Math 670 : By 37.

Eqilatnl pkgs & con needs to syapbek geometryF-

Suppa instead we are mtnstd in udrstdy quiktnl plan in R3 ( Sheth . } B the not cannon polymer mold)

Again treaty the edgy as beets,

we want pohb ( I , .

,in ) EGY '

st . EE = s.She thank is

really 3 qwths 11 freeh come ),wire Why uht a ( h - 3) - din 't spare or

,

after noddy at hy Sol } ), a

12h - 6) -din 'l space .

up in the qualm .mx world,

will see a prdt f 3 . sphs : 8 ( i ) x . .× 5 ( D

,but this dos net tm at to

be a patilf pohtu approach.

Instead,

it tens art to be mh both to use the sympbdk gung f $ "

.

Def : A synpkck stake on a 2nd in H nfd is a closed,nndgnete 2- fm wail m)

.

He non . degroote means tht if 7 XETPM st . wplk 'D= 0 t YETPM ,th X=o ar

, quubty ,tht

W" "

= w ^. . ^ w

B no ihe - vanish if .

was

by : ¢ 4 w = dx , ndy ,t . .

. + dxnndyn .

hg. Any oriented Swfe.

The area form is the syybdnfom .

e.g. ,5 w / nee fm w sin by wplu , D= ( p,

uxv > for all u, vttps !

Df; wt (M, ,w

, ) ,(Mr

,we ) he Synphetl 2h nfds & let 4 : M, → Make a diffonuphsm

. y is a synpbdomwphym

if 4 To2

= w, .

qRh

Thm(Dahou×j . let IMY w ) be sypbk kit PEM .this card . Cht 4 : u → Mohd at p st .

4*w = ÷z,dxindyi

.

In other words,the B no interest local syfbdi footy .

Hamilton

ht (M,w ) be sypbdn & sppse Hi M→R is smooth .

Then dHER ' M ) ; by nmdgnpg ,F '

. XAE HIM) st .

lx.tw = whit,

' ) = DH.

let yj .

M→M be the tpwtrfmiff dittos grated by Xp ; ie,

Yeidm & dah÷ olli'

=X*

bmj. HE w = w,

so Hindus a family fsyykedomphms .

Prof : Ft Atw ; ↳,+w

Fdlx,+w + '

x,+dd= 0. as

def of Lieder . (ufns ¥1 °

I see Day 13 ) M÷Fmh

he uectrfdd XA Balled a Hamilton in veetr field with Hilton film H.

PrtmtE×gT Grow 15,do ndh) & tit It '

.5→R be the heghtfetm ; i.e.

, HIQH =h.then

dH=dh= ixnldondh) = do nah ( X,+;) ⇐> X ,+= to,

so the flow is yd 0,4=10-+44 ),rokk and the urklaxi

,whih food press A

.

In fat, the lot th f ay Hmiltmin rectrfdd always casing the associated Hani Amin (moth nods,int I cans are antis in level at ) :

adel HoledD= dHH#)=↳*wkA= HXA,×H=o

.

hx(HmiHuinMeoh=) : Casar IR" wl cards

.

(q , . , son , p. , - ,PD & w = Edgin 93

'

.Then far g Kilton vf . Xp

,

At = HttpHI Ban into are ⇒ { dh÷tH=s¥i

d¥nH) =. g÷ ,

( Hamilton gets)

indeed,if X#= ZITI B:

- ¥E£D,the

law = E lait dqj ' ' Pi ) = E [ ( ix , D8;) ndpj - dq; n 1(× ,+dp;D = E ( ftp.dpjts#g.dgD=dH

Now,asibr n =3 & a Pahk f mss m moves war a pthl Had Hy acne qlt) st .

matte = - ovb ).

Intone waned pi = m date & the enyyfnk A ( pi ) = knlptt VG ).let 1124 = T +1123 he the wrong phse spa .

Then

Newts send law Bquiraht to the kilt qualms :

doff - tmpi =¥E{dap÷=majq÷= . ¥,

= .s¥.

ad so he • let the ewy A Bansmdytephysilmtn .