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@DVÍXUIN 4

Lnáões jäsg`ds HE Xenrgd hns sgstemds

4.4) LNXDÁÀN GLHG@GDI

Em dläigse telsnrgdi mugtds vezes estdmns glteressdhnsem estuhdr vdrgehdhes `nmpnstds pnr `nieáões he L eiemeltns.

Leste `dsn n usn he ietrds hgstgltds `nmn ( x, y, z ) que`drd`tergzdm n E 0 ç glecg`dz, selhn le`essärgn dhntdr umdlntdáàn mdgs aerdi? d lntdáàn glhg`gdi.

@nlsgheremns glg`gdimelte n E 0, pnhemns hesgaldr nseiemeltns heste espdán pnr 4 : 0, ,g x x x x , nlhe ç väighd d`nrrespnlhïl`gd?

4

:

0

x x x y

x z

(4.64)

Xdmjçm pnhemns hesgaldr ns eiemeltns he E 0 pnr 4 : 0, ,g x x x x , nlhe ç väighd d `nrrespnlhïl`gd?

4

:

0

x x x y

x z

(4.6:)

Ln lnssn exempin, d pnsgáàn hn ílhg`e supergnr(snjres`rgtn) nu glcergnr (sujs`rgtn) làn dpreseltd lelbumdhgcerelád, ln eltdltn, `nlcnrme veremns, d pnsgáàn hns ílhg`esln estuhn hns telsnres glhg`d qudltghdhes he ldturezd`nmpietdmelte hgstgltds.

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Hevemns njservdr tdmjçm que ln `dsn hns ílhg`essnjres`rgtns làn estdmns helntdlhn pntïl`gds, leste `dsnusdremns n usn he pdrïlteses pdrd hgcerel`gdr, `nmn mnstrdmns

djdgxn?

:g g g x x x (4.60)

D lntdáàn glhg`gdi pnhe ser ÿtgi ld represeltdáàn hevetnres e diauls `dsns espe`gdgs he mdtrgzes.

Peod n espdán hds mdtrgzes `niulds hel eiemeltns snj

um `nrpn he es`didres `nmpiexns 4ld . Vnhemns represeltdr, pnr um djusn he iglaudaem, ds mdtrgzes `niulds pnrg x , `nm gvdrgdlhn he 4 æl.

4 : ... X

g l x x x x (4.68)

Danrd `nlsghere n espdán hds mdtrgzes iglbds heleiemeltns snj um `nrpn he es`didres `nmpiexns 4ld .Vnhemns represeltdr, pnr um djusn he iglaudaem, ds mdtrgzesiglbds pnr g x , `nm gvdrgdlhn he 4 æl.

4 : ...g l x x x x (4.65)

Essd lntdáàn tdmjçm pnhe ser usdhd pdrd represeltdrvetnres4. Pe n dutnr tgver diaumd cdmgigdrghdhe `nm Me`êlg`dTuêltg`d, cg`d evghelte que ds `nmpnleltes he xg (que

4 Gssn pnhe ser evghelte se n iegtnr estgver cdmgigdrgzdhn `nm Äiaejrd Igledr, pngs exgste um gsnmnrcgsmn eltre n espdán hds mdtrgzes iglbds (nu `niulds) ens vetnres njiíquns.

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`bdmdremns he g ) se dssn`gdm dnket, elqudltn ns `nlouadhns`nmpiexns hds `nmpnleltes xg (`bdmdhns he g

) se dssn`gdmdn jrd. Dssgm temns?

g

g

(4.63)

4.:) Xenrgd hns Pgstemds

Um Pgstemd ç hecglghn `nmn selhn um `nloultn hees`didres, helntdhn pnr umd ietrd nu um símjnin e um `nloultnhe ílhg`es (snjres`rgtns e/nu sujes`rgtns).

N lÿmern he ílhg`es hecgle d nrhem hn sgstemd. Vnrexempin, se n sgstemd pnssugr qudtrn ílhg`es, glhepelhelte hd pnsgáàn, n sgstemd ç hgtn he qudrtd nrhem.

D pnsgáàn hns ílhg`es hecgle n tgpn hn sgstemd,dhntdremns d seauglte lnmel`idturd: , um sgstemd que pnssugdpelds ílhg`es supergnres ç um sgstemd supergnr, hn mesmnmnhn que um sgstemd `nm dpelds ílhg`es glcergnres serä`bdmdhn he sgstemdglcergnr, e um sgstemd `nm ílhg`essupergnres e glcergnres serä `bdmdhn he sgstemdmgstn. Dssgmtemns?

Pgstemd Pupergnr he Nrheml ...go l D Pgstemd Glcergnr he Nrheml ...go l D

Pgstemd Mgstn he Nrhem (m+l ) ......

go mki l D

: Làn bä ld igterdturd umd lnmel`idturd espe`ícg`d pdrd ns tgpns he sgstemd,essd `nlveláàn ç he es`nibd hns dutnres.

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D tenrgd aerdi he sgstemds requer mçtnhns dvdládhns heäiaejrd djstrdtd, telsnrgdi e extergnr. Vnr gssn lns igmgtdremns ddpreseltdr ns dspe`tns jäsg`ns.

4.0) Pgstemds EPVE@GDGP

Ld tenrgd hns sgstemds bä trïs sgstemds espe`gdgs que pnhem ser represeltdhns pnr nutrds qudltghdhes0 `uods prnprgehdhes diaçjrg`ds sàn `nlbe`ghds sàn ns sgstemds henrhem zern, nrhem 4 (nu prgmegrd nrhem) e nrhem : (nu

seaulhd nrhem).◎ Pgstemds he Nrhem _ern ‐ ( D)Estes sgstemds sàn helntdhns sem ílhg`es e pnhem ser

represeltdhns pnr um `nrpn he es`didres, berhdlhn tnhds ds suds prnprgehdhes diaçjrg`ds. Ns sgstemds he nrhem zern, pnstergnrmelte, estdràn dssn`gdhns dns Glvdrgdltes, um `nl`egtn

he sumd gmpnrtêl`gd heltrn hd dläigse telsnrgdi e ld Císg`d.Pàn exempins he sgstemds he nrhem zern?

d) Espdán vetnrgdi hns lÿmerns rd`gnldgs snjre n`nrpn he lÿmerns rd`gnldgs.

j) Espdán vetnrgdi hns lÿmerns redgs snjre n `nrpn

he lÿmerns rd`gnldgs. `) Espdán vetnrgdi` hns lÿmerns redgs snjre n `nrpn

he lÿmerns redgs.

h) Espdán vetnrgdi hns lÿmerns `nmpiexns snjre n`nrpn he lÿmerns rd`gnldgs.

0 Hevghn dn um gsnmnrcgsmn eltre estes sgstemds, que `nmn veremns cnrmdum espdán vetnrgdi, e ns espdáns vetnrgdgs dssn`gdhns æs qudltghdhes emquestàn.

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e) Espdán vetnrgdi hns lÿmerns `nmpiexns snjre n`nrpn he lÿmerns redgs.

c) Espdán vetnrgdi hns lÿmerns `nmpiexns snjre n`nrpn hns lÿmerns `nmpiexns.

Ln he`nrrer hesse `dpítuin dssumgremns que ds`nmpnleltes he um sgstemd sàn lÿmerns `nmpiexns, leste `dsn,ns sgstemds he nrhem zern, estàn he d`nrhn `nm tgpn he espdánexempigcg`dhn em (c).

◎ Pgstemds he Nrhem 4 ‐ ( g D ; g D )

Estes sgstemds sàn helntdhns pnr um ÿlg`n ílhg`e,supergnr nu glcergnr e se dssn`gdm dns vetnres njiíquns hel E , ens espdáns gsnmnrcns d eies.

Vnr `nlveláàn8 dssumgremns que ns sgstemds supergnresestàn reid`gnldhns dn `nloultn hds mdtrgzes iglbds, ns vetnres`nltrdvdrgdltes e n espdán hnsjrd . Ns sgstemds glcergnres estàn

reid`gnldhns estàn reid`gnldhns dn `nloultn hds mdtrgzes`niulds, ns vetnres `nvdrgdltes e n espdán hnsket.

4 :

4 :

...

...

X

g l

g l

x x x x

x x x x (4.61)

Ns sgstemds he prgmegrd nrhem se reid`gnldm peid reard?

4 :

4 :

...

...

X lg

gl

x x x x

x x x x

(4.62)

Nlhe n dstergs`n glhg`d que tnmdmns n `nlouadhn`nmpiexn.

8 Essd `nlveláàn ç ÿtgi, pngs qudlhn trdtdrmns he sgstemds que setrdlscnrmdm `nmn telsnres he prgmegrd nrhem, essd `nlveláàn serä umdgmpig`dáàn hd ieg he trdlscnrmdáàn.

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◎ Pgstemds he Nrhem : ‐ ( go D ; go D ; g o D )

Estes sgstemds sàn helntdhns pnr um pdr he ílhg`es que pnhem ser he trïs tgpns, supergnr, glcergnr nu mgstn. Estàndssn`gdhns dn espdán vetnrgdi hds mdtrgzes qudhrdhds.

@nlcnrme ns tgpns he ílhg`es tïm ds seaugltes mdtrgzes?

44 4: 4

:4 :: :

4 :

l

l

go

l l ll

d d d

d d d D

d d d

(4.67)

44 4: 4

:4 :: :

4 :

l

l

go

l l ll

d d d

d d d D

d d d

(4.46)

4 4 44 :: : :4 :

4 :

l

g l o

l l ll

d d d

d d d D

d d d

(4.44)

Ns sgstemds he nrhem supergnr, mdgnr nu gaudi d trïs, làn pnhem ser represeltdhns pnr nutrds estruturdrds mdtemätg`ds, pnr essd rdzàn `nmpiexghdhe diaçjrg`d lestes sgstemds ç jem

mdgnr.

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4.8) VZNVZGEHDHEP DIAÇJZG@DP

Vdrd d estruturd Pgstemd pre`gsdmns, glg`gdimelte, hecglgrhuds nperdáões eiemeltdres, umd nperdáàn glterld helnmgldhddhgáàn, umd nperdáàn exterld helnmgldhd prnhutn pnr umes`didr e trïs trdlscnrmdáões? prnhutn extergnr (nu hgretn), prnhutn gltergnr (nu glterln) e d `nltrdáàn.

◎ Dhgáàn

D dhgáàn he sgstemds, helntdhd pein símjnin (+), ç umdnperdáàn glterld e jglärgd, gstn ç, pdrd hngs sgstemds hd mesmdnrhem e hn mesmn tgpn, dssn`gd d um ÿlg`n sgstemd he mesmdnrhem e tgpn `nlcnrme d reard?

... ... ...... ... ...

... ... ...

... ... ...

,go m go m go mki l ki l ki l

go m go m go mki l ki l ki l

D J @

@ D J (4.4:)

Essd nperdáàn sdtgscdz qudtrn prnprgehdhes?4) D dhgáàn he sgstemds ç `nmutdtgvd:) D dhgáàn he sgstemds ç dssn`gdtgvd0) Exgste um sgstemd npnstn e eie ç ÿlg`n

8) Exgste um sgstemd leutrn e eie ç ÿlg`nPnjre estds prnprgehdhes hevemns elcdtgzdr que?

d) Hdhn um sgstemd drjgträrgn......

go mki l D , exgste um ÿlg`n

sgstemd npnstn he ......

go mki l D helntdhn pnr ...

...go mki l D .

j) N sgstemd leutrn ç n mesmn pdrd tnhns ns sgstemds,

glhepelhelte hd nrhem e hn tgpn, pnr essd rdzàn eie serähelntdhn pnrN .

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`) Xnhn sgstemd snmdhn dn sgstemd leutrn ç n prðprgnsgstemd, em nutrds pdidvrds,... ...

... ...go m go mki l ki l D N D .

h) D dhgáàn he um sgstemd drjgträrgn......

go mki l D pnr seu

npnstn, ......

go mki l D , resuitd ln sgstemd leutrnN .

e) Hemnlstrdáàn hd ulg`ghdhe hn sgstemd leutrn.

Peod d seauglte equdáàn?

... ...

... ...go m go mki l ki l D D (4.40)

Zees`revelhn d equdáàn usdlhn n sgstemd leutrnN .

... ...

... ...go m go mki l ki l D D N (4.48)

Pupnlbd que exgstd um seaulhn sgstemd leutrn,drjgträrgn, helntdhn pnrN , lessds `nlhgáões, pnhemnsrees`rever (67) `nmn?

... ...

... ...go m go mki l ki l D N D N (4.45)

Pnmdlhn ns hngs idhns pnr um sgstemd npnstn he......

go mki l D

5

njtemns?

... ... ... ...

... ... ... ...go m go m go m go mki l ki l ki l ki l D D N D D N (4.43)

5

@nmn dglhd làn hemnlstrdmns d ulg`ghdhe hn sgstemd npnstn, estdmnsdssumglhn d pnssgjgighdhe he exgstgr nutrns sgstemds npnstns he...

...go mki l D .

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Veid njservdáàn (h) `nl`iuímns que3?

N N N N (4.41)

Nlhe N ç nutrn sgstemd leutrn drjgträrgn. Veidnjservdáàn (`), `nl`iuímns que?

N N (4.42)

Tue hemnlstrd d ulg`ghdhe, pngs se supnrmns dexgstïl`gd he um sgstemd leutrn drjgträrgnN `beadmns d`nl`iusàn que qudiquer que seod este sgstemd eie ç gaudi dn çgaudi dn sgstemd leutrnN .

c) Hemnlstrdáàn hd ulg`ghdhe hn sgstemd leutrn.

Peod d seauglte equdáàn?

N N (4.47)

Estd equdáàn pnhe ser rees`rgtd usdlhn ns sgstemdsnpnstns he ...

...go mki l D .

... ... ... ...

... ... ... ...go m go m go m go mki l ki l ki l ki l D D D D (4.:6)

Nlhe ......

go mki l D ç um sgstemd npnstn drjgträrgn he...

...go mki l D ,

snmdlhn ns hngs idhns pnr um sgstemd npnstn he......go mki l D njtemns?

... ... ... ... ... ...... ... ... ... ... ...

go m go m go m go m go m go mki l ki l ki l ki l ki l ki l D D D D D D (4.:4)

3 @nmn làn pnhemns dssumgr d ulg`ghdhe dglhd, es`nibemns um sgstemdleutrn qudiquer.

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Veid njservdáàn (h) `nl`iuímns que?

... ...

... ...go m go mki l ki lN D N D (4.::)

Veid njservdáàn (`) seaue que?

... ...

... ...go m go mki l ki l D D (4.:0)

Tue hemnlstrd d ulg`ghdhe, pngs dn supnrmns dexgstïl`gd he um sgstemd npnstn drjgträrgn he......

go mki l D `nl`iuímns

que este sgstemd npnstn ç gaudi dn sgstemd......go mki l D .

Vnr sdtgscdzer ds qudtrn prnprgehdhes hd dhgáàn n`nloultn he sgstemds he mesmd nrhem e mesmn tgpn `nlstgtuemum arupn snj d dhgáàn.

◎ MUIXGVIG@DÁÀN VNZ UM EP@DIDZ

D muitgpig`dáàn he um sgstemd pnr um es`didr, helntdhd pein símjnin ( ), ç umd nperdáàn exterld e jglärgd, gstn ç, hdhnum es`didrk, he mesmd ldturezd hds `nmpnleltes hn sgstemd, ehdhn um sgstemd he `ertd nrhem e `ertn tgpn, essd nperdáàndssn`gd d um ÿlg`n sgstemd he mesmd nrhem e tgpn `nlcnrme dreard?

... ...... ...

... ...

... ...

, go m go mki l ki l

go m go mki l ki l

k D J

J k D (4.:8)

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Essd nperdáàn anzd he qudtrn prnprgehdhes1?

5) Hdhn hngs es`didres4k e :k e um sgstemd ......

go mki l D ,

vdie d seauglte prnprgehdhe? ... ...

4 : ... 4 : ...go m go mki l ki lk k D k k D .

3) Hdhn hngs es`didres4k e :k e um sgstemd ......

go mki l D ,

vdie d seauglte prnprgehdhe? ... ... ...4 : ... 4 ... : ...

go m go m go mki l ki l ki lk k D k D k D .

1) Hdhn um es`didrk e hngs sgstemds ......

go mki l D e ...

...go mki l J ,

vdie d seauglte prnprgehdhe?

... ... ... ...... ... ... ...

go m go m go m go mki l ki l ki l ki lk D J k D k J .

2) Exgste um ÿlg`n es`didr 4 que sdtgscdz d reidáàn?... ...... ...4 go m go m

ki l ki l D D

Pnjre estes pnstuidhns hevemns d`res`eltdr hudsnjservdáões?

a) N es`didr luin, helntdhn pnr 6, ievd um sgstemd......

go mki l D

dn sgstemd luin `nlcnrme seaue? ......6 go m

ki l D N

b) Hemnlstrdáàn hd ulg`ghdhe hn es`didr 4.

Peod d seauglte equdáàn?

... ...

... ...

go m go m

ki l ki l D D (4.:5)Nlhe ...

...go mki l D ç um sgstemd làn luin. D equdáàn pnhe ser

rees`rgtd usdlhn n es`didr 4 e um es`didr que sdtgscdz d prnprgehdhe (2)?

... ...

... ...4 go m go mki l ki l D D (4.:3)

1 Vrnprgehdhes exterlds làn sàn lnmedhds `nmn n`nrre `nm ds glterlds.

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Pnmdlhn ns hngs idhns hd equdáàn pein npnstn he......

go mki l D , `nlcnrme d njservdáàn (h) temns que?

... ...

... ...4 go m go mki l ki l D D N (4.:1)

Usdlhn d prnprgehdhe (1), pnhemns rees`rever (::)`nmn?

...

...4 go m

ki l D N (4.:2)

@nmn n sgstemd ......

go mki l D ç làn luin, `nl`iuímns peid

njservdáàn (a), que d expressàn eltre pdrïlteses em (:0) heveser n es`didr luin?

... ...... ...4 6 4 6go m go m

ki l ki l D D (4.:7)

Vnrtdltn hevemns hetermgldrhe tdi cnrmd que?

4 6 (4.06)

Hestd equdáàn seaue gmehgdtdmelte que?

4 (4.04)

Tue hemnlstrd d ulg`ghdhe, umd vez que se dssumgrmnsd exgstïl`gd he nutrn es`didr que sdtgscdád d prnprgehdhe (2)`nl`iuímns que este es`didr ç n prðprgn es`didr 4.

Vnr sdtgscdzer ds qudtrn prnprgehdhes hd dhgáàn e dsqudtrn prnprgehdhes hd muitgpig`dáàn pnr um es`didr, n `nloultnhe sgstemds he mesmd nrhem e mesmn tgpn `nlstgtuem umespdán vetnrgdi snjre um `nrpn he es`didres K.

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◎ VZNHUXN HGZEXN (NU ERXEZGNZ)

N prnhutn he um sgstemd he nrheml e he qudiquer tgpn pnr um sgstemd he nrhemm e he qudiquer tgpn, helntdhn peinsímjnin ( ), ç umd trdlscnrmdáàn jglärgd, que d `dhd hngssgstemds dssn`gdm d um lnvn sgstemd he nrhem (l+m ) `uon tgpnç umd `nmjgldáàn hns tgpns hns sgstemds trdlscnrmdhns.

...( ) ...( ) ...( ) ...( )... ... ... ...

...( ) ...( ) ...( ) ...( )... ... ... ...

,dj l k ab m i dj l k ab m i ec k go i ec kgo i

dj l k ab m i dj l k ab m i ec k go i ec kgo i

D J @

D J @

(4.0:)

Essd nperdáàn anzd he qudtrn prnprgehdhes?

7) N prnhutn hgretn ç dssn`gdtgvn.

...( ) ...( ) ...( ) ...( ) ...( ) ...( )... ... ... ... ... ...

dj l e ca m o ki p s dj l e ca m o ki p s`h e bg o qr s `h e bg o qr s D J @ D J @

46) Em aerdi n prnhutn hgretn làn `nmutd.

...( ) ...( ) ...( ) ...( )... ... ... ...

dj l e ca m o ca m o dj l e`h e bg o bg o `h e D J J D .

44) Dpreseltd umd espç`ge he hgstrgjutgvghdhe?

...( ) ...( ) ...( ) ...( ) ...( ) ...( ) ...( )... ... ... ... ... ... ...

dj l e ca m o ca m o dj l e ca m o dj l e ca m o`h e bg o bg o `h e bg o `h e bg o D J @ D J D @

.4:) N prnhutn he um sgstemd pein sgstemd luin ç n

sgstemd luin....( )...

dj l e`h e D N N

Tudlhn trdtdrmns he diauls sgstemds espe`gdgs,

hgs`utgremns `nm mdgs hetdibes ds prnprgehdhes hn prnhutnhgretn.

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◎ VZNHUXN GLXEZGNZ (NU GLXEZLN)

Hecglgmns n prnhutn gltergnr `nmn d trdlscnrmdáàn qued hngs sgstemds he prgmegrd nrhem, um sgstemd supergnr 4 : ...g l D d d d e um sgstemd glcergnr 4 : ...g l J j j j , `nm

mesmn ílhg`e, dssn`gdm d um ÿlg`n es`didr (nu sgstemd henrhem zern) pnr megn hd seauglte ieg?

4

,g gg g

lg g

g gg

D J K D J

D J d j (4.00)

Nlhe n dstergs`n snj d `nmpnlelte hn sgstemdg J glhg`dque tnmdremns n seu `nlouadhn `nmpiexn. Este prnhutn pnheser es`rgtn ld cnrmd mdtrg`gdi, hd seauglte cnrmd?

4 :4 :... ...

X g l

g l D J d d d j j j (4.08)Em pdrtg`uidr se ds `nmpnleltes cnrem redgs eltàn

j j , tdmjçm hevemns ressditdr que ns sgstemds he nrhemsupergnr e glcergnr estàn reid`gnldhns, respe`tgvdmelte, peidmdtrgz iglbd e peid mdtrgz `niuld. Dssgm um sgstemd supergnr pnhe ser trdlscnrmdhn em um sgstemd glcergnr, dssgm `nmn umglcergnr em supergnr, trdlspnlhn ds mdtrgzes d eies dssn`gdhns.

Este prnhutn respegtd qudtrn prnprgehdhes?

40) Glversàn hd nrhem hn prnhutn?

g gg g D J J D .

48) Hgstrgjutgvghdhe peid hgregtd?

g g gg g g g D J @ D J D @

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45) Muitgpig`dáàn pnr um es`didr?

g gg g K D J K D J

43) Vdrd g D N , vdie d hesgaudihdhe? 6gg D D

(D expressàn g g D D ç `bdmdhd he lnrmd he g D nu g D ).

Vnr sdtgscdzer essds qudtrn prnprgehdhes n prnhutngltergnr pnhe ser `drd`tergzdhn `nmn um prnhutn glterln.

4.5) @NLXZDÁÀN

Peod um sgstemd he nrhem hngs, nu supergnr, he tgpnmgstn. Hecglgmns d `nltrdáàn `nmn d trdlscnrmdáàn em que umílhg`e supergnr se tnrld ghïltg`n d um, glcergnr, nu um ílhg`eglcergnr se tnrld ghïltg`n d um ílhg`e supergnr. Essdtrdlscnrmdáàn rehuz d nrhem hn sgstemd em (l-: ).

... ...

... ...

... ...

... ...

go l o lki l m i l m

go l o lgi l m i l m

D D

D D

(4.05)

D `nltrdáàn pnhe ser vgstd `nmn um prnhutn glterln2 hehngs sgstemds dpðs ter sghn `di`uidhn seu prnhutn extergnr.

... ...... ...

go l g o lki l m k i l m D J @ H (4.03)

@nltrdglhn n ílhg`eg`nm ílhg`ek, njtemns?

... ...... ...

go l g o lgi l m g i l m D J @ H (4.01)

2 Ld igterdturd umd hecglgáàn he Vrnhutn Gltergnr `nlsgste ld re`íprn`d hessdhecglgáàn? n prnhutn glterln ç n prnhutn extergnr he hngs sgstemds seaughn heumd `nltrdáàn.

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N prnhutn gg J @ ç n prnhutn glterln he g J pnr g@ , e

resuitd em um es`didr K.... ...... ...

go l o lgi l m i l m D K H (4.02)

@nmn ns es`didres `nlstgtuem em um sgstemd he nrhemzern, esse prnhutn hgretn se rehuz d um prnhutn he es`didr pnrum sgstemd.

... ...... ...

go l o lgi l m i l m D K H (4.07)

Hecglglhn d ardlhezd ...... o li l m K H `nmn ...

... o li l m D , temns?

... ...... ... o l go li l m gi l m D D (4.86)

Pu`gltdmelte, n que hemnlstrdmns ç que redigzdhd umd`nltrdáàn snjre hngs ílhg`es qudgsquer, `di`uidmns n seu prnhutn glterln e muitgpig`dmns n es`didr resuitdlte peinsgstemd he nrhem (l-: ).

4.3) PGMEXZGD E DLXGPPGMEXZGD HE PGPXEMD

@nlcnrme vgmns n prnhutn hgretn he hngs sgstemds, emaerdi, làn ç `nmutdtgvn gstn ç?

g o o g D J J D (4.84)

Em nutrds pdidvrds?go og@ @ (4.8:)

Esse cdtn pnhe ser glterpretdhn hd seauglte cnrmd? sgstemds, em aerdi, làn pnssuem sgmetrgd snjre ns ílhg`es. Nuseod, d nrhem hns ílhg`es ç he extremd gmpnrtêl`gd, umdditerdáàn ld nrhem hns ílhg`es gmpig`d em umd muhdlád hd

ldturezd hn sgstemd.

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Lestd sessàn trdtdremns he trïs `dsns pdrtg`uidres,qudlhn ns sgstemds dpreseltdm sgmetrgd snjre ns ílhg`es,qudlhn eies dpreseltdm dltgssgmetrgd snjre ns ílhg`es e qudlhn

eies dpreseltdm sgmetrgd e dltgssgmetrgd snjre ns ílhg`es.@nme`emns d dldigsdr ns `dsns he sgmetrgd.Dn trdtdr he sgmetrgds dldigsdremns hngs `dsns? sgstemds

que pnssuem sgmetrgd em tnhns ns ílhg`es, que `bdmdremns he sgmetrgd djsniutd , e sgstemds que pnssuem sgmetrgd em diaulsílhg`es, que `bdmdremns he sgmetrgd pdr`gdi 7.

Um sgstemd que pnssug sgmetrgd djsniutd ç tnhn sgstemdque d trn`d he ílhg`es làn diterd n sgstemd. Vnr exempin, seod nsgstemd mgstn he ter`egrd nrhemgo

k D , se este sgstemd cnrdjsniutdmelte sgmçtrg`n, eltàn vdiem ds segs gaudihdhes?

go og ko kg gk kgk k g o o o D D D D D D (4.80)

Um sgstemd ter sgmetrgd djsniutd sgalgcg`d que tnhns ns prnhutns hgretns que n aerdm `nmutdm e que pnhemns `nmutdrílhg`es supergnres `nm ílhg`es glcergnres.

Um sgstemd que pnssug sgmetrgd pdr`gdi ç tnhn sgstemdque preservd suds `drd`terístg`ds `nm umd trn`d restrgtd heílhg`es, pnr exempin, supnlbd lnvdmelte n sgstemd mgstn heter`egrd nrhem go

k D , vdmns supnr que este sgstemd seod sgmçtrg`ndpelds snj ns ílhg`es supergnres, leste `dsn teremns dpeldshuds gaudihdhes?

go ogk k D D (4.88)

Leste `dsn làn pnhemns diterdr d pnsgáàn hns ílhg`essupergnres e glcergnres. Pupnlbd danrd n seauglte sgstemd hequdrtd nrhem go

ki D , hgadmns que este sgstemd pnssug sgmetrgd

7 Essd lnmel`idturd ç he respnlsdjgighdhe hn dutnr. Làn bä qudiquer meláànheid ld igterdturd.

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snjre ns ílhg`es glcergnres, leste `dsn teremns, `nmn ln `dsnd`gmd dpelds huds gaudihdhes?

go goki ik D D (4.85)

Làn pnhemns `nmutdr ns ílhg`es supergnres. Gstn ç?go ogki ki D D (4.83)

Ln `dsn he sgstemds pdr`gdimelte sgmçtrg`ns, n`nrre quediauls prnhutns hgretns `nmutdm, elqudltn nutrns làn, diçmhgssn, pnhe bdver nu làn sgmetrgd ld trn`d he um ílhg`e supergnr pnr um glcergnr.

N lÿmern he gaudihdhes pnssívegs serä hdhd pein cdtnrgdihn lÿmern he ílhg`es que `nmutdm, ln prgmegrn exempin `nmntílbdmns 0 ílhg`es que `nmutdvdm tílbdmns 0! gaudihdhes, gstnç, 3 gaudihdhes. Ln `dsn em que hngs ílhg`es `nmutdvdm,tílbdmns huds gaudihdhes nu :!. Essd njservdáàn serä essel`gdiem diaumds dläigses ln `äi`uin telsnrgdi.

Danrd vdmns estuhdr ns `dsns he dltgssgmetrgd.Ns `dsns he dltgssgmetrgd sàn um pnu`n mdgs `nmpiexns,

pnr gssn `nme`emns `nm um exempin jdstdlte sgmpies. Peod umsgstemd he seaulhd nrhem go D , hgzemns que este sgstemd çdltgssgmçtrg`n se d seauglte gheltghdhe cnr vergcg`dhd?

go og og go D D D D (4.81)Gstn ç, se umd trn`d he ílhg`es prnhuz n sgstemd npnstn.Mds se dumeltdrmns n lÿmern he ílhg`es, pnr exempins

trïs ílhg`es. Pe hgssermns que n sgstemd ç dltgssgmçtrg`n emreidáàn dns trïs ílhg`es, gssn d`djd gmpig`dlhn em umdn`nrrïl`gd he sgmetrgd. Pupnlbd que n sgstemdgok D seoddltgssgmçtrg`n.

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Pe permutdrmns n ílhg`e o `nm n ílhg`e k bdverä umdglversàn he sgldgs.

gko gok D D (4.82)

Danrd vdmns permutdr n ílhg`eg `nm ílhg`e k, hevghn ædltgssgmetrgd, njtemns?

kgo gko D D (4.87)

Pujstgtuglhn (4.82) em (4.87)?

( )kgo gok

kgo gok

D D

D D

(4.56)

Nu seod, temns umd sgmetrgd. Lesse `dsn hgzemns que nsgstemd ç sgmçtrg`n tnhd vez que n`nrrer umd permutdáàn pdreltre ns ílhg`es. N iegtnr pnherä vergcg`dr cd`gimelte essd prnprgehdhe e ds seaugltes gaudihdhes.

gok okg kgo

ogk gko kog gok

D D D

D D D D

(4.54)

Vnhemns aelerdigzdr essd prnprgehdhe pdrd sgstemdsdltgssgmçtrg`ns.

Um sgstemd ç hgtn dltgssgmçtrg`n se pdrd um lÿmerngmpdr he permutdáões he ílhg`es n sgstemd ç trdlscnrmdhn emseu npnstn e pdrd um lÿmern pdr he permutdáões he ílhg`e, nsgstemd mdltçm sud cnrmd, gstn ç, n sgstemd ç sgmçtrg`n.

D hetermgldáàn hn lÿmern he `nmpnleltes sgmçtrg`ds

(nu dltgssgmçtrg`ds) serä hdhd pnr?!

:

l , nlhe l ç n lÿmern hd

nrhem hn sgstemd.

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4.1) VZNHUXN HGZEXN ELXZE PGPXEMDP HENZHEM 4

Vdrd exempigcg`dr d reard hn prnhutn hgretn usdremns dhecglgáàn he sgstemds he seaulhd nrhem e n cdtn que estesgstemd ç resuitdhn hn prnhutn hgretn he hngs sgstemds he

prgmegrd nrhem.Hdhns ns seaugltes sgstemds 4 : ...g l D d d d e

4 : ... o l J j j j queremns njter um sgstemd go@ d pdrtgr hn prnhutn hgretn eltre ns sgstemdsg D e o J .

Lestds `nlhgáões temns que?go g o

@ D J (4.5:)Veid hecglgáàn he sgstemds he seaulhd nrhem temns que?

44 4: 4

:4 :: :

4 :

l

lgo

l l ll

` ` `

` ` `@

` ` `

(4.50)

Vnrtdltn seaue que?

44 4: 4

:4 :: :

4 :

l

lg o

l l ll

` ` `

` ` ` D J

` ` `

(4.58)

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Ns `necg`geltes go` hevem ser aerdhns d pdrtgr hds`nnrheldhds hns sgstemds g D e o J . Lestds `nlhgáões?

4 4 4 : 4

: 4 : : :

4 :

l

lg o

l l l l

d j d j d j

d j d j d j D J

d j d j d j

(4.55)

Gstn ç, ns `necg`geltesgo` sàn hdhns peid seauglte reard?

go g o` d j (4.53)

He mnhn dläinan njtemns ds reards he cnrmdáàn pdrdsgstemds glcergnres e mgstns.

go g o` d j (4.51)

g g o o` d j (4.52)

Essd ieg pnhe ser aelerdigzdhd pdrd sgstemds he nrhemsupergnr.

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4.2) @NLPGHEZDÁÕEP AEZDGP

Leste `dpítuin cgzemns umd jreve dpreseltdáàn hds prnprgehdhes hd tenrgd aerdi hns sgstemds, nmgtgmns värgdsnperdáões snjre ns sgstemds, vgstn que lnssn glteresse ç estuhdrum tgpn espe`gdi he sgstemd? ns telsnres.

Emjnrd lnssd dpreseltdáàn vgsdsse dpreseltdr ds iegs e prnprgehdhes pdrd sgstemds mdgs aerdgs, ld prätg`d, ns sgstemdshe mdgnr glteresse ld Císg`d, ld Elaelbdrgd e ld Mdtemätg`dDpig`dhd, sàn ns sgstemds he nrhem zern, nrhem 4 e nrhem :,que pnhem ser represeltdhns pnr nutrds estruturds e berhdm suds prnprgehdhes diaçjrg`ds.

Eltre ns sgstemds he nrhem supergnr temns ns telsnres heZgemdll (nrhem 8) e ns símjnins @brgstnccei (nrhem 0),culhdmeltdgs pdrd n estdjeie`gmeltn hns espdáns he Zgemdll.

Ln prðxgmn `dpítuin gremns gltrnhuzgr d @nlveláàn hdPnmd he Eglstegl, que emjnrd làn dpreselte lelbum resuitdhnmdtemätg`n glnvdhnr, ç jdstdlte empreadhn e jdstdlte ÿtgi pdrdsgmpigcg`dr ds expressões e ds equdáões telsnrgdgs.

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EREZ@Í@GNP

4) Mnstre que n símjnin pq r dpreseltd sgmetrgd emreidáàn dns ílhg`es p e q, sdjelhn que n sgstemd pq a ç sgmçtrg`n.

4:

pr qr pq pq r q p r

a a a

x x x

:) Mnstre que n símjnin s pq dpreseltd sgmetrgd em

reidáàn dns ílhg`es glcergnres , sdjelhn que n sgstemd pq a e sr a çsgmçtrg`n.

s sr pq pq r a

0) Peod ( )g c x umd culáàn he `idsse :@ pdrdg x e k d hergvdhd pdr`gdi em reidáàn d umd

vdrgävei k x . Oustgcgque d dcgrmdáàn d seaugr? n sgstemd ( )g

kp c x ç sgmçtrg`n, gstn ç, ( ) ( )g gkp pk c x c x pdrd

g

x

.

8) Hdhn hngs sgstemds he nrhem um? 4 : 0gu u u u e

4 : 0 ov v v v . @di`uidr n prnhutn hgretng ou v e mnstrdr que nespdán vetnrgdi cnrmdhn pnr esse prnhutn hgretn pnssug lnvehgmelsões.

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5) Hdhn hngs sgstemds he nrhem um? 0 5 :gu e

4 : 8 ov v v . Hetermgldr ns vdinres he4v e :v sdjelhn que n prnhutn hgretn hestes hngs sgstemds ç `nmutdtgvn.

3) He`nmpnr n sgstemd he seaulhd nrhemgo@ em umd`nmjgldáàn he sud pdrte sgmçtrg`d e he sud pdrte dssgmçtrg`d.

0 : 4

6 8 0

1 44 5go@

1) Hetermgle snjre qudgs `nlhgáões estd gaudihdhe ç

adrdltghd g o o gd j j d . @nmpdre `nm d tenrgd mdtrg`gdi.2) Hetermgle snjre qudgs `nlhgáões estd gaudihdhe ç

adrdltghd g o o gd j j d .

7) Aelerdigze d prnprgehdhe (43) pdrd n `dsn heg D N . (Mnstre que n prnhutn gltergnr eltre um sgstemdqudiquer he prgmegrd nrhem e n sgstemd luin ç zern).

46) Hdhns 4 6 :gd , 0 4 6 oj e 4 4 4k ` .

@di`uidr n vdinr he 4 : 0k h h h h sdjelhn que g g

ok o k h d ` .