Leave a on exercise .
Prop.4.23-supg.ae D is o
grad deiveta of degree ref (riley, )
① D is a Loud operator : If WE R"
LM) valides
rdeulrudly an
on
open subset
of M
,
then so does Dw .
② It Ä is awww
groded deine Hen
of degree r Sun Kid Dlf
) =
Dlf ) and ⑦ (df) =D Idf ) tf ECR).
then D= 5 .
PI .
Exercise / Tuterid .
Prop.h.24-Mmtd.ih.ME t HM ) '
⑥ [
is.iq] =
① [ d. d) =
d. L
,
L;D
= 0
istoigtiyois
-
O
② Id
,
i
,
] = doi
,
t i
>
od =
Lg
③ [ d. d ] =
Zdz =D PRI Exercise)
④ tds.ly] =L
, .dz -
Lg-2
,
=L
,
.gg .
thhaid .
⑤ Eds ,
i
) = h
, .iq
-
iyod,
=
ihn]
Reinen d is there here unique groelend dein
of degree 1 s .
l .
Df = df and D (df ) =D .
L
,
is the
unique grodud dein .
of →,
- O S.
L
2 ,
f = dfls) =
s .
f
2. (df ) =
dlsf) .
5.lu/-egrationoum-n
Recall tho transformation formula for multiple integrals :
Suppe UEIR"
is on
open
Sunset and § . U →
0/10)
a olitteau . between
open
kebsets of IR"
.
het f :
Ötv) → IR he a G- fctwihloupodsurr.at
↳¥,
=
{ ( too ) Idet Dot (*) .
↳oks like the transformation
of n torus
on on man ! fdds
of dimm :
Suppe M
mfd .
,
deine ( M) -
h
,
we R
"
LM )
and LU , n) is n
Chart of M :
Then
wtu-wi.eu dein . . ndu "
w
!. .
wtf ,
- jun)
.
E
CO, R) .
Suppe v :
U → v LU) is on .
der Chart for Mae U .
Then
WI
"
=
Wi. . n
dir . . ndv "
w!.
nluüly) ) =
wlüily) ) (In
-
'
es ,
. .
.
Tyeiiea )
wi.nl#nEg)--wlv-ily)LEv1es ,
. .
,
Iv een
)
Y C- u LU) ZE v ( U) .
Now let OI : ulo) → v (U ) he ¢ : - v. u '
= , v. ! d = ü '
and
Tyü
'
=
Toll)
-
to
Id
-
w:.
. Lily)) wlüly))
( Tyne. .
. . . Inne.
) 4)EilHyde,
. . .
=
wlu"
(y ) ) (Tg) :
Typen , . . .
,
Taiji!
Idk )
"
Iden)
= det (
Dg) wlülg) ) ( Thünen .
. .
.to/yjien)=detlDgo)wi.ndly) ) .
We ! ulwtly) ) = det (Dyd) w
!. .
dolly) ) .
H O
Hye ulu )
If we
oss neue U hohe wonne und und Lance so is
'
n (U),
then
sign of der ( by 4) is
eihwdwoy, peilte er
negative .
(*)
Says that the integral over the Loud werdende
Express) on
of here for .
wn . .
"
( Ws -
now
'
: ulot →
R )
Is up
to a
Sign Independent of cuoice
of the Eat .