Math 479 :DiProf : If fi UER2 lR B a different film ,

then the graph

¢x, y, flay)) : IX , DEU} B a rgkswke .

Prof : Paawtnehg Ilxy ) = lxiyttxiy)).

thus Dato,bjeelme

,& banks inverse IYY , yftxidtlxiy ) .

Macao,

dean,

= ( pbxit,) ↳ tif M

.

columns.

*

So of case graphs gme us ntnitef mayexapbs f rgdr sorts :

y= smkdsmly )

gitlxid:* 42

€ #team ' ftp.A.uaIn ft

,the slyhtf amogthifsthtayrglrsrte D loaf a graph faflm .

Inverse Fmdmthm : At F : HER" IR " be a difhdork mp & sppsetht at § Eu the toll donate# ; pi Rn

is an romp him lie, msingwmtix) .

Tin 7 nhl V of § & W f Ftp) st .F-li W V , } a well . defund smooth

mp .

he

:HP#÷f÷÷¥fp=[ ftp)] is an ionuphm iff ftplto .

pqj . ht

EcIN be a rglr safe & ht

FEE.then J nld V f § so Hit V n E is the graph of . diff

.

the film

of the from zflx , y) or ytglx ,⇒ or x= Hy,z).

Proof :htp

EE.

Same E 13 a 7h Satu,the B a pannetiath I : UER2 lR3 her § & won write

If u ,ul = ( xkiul

, yk , u),zlu , d)

.

Now,by

�3� vekmuthtxuxxv t & so at lest me f the coordnts D non oeo .

Assume its the z - cordite ↳ . ne

we anist mdity the follow y approprhtf it itsthe × or y) , which is :

EE⇐¥t*Now

,upjatfmtheiyefx to the xy - plane :

zn

✓

:JYB I z

@"to÷#,

L

By anposith , we get a mp toxin 112,2o€ $"×µdttoxhdtdxtoi : )EEhf¥÷)=lF¥ KEE)in#.

Bt he hwth } B a msiywmtnx , so he xr

an meth Iwse Function than to see tht Z a nhd Wf TH ) & a nhl Vf IYP) st. the is . different inverse

mp Gtoxlt : W V.

Then I daintht , near § , E B the graph of the tnth Zoltoxlt, namely { ( x ,y , lzoltoxltkx ' 4) : lay) EW }.

Of case ,he need to hwthtpnbm the gzhralf reports on the swke : for knew

,mhw

Holton) ' ') ( xiy ) D on Z,since ltoxl

' '

lay) EU .Bit new we expdtht port out as

( × ( htoxtllxiy )),y( ltoxtllx , H)

,zlttoxtllx ' H) )

.

Bt new # fxoltoxltlxiy )) = ltoidoltoxl' '

1x ,y1= 1× , y),

so re hw x( koxttlxiyl ) =x & y( ltoxltlxn )) =Y,so ( x , y, Holtoxl

' ') lay)) EZ, as needed

.%