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Avn - uniroma1.it · Avn '÷ Ogg ' esercitazione or 14:30 > Aula Arnoldi Prof. Montefusco °...

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Avn Ogg ' esercitazione or 14:30 Ⱦ Aula Arnoldi Prof. Montefusco ° Complain I questions ' sir Corsi . Martedi prossim orsio 12 :b Ⱦ 14%50 .
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Avn '÷Ogg '

esercitazione or 14:30Aula Arnoldi Prof. Montefusco

° Complain I questions'

sir Corsi

. Martedi prossim orsio 12 :b 14%50.

TEOREMI.CC ambidmenbo di variabili negli iutymli doppi ) .

Siano T,D due domini ryolan

' di R2, e sia

¢ : T D una fuuzioue tale che

1) ¢ bieltivs ¢ : T - D

2) $ £ CYT ;D ) @,v) to 5f@,v)=kH4,g@,v ))3) lo jacobian di$47 ¢ e-

[email protected]

.

T e D.

01 se 01 e- uudiffeomofisoAtlas

,H fruition fcxiy ) continua in D ¢ ' 1 I un diffeomofismo

si ha : ( segue del tonne d- inverts.

locale ).

[email protected])| dudv.

T "

Eta fcxeixiyeio))

l¥aqi¥μat HEfdseusfTE0ReMA_ Siano T e D due domini normal '

ugolarr risp .

di

[ o , too )x[ ort ] e R ?t.c.la frost .

0/4,0 ) =q@DigseuduerifichiIoH-D.AUorasihaffjfCxiy1dxdy-ffjfCgcosQgseuojgdgdd.V

- f continua in D.

ESERcisio_CIeolareffxsdxdy.doveEi1aponionedipianracchiusedallecurvag-TZ-fIaOeyE.cosO@TFEtooa.scwtnrcoiO.s

kg /tfxsdxdy = FEE

.

fdq paid¥,§"ks4aa;kPftdo=

0

= {

ftp.#Ddo=zf*E4as2o-4+@t)doa....

.

Cos ? I

Estrin Calco lane 11 area di E dell '

eserciao precedentM¥a

Area At = As dxdy =μ¥dO fdp g= Concludedda soli )

.

0

Significato dell jacobian uelcambiamenodi coordinate

=#kM,¥I¥¥**"×

5f@,v)=@@,v), y(uμ )

lngrandisco .

°&4

/ Q3

BCUIHAD .k,(u+su,v+AD

HEIKE,

a :* ,oi

Qi = §# ) i=s. . .4

.

01ga

.qX%

Jf¥fh*q,

Betsypause,v+aD

\y⇒&sir. =me. air ,

'

%Pμ,v)Kutaum

)

¥uezDdi×

Qi = § Cpi ) i=s. . . 4

.

as =§@z ) = [email protected] ) = ( x ( utan ,v ) , y(u+au,vD

du"molto piccolo

"

× ( u+su,v ) x x@,v)[email protected]

) + gender ) su .

[email protected]

⇐':# avi)Q4=$(Py)= ( xcutsu,vtaD

, y¢+su,*+sv )) = * )

Formula di Taylor del tontine :

[email protected] ) +

fuI@v7bu-igutui4sv.aaxCxHdtQoueHsutgkvcuiDdv.ycuirhautcuiddutgytcuidsy =]

In coordinate

planes7

f⇒-p

ar:

r x

.K terming delle jacobian com

.

spade a far peoare dioersamate

region 'une quelle rossa e quelle verde

,che comisfondew a aoee ben

diverse nd piano ( x ,y )

agxCxHvkQouHvButgkvaDdv.y@vH0auICuiDdutffICuiDby-tQtfEein.afucu.vDsutGCuH.aEuixav.AreaCparallebgoammaQnQrQ3QeD-areadelparaAebgoomma6struitosuidueueuoriwsfoEi0fDauwz-fEnfuTsr.xWzaatetfwliYwiHttoetE.E

:÷£IH==**1detfEyEgfefl

Al uttangolino di area Dnudr corrispoude ,mediant lathes format . $ ,

un parallelogram ma di area duiv / det 5&j¥|

twp '

( dir )D= domino normale risotto al piano xy =

= { &,y,z)€R3 : Cxiy ) EE, xcxy ) = z = pay ) }

dove E £ un dominie normale del piano xy ,

Xkay ) , Play ) : E - R continue e tali che Kay )< Pcxiy )

÷⇒- * Hoift - tax ,y )

D ->y to !D=μ§kytxHyDdxdy€f IDI

:Sid f ( xiyit ) una fusion definite in D e limitata

Data unapartitionedD in n domini nemali {Dilia

. . in

a intemi disgiunti , powered

$ (P) = €yae(Di ) supfcayiz )Qiyit)£Di

six ) = favor CDD in,f f

Preudiam sup s(P ) e inf $4Ppartiz .

di D Pparhz . did

Sidim.

che sup SCM = inf SHP P

Se sow ugnali , f si dice Riemann- ihtegsbik in D e si pone

[email protected]

' Riemann - latepabile

2) Ll integrate gode delle popoiehi de ciaspeltiamo .

a : lineanhib)

monotonyffc dxdydz = C roe D.

d) addition '

rispeth al domino d '

htegsse)

dis.tnango1areIffff@yiHdxdyokfafffdfIGyizfdxdydtYkoremadellemedrvotDlinff-fffyfx.y

,Ddxdydt = uol.to )sgpfD

Formula diridunSe D e

'

come sofas , e f e- uwhwua in D,atlas

[email protected]

) : a <xeb, JH a- y ⇐ TH ) [ iutyssiae per

= t.am?ttdKeIte..D=to

= μd* μ dydz fey,z )interior per faire .

Ex

dove Ex ={§H : @yx⇒eE }

ESEP.ci#Cdeo1arfff..togCztrf+3)dxdyd.t ,dove Tiilktraedo

di vatici ( 0,0,o) , (1,1-0) ,(0/1,0) , @, 1

,-1 )

.

^Z

x

• 01 f continua int

[email protected]

of K' ,'D Cas ) -°(q )

T= { Eiyia ) : Cx,y)€E : x . y

±z=o} %

Z = axtby PqssagyopgKai ) of atb

@, 1,

-1 ) -1 = b

z= × - y

E = / Kyla o eyes , o=×=y }

ftp.hgcztytstdxdydt-ff.dxdy/dytbgG+y+3)=H'

fbgtdt = thgt - ttc = t ( left - a )

a = ffedxdy ( HytD( lgzttyts) - s)¥I×°y =

ftp..bg#ytstdxdydt=ff.dxdy/dtybgC?+y+3 )= * '

fhgtdt = thgt - ttc = t ( left - i )

a = ffedxdy [ Etty+37 ( lofty +3) - I)¥}y =

=

ffedxdy $+3)¢g(y+3) -D - ( x⇒)@ge⇒ ) - DF -

= footy ( μ× kytDkyefs) . e) - [email protected] )] ] =

=

[email protected]'t tgt-a) - t÷=

= t÷( 26Gt - 3) + a.

= § of [ylyts) @g(y⇒) - a )

*&t¥@lgEyt's ) - D+jE(Hg3 -If

= dnalisi I.

.


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