*(il ffi-,

(1)

tsFR ctzl tslrrltots

{rerèseg

caebrrire

d:= ^\r<4 ₌

t@7

(il ^ù'= _#,ùw= {*-,

h(-ù ^Uq= w^Y'e ^fi'vr

d*.".;h alte pf,ne "ll\-l$ Ael lib.o ^"

'Wt. Vr*i'X/ I

_ì

é

w2

!*' ffi

/ v2/,È\

=(r/El

tttttl

= ^(J"ù

\u/

Mah,c; e V&rr;' DIL E _DEL tlttllots

@ ^5; ^con:i deri su lR* it yodohs"araie , ( _(*) _, _(V,)> " j(x,fi+x,f)rhfir\yq

Trcrrarre unà kse Ortonormale !rr,ur,

^rta

,uq] ^riyeko ^al ^prodo ff, ^salxe

consi derabo 6ale <he L;n

^(u

r) ⁼ ^L; n (v1\ , ^Li ⁿ

^1u

^o,u") ⁼ ^Lin ^ur,va)

Lln (ur,ùr,dr)- _Lin (vr,rr,4) e ^Lirt (utr,ùr,ùs,r+)= L,.n(vr,ur,v,rv+)= _lR4

-=({) ^v4=(fl

cos: _\ar: (:i ^u,=()

Èr p.,rnq ^cc6a usia/tro ['algor)trno d; &ram- Srhmidt _per

una 6ase ,É"6onal< [*r, ^W,%, %] ^;

( r<tr,t,

_>

WM

1wt,4)=c e ,'x/ztVù= ^O w+= v+-

ffi-,

=- v. '+- - ^{-//2 ..,.} _two

A

clesso Ld6h _prerrdere

:

u,= ^tl -1 ^(t\/t,vrl ^w1 ⁼

- _(wr, ^(w',%),

trt\

4/3

W ^v\/2_

LU

t ^'Wg=

^VJ

{\Xh,V47 W, ₌

14tW) ^r

= /+***r-jW-!u,=

Qiesfo proced; mentc ^e

(2)

@ ia r,R'-"rR3 Édre .he ^r-A)=

$irr*) ^{Mo^brxe} ^de ^F

e, ^d;g,onal ;.zabile e deÉerymin&re un& _base d, ^a.,,6o ^vr,hor; .

^.^,

lndi.aÉa can A la ^rna6r;ce cfu rdpples mp F r;ageft"o alla

base cshopigs _u= [€t,€2,e\ ^eal.olare ^Ah ^pe] fn, ^'rÉ Z e ^qU)

con _g(e1= ^(t'-3Ò ⁺ l)(6'-3É*J)

"

&e*au"'t

A = ^Me (F) E ( \1 i ^-1 \a'ù, ^2J ^s;§ Fn (0 _il _sw _yl;,wnio ,ardk*;st,u

patt)= _det (,i _\/ _{-1 2-t/} *i) ⁼ ^É3+ ^+t ^-5t+2= ^(t-e)(t-4r

A e d,ryonahzza6;/e _se e

So

/o 5e

^I

^'arbo ^raloÉe ^{mv lt} ^iplo t, I ^ha

nr(1)=2 s ^ihfà,(t; ^,

^ ^ltnptc; ^bÀ' ^gewnebr

^;

^ca

\(4=di",(k. (A-r\=l- _rg _(A-r)= _3-r3

f il) ^=,

5b (!) . ^Ke.(É--I) ail'r6 -a+b-c=e. ^A/e _{6egv._} _che ; ^dre

veffori vz= (L) e _V= G) ^,%,.nuno ^um. ^Àaae pe, l'a,bosp,zio

d, *ràornlore 1.

lnolb,e ' _f§) ^, ^Ke,(A-fr) ^5e _" ^solo ^se (:Ì{.il(,r)=(il ^da ^rr; ^À=A=-r.

FUbr#o _W=

0 ^fleru lrautospoao d, arboralore 2

_"

e ^und. hs. d, arto"e,bsr; per ^F

rmtr; .e c he ..pf ^reeenta ^F riep1ffn * llu he ^ò

À = $'r, ^nr, _{r, J}

f4,(F) = (Éj;)

C.! = ^(V|tz2 ^lvu) '

e

-t

lat

t11-1\

lt t ^ol

\{ o t) ê ^h ^nnabri.e d;.arnbiw{o d, br, ^da ê â ^b.

(3)

Dalla relae;.ne f ⁼ ^M* ^(r)= ^{eL ^M6 ^(F) ^(uu ₌ ^Cru ^YaG) G} ^si ^À*

^,

che ^,

An= ceb"(Mr(r))nG;' _' = (il _t)(È' j _iluii) ^' ⁼

\-i ^o rt

ltn 1_1\/_t ^t ^_1 \ ^/_2n+2 2nl {"+l\

= (# t ^?/( ^1t ^? ^[)= ⁽ -i::i ì:t ^';r'J

Pe, col.olare 1(n) ^con ^?fb) = (e'-16-/) lf'-:é+l) ^osserr;Nno

g(É) = ( t'-tt+\t- ¹ = ^G- ^t1' {t-t1" ^- I = ^(t- t) pofl ^- I kr ^;l ^tea*vna d; Gyt"l-H»n;lbon p^ (,t) = o _?uind;

,

?(A) = (lr-a1 _paG) -r _=-J

C ^D;re

S.*,,ew

er

X+

x+,

'yr ls;

r*

/:

(;

U

-x

5. Y

r)+

yf

r

+2u

r 2'i

&,h)

'ernà

v tt

2(

f-€

eJ

X.

ry v'

€or

Xt,

{;

6,

real

/

\+f-x

, 1r,

fU7 +) +2y

)

ianìo

oral; ^X

x+2 .fy+yl

iyr2l

ìt

Cvi

=x

2

I

)

2 la seg»ube rvwbr;n sìmwe€.tric4

e deF,oiba ysiti,ru

^.

d, Pil)ina "|ry del l;brc"1l'latar; e Wer,'

Denu ^essero ^:

i) x+2zo da

.,\ lxt.l x+2

^I

tt) lx+z lrrl

x>- ₂

y ^(x+t) >o oa êv) tr"n,t".t, ^d; ù ^y> Ô ê ^xù

,,',')

I ^x+Z

I ^x*z

I ^xy+zy f,l;; yigril'§iì++

2y) /, "

H.' éo,, 3 /ro ,)1 ^,§-1-, ^{p^l k+4)} ^o,

y(x+2)

^/.T-4-^l Ii rfiii

(x*4(1-x)

corfro d;

In cohcfu5ione

da ^cvi x+2 x+2 o

; ^§r1.rl ^> ^o ^, 6+ z)f f-x) '/ ,o ^du ^L,i ^tenedo ^ao/',e

i) e i") 5; ha ^1-x> o ^c iod x< ⁷

Jet 5>O ^ovwto

5 ^g ^dqF;n;Òe ^ps;f;,ra ^pe,t- y>O e =2<x<7 ^rn& x*O

(4)

m J;anc attilzt--.,&", n ^nuffiri ^heali ^dis6ir,Òi.

v

-t -/ l-'" f 1,v. 1,>,, ,vur' llrr\]LrrJvl

J;a kr. 1;rÉero" kflin;an,o G=h e Cx= *=,^§ m=l

n ^j< ^Ò

bi.nor6.are che ^/d ^rmbrice simrnetr;,, ì"1 ^bale cfu _6{ _)" ₌ _6r,J-*

e ^def'n;6a ^po

^r

^b;vs

,

s"&ÀÉo,^, tVe I ^fo6l; ^delle ^sse ^rci ^baz',o,t; ^de _I ^l4 _I ^to /lots ^e

erd ^jÉdÉa de},i;ta la _rrntriw d; vandormonde 't'h1,

vh,, _, aò = (ri{* ... * ) ^con determ;*te eletv6,.-,ùn)=

\dl'*r' ^-, #'/

Osservi»rw.b (v@" _x1 ^. ₌ (^jiti _,q

Ne segue ,jre (V "'V)i1 ₌ _firr

,= (N),, _e

fY é defin;ba _gasibiua se e ^solo ie

riattrta ^,z *z ^>o . ^Ma , P ^Pni

del _U _/ro lur:

"-, tu) _l

r>§ rT (a' - ar)

rY _(ar..

u^)), /dJ-rl _^

,)/al:' l= z*'u

l;l ^n=t

\a;-'l

A'/= Y.V

fion ^nvìlo

L' ulb,raq dis*5orylian a è ^dovt)ba _al Bfr"

sono dietinb; ri:ulba detY ₌ _deb V * o

Ow _.ie _z+D 'Vz _e q ^lre,ffore _m _nuilo"

dunl,e

Z€ _/Rh

.k _&t, _àr, _--., _a,) e flvngw, _Ken(fl ₌

[oJ

Dab; i ^votrari ^nr= _(il _e _o= _(il _wtrg.a.e _cAe _.sor,o

e ^Scri ^vete

^1t

eg ^..6» one del _p)ano d, ^.Ae _6reneranO

"

carcoh.e le ,i^bo,r, (r,yeth" _altq Às6e .66|aica d; R*) proiezicne ^orÒgonale e ^d;' r;flessian? t;Te-fto Od ^o(.

orbynal;

J

feen& 5;a I ^: ^d.x+bv ⁺ ^cZ= 0 " Êxendo ^vt ^cd. ê ^W. êd. ^dnng

eixie.{àlit::::B' ^da ^evi §=-àt=Ì.

D( ln _guind; _€Jueziorp 6x- _3y+Qz - _O"

Attn nd, gt ^ohwre ^[, ^sfe.so ^pis,lbabo € ^4)ierc ^w _$ ^Lin ^(t,vù

^,

(5)

flna

ryx;b;le ^s.elÒa é ^*= (l) . ^Etsendo ^(Wtvr1= ^O _Tf/, ^.h;amo

;l proced;rrre,to ol, ^Grorn- hnidb ale ^bse {'o, (r,w} _in _rnodo

da' offienerne una or8o6>onole {nr, ⁿ ^,vs J

^;

%= w'ffivt-ffi,vz= _(é)-l,(t _àff) _=#,&)

i vekori d; ^,4, Linfuu,vr) *,na tuh; e tol ; ^votol; ^orÉo6o ^nali ^a W'

lnattrcparore (t)-c,. &esotaE. *,<(!) ,fi)r=o ^do

cvi riobw;w)o l'*qu&>ione d; s4'6x'9y+(z=O"

IVo"r,^,61, *anda i vebor; ^\4 t ^vz e 6 ^a ^fum;an:rc ^la ^base ortornrnale b= rut,t)2,,,7! con t)t=

h(i) ^, ^ùz= *(i) ^u,= ^#rfi)

La p.o iez)one ortynale _P sul _p;ar-ta d, e ^tal" ^cÀg

^,

P ( & w+ ^Kzuzt ^,§ _':r) ⁼ ^Kt ^ut ⁺ ^Ktùz

La ms..hri<, Pb che rappresrri^ P nape.,to a/i* bt t

{nd;.aÉa @n .= _[ *,et,e] h ^òaae ^(rytonlca Mx ^da

Cel _Ia n ro.bryce di corn\ianuto ^d;

ce^ = ⁽ ^r, _I ^o, I u,) ₌ ({/É ^y,P, :{ffi)

\tltq "ii,Èe

41,ffi1

r /'-t_Tr

Cb. = Gu ^* = ^'( eb gerché ]n àaoe o,J beorerra

4$ def ^l;bro *f\4s$1;.; e Vffir; ^" , ^la rnabu<e _Cet

d;

é

(ovrero /rq ^J'in l'etra ^cÀe <oincrde coo la Étaspoebu)

_.

kfta % ^I* ^maÉr,.e ^d; ^g,oteaffE

e, ^Po ^=(tr?A)

;, e ;,tu ^'or

§. l, ^ri5 ^ctiba

^t

5,5 § _pqpi/?a e Orbo5,onille

I

133 iu d tisgko

i 2v ⁵⁴ ^-24

(4 52 ₃₆

k4 ¹⁶ ^41?

Pe = ^Cg5 ^P5 C5. = ^{Cet Pl} f ^(*r)

alla 6ar ^cprpnica ^si ^ha

(6)

i'fu*.?f;'ir,,.J'.i;-ri: ;r#!:" ^rn prososizwÉ s $

, ^(!) ⁼ ^< ^(!), ^ùt) ^ut+, (!),,,2) ⁰²⁼

7\+).,,+?- ll\

= trG,.ry(q=*&it#i:#;,)

La r;flebr,orÉ R rmpeff" ad or € ^fale ^cÀe ^R(tr) =sr) ^R ^(ur) .ù,

r ^R{ut),-ur.

Pert*b, la r^ab.;oe & ^d; ^R ^riV&o ^allu ^hse ^b e ^Rr= _({|-fr].

Essend, RÀ = zPb-I ^Re ⁼ _Gl ^Rr teo = eaQRr-r)(u, = Zpe-I

dn Lv) p ìe= ìi.(* 4 / 61 _{Ì; iù} tol -4s

@ ^ln ^r; ^fer;mer{n ^{all 'ese} ^rdzio ^ptewdente ^5,a ^§= _!,,\ ^,

Toue.ne ; ve,hon t C ^tonghey6, _-tVD, _cAe _fon)'!*, _e,,,

s ^un ^Brb p, ^a,, &rccos *h e ^bal; ^{.he /e} ^pn ^,€z)oni

d; 5 e b sul _gtano c( d,ano pet^fend;.olari

^,

s-r§= (ò ;(t= _iH) = _ryu3 =ff *

rVe segrte che _s ₌

,ffi ^Ur ⁺ ,lT ^u,

5,u b=Pr**dr)z*§,ds.

Ù"É€ P(r) ( _Pfe) le gro;ezion) d, ⁵ e _b _au d. , ^drn*do

etsere _< P(t), P(t)) = c ^si ^ha _{t|ur, Fvr+ Yùr)= \E ^É -c

da cui _P=o"

Pe"banbo b e ^Ln ^vp,hore ^dsilu ^Èrnu t= _{Yùr+ §u:}

(7)

5; ha ^lsl= 6 ^lb1= 'lre ^(5,É ⁾ ⁼ ^,lg ⁶

il cùbeno de// 'aqplo corwytrexa flra C e lI _brrtlafo _dU Jue"

vChu 5 eb dgar; ^Ò. , ffi

(S; _veda la deDnr>ione _{S.1T d,} _p?Jr,ne l,lg ^det ^t;&o ,,f,4atri";

" ^V#u,,.)

Pe.àanfo, ;n òaee "rlle ri(h;esÉe detl)esercie)o o/eye esserg _;

/" ,-___

) _{ì r--} ^Vtri6'= ^3ilx,

oa ^oui

| ^\t ^te/r ^{ ^_ ^vo

\ \re 3,8

ci sono gv;nd; dre aalaz_;on)

^,

bt.- ilil uz+,ftr ur = _(il.({) ^: ({) ^e

br=\mUzt,mur= _({) -(i)= (i)

f ^d- ⁼ ^f?Jg

( { r= t1ffi

*(il ffi-,

tsFR ctzl tslrrltots

{rerèseg

caebrrire

d:= \r<4 =

t@7

*(il ù'= #,*ùw= {*-,

h(-ù Uq= w^Y'e fi'vr

d*.".;h alte pf,ne "ll\-l$ Ael lib.o "

'Wt. Vr*i'X/ I

é

w2

!*' ffi

/ v2/,È\

=(r/El

tttttl

= (J"ù

\u/

Mah,c; e V&rr;' DIL E DEL tlttllots

@ 5; con:i deri su lR* it yodohs"araie , ( (*) , (V,)> " j(x,fi+x,f)rhfir\yq

Trcrrarre unà kse Ortonormale !rr,ur,

,uq] riyeko al prodo ff, salxe

consi derabo 6ale <he L;n

r) = L; n (v1\ , Li n

o,u") = Lin ur,va)

Lln (ur,ùr,dr)- Lin (vr,rr,4) e Lirt (utr,ùr,ùs,r+)= L,.n(vr,ur,v,rv+)= lR4

-=({) v4=(fl

cos: \ar: (:i u,=()

Èr p.,rnq cc6a usia/tro ['algor)trno d; &ram- Srhmidt per

una 6ase ,É"6onal< [*r, W,%, %] ;

( r<tr,t,

WM

1wt,4)=c e ,'x/ztVù= O w+= v+-

ffi-,

=- v. '+- - -//2 ..,. two

clesso Ld6h prerrdere

u,= tl -1 (t\/t,vrl w1 =

- (wr, (w',%),

trt\

4/3

W v\/2_

t 'Wg=

{\Xh,V47 W, =

14tW) r

= /+***r-jW-!u,=

Qiesfo proced; mentc e

@ ia r,R'-"rR3 Édre .he r-A)=

$irr*) Mo^brxe de F

e, d;g,onal ;.zabile e deÉerymin&re un& base d, a.,,6o vr,hor; .

lndi.aÉa can A la rna6r;ce cfu rdpples mp F r;ageft"o alla

base cshopigs u= [€t,€2,e\ eal.olare Ah pe] fn, 'rÉ Z e qU)

con g(e1= (t'-3Ò + l)(6'-3É*J)

&e*au"'t

A = Me (F) E ( \1 i -1 \a'ù, 2J s;§ Fn (0 il sw yl;,wnio ,ardk*;st,u

patt)= det (,i \/ -1 2-t/ *i) = É3+ +t -5t+2= (t-e)(t-4r

A e d,ryonahzza6;/e se e

/o 5e

'arbo raloÉe mv lt iplo t, I ha

nr(1)=2 s ihfà,(t; ,

^ ltnptc; bÀ' gewnebr

ca

\(4=di",(k. (A-r\=l- rg (A-r)= 3-r3

f il) =,

5b (!) . Ke.(É--I) ail'r6 -a+b-c=e. A/e 6egv._ che ; dre

veffori vz= (L) e V= G) ,%,.nuno um. Àaae pe, l'a,bosp,zio

d, *ràornlore 1.

lnolb,e ' f§) , Ke,(A-fr) 5e " solo se (:Ì{.il(,r)=(il da rr; À=A=-r.

FUbr#o W=

0 fleru lrautospoao d, arboralore 2

e und. hs. d, arto"e,bsr; per F

rmtr; .e c he ..pf reeenta F riep1ffn * llu he ò

À = $'r, nr, r, J

f4,(F) = (Éj;)

C.! = (V|tz2 lvu) '

e

lat

t11-1\

lt t ol

\{ o t) e h nnabri.e d;.arnbiw{o d, br, da e a b.

Dalla relae;.ne f = M* (r)= {eL M6 (F) (uu = Cru YaG) G} si À*

d:= ^\r<4 ₌

(il ^ù'= _#,ùw= {*-,

h(-ù ^Uq= w^Y'e ^fi'vr

d*.".;h alte pf,ne "ll\-l$ Ael lib.o ^"

= ^(J"ù

Mah,c; e V&rr;' DIL E _DEL tlttllots

@ ^5; ^con:i deri su lR* it yodohs"araie , ( _(*) _, _(V,)> " j(x,fi+x,f)rhfir\yq

,uq] ^riyeko ^al ^prodo ff, ^salxe

r) ⁼ ^L; n (v1\ , ^Li ⁿ

^o,u") ⁼ ^Lin ^ur,va)

Lln (ur,ùr,dr)- _Lin (vr,rr,4) e ^Lirt (utr,ùr,ùs,r+)= L,.n(vr,ur,v,rv+)= _lR4

-=({) ^v4=(fl

cos: _\ar: (:i ^u,=()

Èr p.,rnq ^cc6a usia/tro ['algor)trno d; &ram- Srhmidt _per

una 6ase ,É"6onal< [*r, ^W,%, %] ^;

1wt,4)=c e ,'x/ztVù= ^O w+= v+-

=- v. '+- - ^{-//2 ..,.} _two

clesso Ld6h _prerrdere

u,= ^tl -1 ^(t\/t,vrl ^w1 ⁼

- _(wr, ^(w',%),

W ^v\/2_

t ^'Wg=

{\Xh,V47 W, ₌

14tW) ^r

Qiesfo proced; mentc ^e

@ ia r,R'-"rR3 Édre .he ^r-A)=

$irr*) ^{Mo^brxe} ^de ^F

e, ^d;g,onal ;.zabile e deÉerymin&re un& _base d, ^a.,,6o ^vr,hor; .

lndi.aÉa can A la ^rna6r;ce cfu rdpples mp F r;ageft"o alla

base cshopigs _u= [€t,€2,e\ ^eal.olare ^Ah ^pe] fn, ^'rÉ Z e ^qU)

con _g(e1= ^(t'-3Ò ⁺ l)(6'-3É*J)

A = ^Me (F) E ( \1 i ^-1 \a'ù, ^2J ^s;§ Fn (0 _il _sw _yl;,wnio ,ardk*;st,u

patt)= _det (,i _\/ _{-1 2-t/} *i) ⁼ ^É3+ ^+t ^-5t+2= ^(t-e)(t-4r

A e d,ryonahzza6;/e _se e

^'arbo ^raloÉe ^{mv lt} ^iplo t, I ^ha

nr(1)=2 s ^ihfà,(t; ^,

^ ^ltnptc; ^bÀ' ^gewnebr

^ca

\(4=di",(k. (A-r\=l- _rg _(A-r)= _3-r3

f il) ^=,

5b (!) . ^Ke.(É--I) ail'r6 -a+b-c=e. ^A/e _{6egv._} _che ; ^dre

veffori vz= (L) e _V= G) ^,%,.nuno ^um. ^Àaae pe, l'a,bosp,zio

lnolb,e ' _f§) ^, ^Ke,(A-fr) ^5e _" ^solo ^se (:Ì{.il(,r)=(il ^da ^rr; ^À=A=-r.

FUbr#o _W=

0 ^fleru lrautospoao d, arboralore 2

e ^und. hs. d, arto"e,bsr; per ^F

rmtr; .e c he ..pf ^reeenta ^F riep1ffn * llu he ^ò

À = $'r, ^nr, _{r, J}

C.! = ^(V|tz2 ^lvu) '

lt t ^ol

\{ o t) ê ^h ^nnabri.e d;.arnbiw{o d, br, ^da ê â ^b.

Dalla relae;.ne f ⁼ ^M* ^(r)= ^{eL ^M6 ^(F) ^(uu ₌ ^Cru ^YaG) G} ^si ^À*

che ^,

An= ceb"(Mr(r))nG;' _' = (il _t)(È' j _iluii) ^' ⁼

\-i ^o rt

ltn 1_1\/_t ^t ^_1 \ ^/_2n+2 2nl {"+l\

= (# t ^?/( ^1t ^? ^[)= ⁽ -i::i ì:t ^';r'J

Pe, col.olare 1(n) ^con ^?fb) = (e'-16-/) lf'-:é+l) ^osserr;Nno

g(É) = ( t'-tt+\t- ¹ = ^G- ^t1' {t-t1" ^- I = ^(t- t) pofl ^- I kr ^;l ^tea*vna d; Gyt"l-H»n;lbon p^ (,t) = o _?uind;

?(A) = (lr-a1 _paG) -r _=-J

C ^D;re

oral; ^X

Denu ^essero ^:

tt) lx+z lrrl

x>- ₂

y ^(x+t) >o oa êv) tr"n,t".t, ^d; ù ^y> Ô ê ^xù

I ^x+Z

I ^x*z

I ^xy+zy f,l;; yigril'§iì++

H.' éo,, 3 /ro ,)1 ^,§-1-, ^{p^l k+4)} ^o,

da ^cvi x+2 x+2 o

; ^§r1.rl ^> ^o ^, 6+ z)f f-x) '/ ,o ^du ^L,i ^tenedo ^ao/',e

i) e i") 5; ha ^1-x> o ^c iod x< ⁷

Jet 5>O ^ovwto

5 ^g ^dqF;n;Òe ^ps;f;,ra ^pe,t- y>O e =2<x<7 ^rn& x*O

m J;anc attilzt--.,&", n ^nuffiri ^heali ^dis6ir,Òi.