MTHGHIDGT

Def: Span Mm & N"

are mnitlds & fi M → N guth . Far pear , the diffmkl f ftp.stle

Liner up dfp: TPM → Tap, N grin g :

For vetpm ,choose a :L - E

,{ ) ' M st

. did =p ,a lot v. ht B.- for 4 define

dfplut BY) .

In cooks :

#eIMiE¥n÷kErHe#@,¥t'oalt ) = kith. ,xmHDf€÷#R"

1Yi , . . , Yn) = 4 ' ' of 06 ( x. , . , xD = ( y , Ix , , ..

,xm )

,, .

,ynlx , , . ,

and)Then

data , P'' D= I -25¥ xi 's , . . .

, I÷xi ' os ) =R÷;) kilos)nxm9matrix

£m -element column vector

" dfo' ' ' '

v"

Def " let f'

.Mm →v

" he smooth .A paint pem is a uitzl point ff if Afp : TPM - Tap, N is not swjectk ; then

ftp.acritilwheff . A point GEN utih is at a oiklphe Barger phe

Cliche Exaple

÷i#¥E±EeH,

⇐ :

f÷E±**± p.

.¥

Thin : Tf qe N B rghr , then ftlq) is a smooth sbmnfblf M f dime in m - n.

Prof : Inwse Fmckm 1hm. %.

Ft . Dpi f : MatnxnHR) → {symmetric nxn metres } by FCAHAAT

.

then df is always sujectk & f ' ' ( I ) = OCD is a

smooth sbmitld f dines in n

'- H±z" =n¥

.

F¥ : Remember or np misdeed" → Rd gin J nlei , .is ) = feet.

then a Era is a rgbuhe when n 23 so

n

the pdygu spare ii' (8) , ] a smith submitted of coding in d

.

Vatr Fields-

:

Df: A vector field X on a smooth nfd M is a smath satin f the tgutbkk TM ; ie,

a smooth mp x : Mhm

Gian how nine detml tot metro,this has a vectw field is jst a kw drink an Colm )

.

Note : If 4 : UEIR"

→ M B a and . oht at PEM,

then

X (p) = E ailp) Zi ,the ah ai : Mk B swth & { Fxi} is the by f Tpm associated 4 In, 4)

.

. As a mp CTM) → CTM ),

a met field ads as

HAIR = E a ill ¥lpuhth notice rally B a smith fth on M

.

Df: If X & Y one smooth met fields on M,the Lie breket f X RY is a met Fedd [ × , 'D defined by

[ X , y ] Ht : = XIY f) - Ylx f)

In cards,

, if X= E aitx ; & Y = E bi ki

,then

[ x ,

' Df = XHH - Ykf) = XIE bi ¥E ) - HE ai ¥i ) = Ina;l5b÷. T÷. this'¥×,

. ) -Fybj l Tx÷¥a.

- ai d¥E

= Ela ;T÷j - bj T÷;)T÷. + Eaib ;k¥¥tgTx¥

= (E. ajT÷; - bj f÷j)s÷÷local card express in far [ X, D

Prof : If X , YZ are smooth new folds a M & a,be R

,fig E CID

,then

a [ X ,Y ] = - [ Y, × ] Lmkconmutiig )

@ [ axtb 4, Z ) = a [ X , ⇒ + KY,⇒ ( limit )

�3� [ [ X , Y ] , ⇒ t [ [ y, ⇒ ,X ) + Kzx ]

, D= o ( Jacobi idntig )�4� [ fx , g Y] = fg [ Xi Y ] tf (Xg ) Y - g( Yf )X