D IANUSMAT IKH

ANALUSH

Ant¸nhc Strèklac

EpÐkouroc Kajhght c

Tm ma Majhmatik¸n

P A T R A 2 0 0 7

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This is LaTeX2e 2003/12/01 (MiKTeX 2.4).Figures by PICTEX.

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EpexergasÐa keimènou: Ant¸nhc Strèklace-mail: streklas@math.upatras.gr

Pr¸th èkdosh: IoÔnioc 2007

2

'Umnoc sthn Rèa

Pìtnia Rèa, jÔgater polumìrfou Prwtogìnoio t' epÐ taurofìnwn ierìtroqon �rma titaÐneictumpanìdoupe, filoistromanèc, qalkìkrote koÔrhm ter Zhnìc �naktoc OlumpÐou, aigiìqoiop�ntim' aglaìmorfe, Krìnou sÔllektre m�kaira,oÔresin   qaÐreic jnht¸n tìlolÔgmasi friktoÐcpambasÐleia Rèa, polemìklone, ombrimìjuke,yeudomènh, s¸teira, luthri�c, arqigènejle,m thr men te je¸n hdè jnht¸n anjr¸pwn.Ek sou gar kai gaÐa kai ouranìc eurÔc Ôperjenkai pìntoc pnoiaÐ te. Filìdrome, aerìmorfe.Eljè, m�kaira je�, swt rioc eÔfroni boul ieir nhn kat�gousa sun euìlboic kte�tessi,lÔmata kai k rac pèmpous' epÐ tèrmata gaÐhc.

Sebast  Rèa, jugatèra tou Prwtogenhmènou polÔmorfikoÔ Q�oucesÔ pou odhgeÐc twn taurofìnwn to ierìtroqo �rmaqtup�c ta tÔmpana, agap�c thn manÐa, kìrh pou bg�zeic q�lkinouc  qoucmhtèra tou basili� Z na, pou katoikeÐ ston 'Olumpo kai fèrei thn aspÐdapanentimìtath me lampr  morf  eutuqismènh suntrìfissa tou QrìnouesÔ pou qaÐresai sta boun� me twn jnht¸n ta frikt� alal�gmata.W Pant�nassa Rèa pou prokaleÐc ton jìrubo ston pìlemo, genaiìyuqhesÔ pou s¸zeic me teqn�smata, pou apolutr¸neic pou eÐsai prwtogenhmènh,eÐsai mhtèra kai twn ajan�twn je¸n kai twn jnht¸n anjr¸pwndiìti apì sèna proèrqontai kai h gh (gh ) kai o ouranìc (pur)o eurÔqwroc pou thn perib�llei kai h j�lassa (Ôdwr) kai oi �nemoi (a r),esÔ pou agap�c na rèeic kai èqeic thn morf  tou reustoÔ aijèra.'Ela makarÐa je� na mac s¸seic me kalìboulh di�jeshfère mac thn eir nh me triseutuqismèna d¸rakai xapìsteile stic �krec thc ghc ta �qrhsta kai tic dustuqÐec.

Orfikìc Ômnoc

3

H Ôlh tou Maj matoc

• DianÔsmata. Eswterikì kai exwterikì ginìmeno dianusm�twn. Par�gw-goi dianusmatik¸n sunart sewn. DianusmatikoÐ telestèc, (Grad, Div,Curl)

• PÐnakec. 'Algebra pin�kwn. DiagwnopoÐhsh pin�kwn. Idiotimèc kai idio-dianÔsmata pin�kwn.

• Seirèc. 'Apeirec seirèc. Krit ria sÔgklishc �peirwn seir¸n. SeirècTaylor kai Maclaurin. Dunamoseirèc. Seirèc sunart sewn. Or-jog¸niec sunart seic. Polu¸numa Legendre, Hermite, Chebyshev,kai Laquerre. Seirèc Fourier. Metasqhmatismìc Laplace.

• Statistik  kai pijanìthtec. TuqaÐec metablhtèc. Katanomèc. RopèctuqaÐwn metablht¸n. 'Elegqoc upojèsewn. Diast mata empistosÔnhc.JewrÐa sfalm�twn.

To biblÐo tou Maj matoc

An¸tera Majhmatik�. M. Spiegel.Schaum’s outline series. McGraw - Hill, 1963

4

Perieqìmena

1 Dianusmatik  An�lush 71.1 Ta Fusik� megèjh . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Ta dianÔsmata . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Oi pr�xeic twn dianusm�twn . . . . . . . . . . . . . . . . . . . 91.4 Ginìmena dianusm�twn . . . . . . . . . . . . . . . . . . . . . . . 141.5 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Dianusmatikèc sunart seic 272.1 Dianusmatikèc sunart seic . . . . . . . . . . . . . . . . . . . . 272.2 To ìrio kai h sunèqeia . . . . . . . . . . . . . . . . . . . . . . 302.3 H par�gwgoc kai to olokl rwma . . . . . . . . . . . . . . . . . 312.4 Ta dianusmatik� kai ta bajmwt� pedÐa . . . . . . . . . . . . . . 362.5 H klÐsh   apìklish kai o strobilismìc . . . . . . . . . . . . . 362.6 To epikampÔlio kai to epifaneiakì

olokl rwma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.7 Kampulìgrammec suntetagmènec . . . . . . . . . . . . . . . . . 442.8 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 PÐnakec, OrÐzousec, Grammik� sust mata 613.1 Oi pÐnakec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 Oi orÐzousec . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3 O Pollaplasiasmìc twn Pin�kwn . . . . . . . . . . . . . . . . 663.4 Ta grammik� sust mata . . . . . . . . . . . . . . . . . . . . . . 703.5 Idiotimèc kai idiodianÔsmata enìc pÐnaka . . . . . . . . . . . . . 723.6 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Seirèc Fouriè 854.1 O dianusmatikìc q¸roc . . . . . . . . . . . . . . . . . . . . . . 854.2 Eswterikì ginìmeno . . . . . . . . . . . . . . . . . . . . . . . . 864.3 Seirèc Fouriè . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4 OloklhrwtikoÐ MetasqhmatismoÐ . . . . . . . . . . . . . . . . . 94

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4.5 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 'Alutec Ask seic 107

Prìlogoc

Oi shmei¸seic autèc eÐnai apì tic paradìseic kai ta frontist ria tou Ma-j matoc Majhmatik� II Mèroc A gia to Qhmikì Tm ma.

ApeujÔnontai stouc foithtèc tou pr¸tou ètouc tou QhmikoÔ Tm matoctou PanepisthmÐou Patr¸n. Ja emploutÐzontai se k�je nèa touc èkdosh apìtic epexhg seic stic aporÐec pou ja diatup¸noun oi foithtèc tou tm matoc.Par�llhla me thn Ôlh twn shmei¸sewn aut¸n oi foithtèc did�skontai stoMèroc B. tou maj matoc k�poiec arqèc apì thn Statistik  kai tic Pijanìtht-ec.

Oi shmei¸seic perièqoun kai k�poiec entolèc apì to upologistikì prìgram-ma Mathematica qwrÐc pollèc epexhg seic. O foitht c gia na katal�beito tm ma autì prèpei na gnwrÐzei stoiqeiwd¸c thn qr sh tètoiwn program-m�twn. 'Ola ta progr�mmata aut� leitourgoÔn lÐgo polÔ me ton Ðdio trìpo.'Eqoun odhgÐec qr sewc pou mporeÐte na tic diab�sete pat¸ntac (sun jwc)to pl ktro F1. MporeÐte na mhn diab�sete kajìlou to tm ma autì, den jasac qreiasteÐ stic exet�seic.

Se aut  thn pr¸th èkdosh pijanìn na perièqontai pollèc ableyÐec, par-al yeic   kai l�jh lìgw thc èlleiyewc qrìnou gia mia deÔterh an�gnwsh. Giautì zht�w prokatabolik� thn sugn¸mh twn foitht¸n. Parakal¸ touc anag-n¸stec na mhn dist�soun na mou upodeÐxoun tuqìn l�jh all� kai paraleÐyeicpou ja upopèsoun sthn antÐlhy  touc. Ja touc eÐmai lÐan upoqrewmènoc kaija ta l�bw sobar� upìyh mou se mia deÔterh èkdosh.

Ant¸nhc StrèklacP�tra 2007

6

Kef�laio 1

Dianusmatik  An�lush

1.1 Ta Fusik� megèjhTa fusik� megèjh eÐnai fusikèc idiìthtec pou qarakthrÐzontai apì poiìth-ta kai posìthta. Ta majhmatik� sÔmbola orÐzoun thn poiìthta tou megèjoucen¸ oi arijmoÐ qarakthrÐzoun thn posìthta tou megèjouc.

Poll� fusik� megèjh mporoÔn na prosdioristoÔn pl rwc apì èna mìnoarijmì. O arijmìc autìc qarakthrÐzei thn posìthta tou megèjouc. Ta megèjhaut� onom�zontai monìmetra   bajmwt�.

H jermokrasÐa, h enèrgeia, to m koc k.l.p. eÐnai megèjh bajmwt�. 'Eqoumegia par�deigma jermokrasÐa 5o KelsÐou   m koc 55 mètrwn. H tim  thcmon�dac eklègetai aujairètwc.

Up�rqoun ìmwc �lla fusik� megèjh pou prosdiorÐzontai apì perissìter-ouc arijmoÔc. Gia par�deigma to mègejoc thc taqÔthtac qarakthrÐzetai apìtreic arijmoÔc pou parist�noun to m koc, thn dieÔjunsh kai thn for� toumegèjouc. H kat�llhlh majhmatik  ènnoia gia na perigr�youme apl� kai mesaf neia tètoia megèjh eÐnai ta dianÔsmata. Ta antÐstoiqa megèjh onom�-zontai dianusmatik� fusik� megèjh. O kl�doc twn Majhmatik¸n pou exet�zeita dianÔsmata kai tic idiìthtec touc onom�zetai Dianusmatik  An�lush.

Tèloc up�rqoun fusik� megèjh pou prosdiorÐzontai apì perissìteroucapì treic arijmoÔc. Tètoia megèjh onom�zontai tanustik�, perigr�fontaiapì touc tanustèc kai o antÐstoiqoc kl�doc twn Majhmatik¸n onom�ze-tai Tanustik  an�lush.

Par�deigma èqoume èna anisìtropo ulikì. 'Ena ulikì dhlad  pou k�poiaidiìtht� tou eÐnai diaforetik  sthn x− dieÔjunsh diaforetik  sthn y− dieÔ-junsh kai diaforetik  sthn z− dieÔjunsh. An mia dianusmatik  idiìthta touulikoÔ exart�tai apì mia �llh grammik� p.q. apì thn sqèsh ~p = m~v tìte top1 den exart�tai mìno apì to v1 me stajer� analogÐac m11 all� kai apì

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ìlec tic �llec sunist¸sec v2 kai v3 me stajerèc analogÐac m12 kai m13.Gr�foume

pj =3∑

k=1

mjkuk

To mègejoc m orÐzetai an prosdiorÐsoume kai ta ennèa stoiqeÐa mjk thcm trac. Gr�foume

m =

m11 m12 m13

m21 m22 m23

m31 m32 m33

To mègejoc autì eÐnai ènac tanust c t�xhc 2. Oi sunist¸sec tou eÐnai oi32 = 9 arijmoÐ mjk, ìpou 1 ≤ j, k < 3. EÐnai bolikì na gr�youme tastoiqeÐa enìc tanust  t�xhc 2 san mia m tra (pÐnakac) 3 × 3. Autì denshmaÐnei ìti ènac pÐnakac me 9 stoiqeÐa parist�nei p�nta èna tanust  t�xhc2. Prèpei na ikanopoioÔntai kai k�poiec �llec proôpojèseic.

Up�rqoun tèloc kai tanustèc me t�xh 3 pou parist�nontai me to sÔmbolomijk (treic deÐktec) me t�xh 4 kai tèsseric deÐktec k.l.p.

Mia perioq  pou se k�je shmeÐo thc antistoiqeÐ èna bajmwtì   èna di-anusmatikì   èna tanustikì mègejoc onom�zetai antistoÐqwc bajmwtì pedÐo  dianusmatikì pedÐo   tanustikì pedÐo. Ta pedÐa aut� eÐnai dunatìn na èqounkai analutik  qronik  ex�rthsh. Gr�foume tìte V (~r, t), ~F (~r, t), Tijk(~r, t) giata pedÐa aut�.

1.2 Ta dianÔsmata'Ena di�nusma parist�netai gewmetrik� apì èna prosanatolismèno eujÔgrammotm ma. O prosanatolismìc tou kajorÐzetai apì èna bèloc ~AB. To A eÐnai harq  kai to B to tèloc tou tm matoc. H dieÔjunsh kai h for� tou tm matocautoÔ prosdiorÐzei thn dieÔjunsh kai thn for� tou dianÔsmatoc, en¸ to m koctou, to mètro tou dianÔsmatoc.

............................................................................................................................BA

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G D

Ta dianÔsmata.

Ja gr�foume ta dianÔsmata me èna bèloc apì ep�nw. Gia par�deigma h taqÔth-ta sumbolÐzetai me ~v. Ta mètra twn dianusm�twn ja ta sumbolÐzoume me tamajhmatik� touc sÔmbola qwrÐc to bèloc apì ep�nw. Gia to mètro thc taqÔth-tac gia par�deigma gr�foume mètro( ~v) = v.

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Oi sumbolismoÐ |~v|   kai ‖~v‖ qrhsimopoioÔntai epÐshc gia to mètro enìcdianÔsmatoc.

(µετρo ~v) = v = ‖~v‖ = |~v|'Ena di�nusma perigr�fetai algebrik� apì treic arijmoÔc. TopojetoÔme to

di�nusma ètsi ¸ste h arq  tou na sumpÐptei me thn arq  enìc trisorjog¸niousust matoc anafor�c. H jèsh tou tèlouc tou dianÔsmatoc perigr�fetai apìtreic arijmoÔc. Oi arijmoÐ autoÐ eÐnai oi suntetagmènec tou dianÔsmatoc wcproc to sÔsthma anafor�c. Me aut  thn ènnoia to di�nusma thc taqÔth-tac perigr�fetai apì thn tri�da twn arijm¸n (v1, v2, v3) pou onom�zontai(Kartesianèc) sunist¸sec.

'Ena opoiod pote di�nusma ~v gr�fetai wc ex c 1

~v = (v1, v2, v3)

o....................................................................................................................................x

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y....................................................................................................................................................................................................

z

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~v

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v3

v1

v2

Oi Kartesianèc suntetagmènec.

Orismìc: DÔo dianÔsmata eÐnai Ðsa an oi antÐstoiqec sunist¸sec touceÐnai Ðsec. 'Ena di�nusma eÐnai Ðso me to mhdenikì di�nusma ìtan kai oi treicsunist¸sec tou eÐnai mhdèn.

1.3 Oi pr�xeic twn dianusm�twn

MporoÔme na prosjèsoume dÔo dianÔsmata ~a kai ~b . To apotèlesma eÐnaièna trÐto di�nusma pou sumbolÐzetai me ~a +~b. H pr�xh thc prìsjeshc eÐnaimia eswterik  pr�xh sÔnjeshc ston q¸ro twn dianusm�twn.

1Στον υπολογιστή ορίζουμε το διάνυσμα με την εντολή῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[42] := v = {v1, v2, v3}Out[42] = {v1, v2, v3}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Για τον ορισμό του διανύσματος χρησιμοποιούμε τις αγκύλες {} αντί για τις παρενθέσεις ().

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Orismìc: To �jroisma dÔo dianusm�twn ~a +~b brÐsketai gewmetrik�apì ton kanìna tou parallhlogr�mmou. Metafèroume ta dianÔsmata se ènashmeÐo O pou sun jwc eÐnai h arq  tou sust matoc suntetagmènwn. Met�sqhmatÐzoume to parallhlìgrammo pou èqei proskeÐmenec pleurèc ta dianÔs-mata ~a kai ~b . To di�nusma ~a +~b eÐnai h diag¸nioc tou parallhlogr�mmou.

To �jroisma dÔo dianusm�twn eÐnai èna trÐto di�nusma me sunist¸sec to�jroisma twn antÐstoiqwn sunistws¸n twn dÔo dianusm�twn. Dhlad 

~a +~b = (a1, a2, a3) + (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3)

'Opwc faÐnetai kai apì to sq ma, to deÔtero di�nusma mporeÐ na metaferjeÐètsi ¸ste h arq  tou na sumpÐptei me to tèloc tou pr¸tou dianÔsmatoc. EÐnaiepomènwc profanèc ìti oi dÔo autoÐ orismoÐ tou ajroÐsmatoc eÐnai isodÔnamoi.

o....................................................................................................................................x

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y....................................................................................................................................................................................................

z

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~a...................................................a3

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..............................~b

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..............................~b

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~a +~b

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.......................................................................................................................................................a3+b3

H prìsjesh dÔo dianusm�twn.

H pr�xh thc prìsjeshc twn dianusm�twn èqei thn prosetairistik  idiìthtadhlad 

(~a +~b) + ~c = ~a + (~b + ~c)

kai thn antimetajetik  idiìthta dhlad 

~a +~b = ~b + ~a

EpÐshc up�rqei di�nusma ~0 tètoio ¸ste

~a +~0 = ~0 + ~a = ~a

To di�nusma ~0 onom�zetai oudètero stoiqeÐo thc prìsjeshc. Tèloc giak�je di�nusma ~a up�rqei èna di�nusma ~b pou onom�zetai antÐjeto tou ~a.SumbolÐzetai me −~a kai isqÔei

~a +~b = ~a + (−~a) = ~0

10

Me thn bo jeia tou antÐjetou dianÔsmatoc orÐzoume thn diafor� dÔo di-anusm�twn, san to �jroisma tou pr¸tou me to antÐjeto tou deÔterou.

~a−~b = ~a + (−~b)

.................................................................................................................................... ~a

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..............................~b

....................................................................................................................................−~b...............................................................................................................................

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~a−~b

H diafor� dÔo dianusm�twn.

Orismìc: To ginìmeno enìc dianÔsmatoc ~a me ènan pragmatikì arijmìξ eÐnai èna di�nusma pou sumbolÐzetai me ξ~a. To di�nusma ξ~a èqei thn ÐdiadieÔjunsh me to di�nusma ~a. To mètro tou eÐnai Ðso me to ginìmeno tou |ξ|me to mètro tou ~a,

‖ξ~a‖ = |ξ| · ‖~a‖An o arijmìc ξ eÐnai arnhtikìc h for� tou dianÔsmatoc ξ~a eÐnai antÐjethapì thn for� tou ~a en¸ gia jetik� ξ ta dÔo dianÔsmata èqoun thn Ðdiafor�.

o....................................................................................................................................x

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y....................................................................................................................................................................................................

z

...........................................................................................~a

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94~a

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−2~a

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A BG

D OBOA

= 94

= ξ

ΓA‖∆B

O pollaplasiasmìc enìc dianÔsmatoc me ènan rhtì arijmì.

H pr�xh aut  eÐnai mia exwterik  pr�xh sÔnjeshc kai onom�zetai exw-terikìc pollaplasiasmìc. Profan¸c 1~a = ~a kai 0~a = ~0. O exwterikìcpollaplasiasmìc ikanopoieÐ epÐshc kai thn idiìthta

ξ(η~a) = (ξη)~a

Algebrik� h pr�xh orÐzetai wc ex c

ξ~a = ξ(a1, a2, a3) = (ξa1, ξa2, ξa3)

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Oi dÔo pr�xeic ikanopoioÔn tic akìloujec idiìthtec

ξ(~a +~b) = ξ~a + ξ~b (ξ + η)~a = ξ~a + η~b

pou onom�zontai epimeristikèc idiìthtec tou pollaplasiasmoÔ wc proc thnprìsjesh. H sqèseic autèc eÐnai aparaÐthtec ètsi ¸ste oi dÔo pr�xeic, h prìs-jesh kai o exwterikìc pollaplasiasmìc na eÐnai pr�xeic sumbibastèc metaxÔtouc.

Orismìc: To sÔnolo twn dianusm�twn ston trisdi�stato q¸ro me ticdÔo parap�nw pr�xeic, apoteleÐ dianusmatikì q¸ro pou onom�zetai trisdi�s-tatoc EukleÐdeioc dianusmatikìc q¸roc epÐ tou R kai sumbolÐzetai me R3.Me ìmoio trìpo orÐzetai kai o EukleÐdeioc migadikìc dianusmatikìc C3 epÐ tous¸matoc twn migadik¸n arijm¸n C.

Orismìc: Ta dianÔsmata ~a1, ~a2 kai ~a3 eÐnai grammik� anex�rthta anh sqèsh c1~a1 + c2~a2 + c3~a3 = 0 isqÔei mìno an kai oi treic suntelestèc ci

i = 1, 2, 3 eÐnai mhdèn dhlad 

c1~a1 + c2~a2 + c3~a3 = 0 ⇔ c1 = c2 = c3 = 0

TrÐa grammik� anex�rthta dianÔsmata ~e1, ~e2 kai ~e3 tou trisdi�statou q¸roueÐnai b�sh tou q¸rou kai k�je di�nusma gr�fetai san grammikìc sunduasmìctwn dianusm�twn aut¸n.

~v = v1~e1 + v2~e2 + v3~e3 =3∑

k=1

vk~ek vk ∈ R

o....................................................................................................................................x

...............................~e1

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y..................................

~e2....................................................................................................................................................................................................

z

........

........

........

..........~e3

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

~v

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

v3

v1

v2

To trisorjog¸nio dexiìstrofo kartesianì sÔsthma suntetagmènwn.

TrÐa dianÔsmata pou den brÐskontai sto Ðdio epÐpedo (  den eÐnai par�llh-la) apoteloÔn èna sÔsthma anafor�c dhlad  mia b�sh tou q¸rou. Gia thnaplìthta twn ekfr�sewn sun jwc qrhsimopoioÔme èna trisorjog¸nio sÔsth-ma axìnwn Oxyz san sÔsthma anafor�c. Ep�nw stouc �xonec Ox, Oykai Oz orÐzoume antistoÐqwc trÐa k�jeta metaxÔ touc kai monadiaÐa (èqoun

12

m koc thn mon�da) dianÔsmata ~e1, ~e2 kai ~e3. Ta dianÔsmata aut� deÐqnounthn jetik  for� twn axìnwn.

'Estw èna di�nusma ~v topojethmèno sthn arq  twn axìnwn pou sqhmatÐzeime touc jetikoÔc �xonec tou sust matoc tic gwnÐec a, b kai c. Oi sunte-tagmènec tou dianÔsmatoc mporoÔn na brejoÔn apì tic gwnÐec autèc. 'Eqoume~v = (v1, v2, v3) ìpou

v1 = v cos a v2 = v cos b v3 = v cos c

Ta sunhmÐtona aut� onom�zontai dieujÔnonta sunhmÐtona.

o................................................................................................................................................................x

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

y....................................................................................................................................................................................................

z

..................................................................................................................................................................~v

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

........

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

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

a

b

c

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

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

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

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

..............................................v1

v2

v3

Ta dieujÔnonta sunhmÐtona.

To mètro tou dianÔsmatoc apì to Pujagìreio je¸rhma eÐnai

v =√

v21 + v2

2 + v23

Ta dianÔsmata thc b�shc eÐnai

~e1 = (1, 0, 0) ~e2 = (0, 1, 0) ~e3 = (0, 0, 1)

Pollèc forèc, gia tupikoÔc lìgouc, qrhsimopoioÔme kai touc sumbolismoÔc

(~i,~j,~k) η (~x0, ~y0, ~z0)

gia ta basik� dianÔsmata ~e1, ~e2, ~e3.'Ena di�nusma wc proc thn b�sh aut  gr�fetai

~v = (v1, v2, v3) = v1~e1 + v2~e2 + v3~e3 = v1~i + v2

~j + v3~k = v1~x0 + v2~y0 + v3~z0

13

1.4 Ginìmena dianusm�twn

Orismìc: Eswterikì ginìmeno twn dianusm�twn ~a kai ~b me suntetagmènecantistoÐqwc (a1, a2, a3) kai (b1, b2, b3) orÐzoume ton pragmatikì arijmì a1b1+a2b2 + a3b3. To eswterikì ginìmeno eÐnai Ðso me to ginìmeno twn mètrwn twndÔo dianusm�twn epÐ to sunhmÐtono thc gwnÐac pou sqhmatÐzoun. Gr�foume 2

~a ·~b = a1b1 + a2b2 + a3b3 = |~a||~b| cos φ

Me thn bo jeia tou eswterikoÔ ginomènou mporoÔme na broÔme thn gwnÐadÔo dianusm�twn. H sqèsh eÐnai

cos φ =~a ·~b|~a||~b|

≤ 1

H parap�nw anisìthta eÐnai gnwst  san anisìthta tou Sbartc.H gwnÐa twn dianusm�twn ~v kai ~e1 gia par�deigma eÐnai

cos a =~v · ~e1

‖~v‖‖~e1‖ =(v1, v2, v3) · (1, 0, 0)

‖~v‖‖~e1‖ =v1

v

To eswterikì ginìmeno èqei thn antimetajetik  idiìthta kaj¸c kai thnepimeristik  idiìthta wc proc thn prìsjesh, dhlad 

~a ·~b = ~b · ~a~a · (~b + ~c) = ~a ·~b + ~a · ~c (1.1)

An dÔo mh mhdenik� dianÔsmata ~a kai ~b èqoun eswterikì ginìmeno Ðso memhdèn tìte ta dianÔsmata aut� eÐnai k�jeta metaxÔ touc. Gr�foume sumbolik�~a ⊥ ~b.

Gia ta basik� dianÔsmata isqÔoun oi sqèseic

~e1 · ~e2 = ~e2 · ~e3 = ~e3 · ~e1 = 0 ‖~e1‖ = ‖~e2‖ = ‖~e3‖ = 1

2Το εσωτερικό γινόμενο῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[46] := a = {a1, a2, a3}b = {b1, b2, b3}Dot[a, b]Out[46] = {a1, a2, a3}Out[47] = {b1, b2, b3}Out[48] = a1 b1 + a2 b2 + a3 b3

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Στο αποτέλεσμα το πρόγραμμα αφήνει ένα διάστημα μεταξύ των παραγόντων π.χ. a1 καιb1 που δηλώνει το γινόμενο των παραγόντων αυτών δηλαδή a1 ∗ b1.

14

Dhlad  ta basik� aut� dianÔsmata eÐnai k�jeta metaxÔ touc kai èqoun mètrothn mon�da. 'Ena tètoio sÔnolo dianusm�twn onom�zetai orjokanonikì.

Orismìc: 'Ena sÔnolo onom�zetai orjokanonikì an isqÔoun oi sqèseic

~ei · ~ej = δij

ìpou o tanust c δij onom�zetai sÔmbolo tou Krìneker kai orÐzetai apì thnsqèsh

δij =

{1 αν i = j0 αν i 6= j

(1.2)

K�je di�nusma tou q¸rou gr�fetai san grammikìc sunduasmìc twn di-anusm�twn miac orjokanonik c b�shc.

~v = v1~e1 + v2~e2 + v3~e3

ìpou eÔkola apodeiknÔetai ìti oi sunist¸sec tou dianÔsmatoc dÐnontai apì ticsqèseic

v1 = ~v · ~e1 v2 = ~v · ~e2 v3 = ~v · ~e3

H qr sh enìc orjokanonikoÔ sunìlou san b�sh tou q¸rou aplopoieÐ tic sqè-seic.

Orismìc: To exwterikì ginìmeno twn dianusm�twn ~a kai ~b sum-bolÐzetai me ~a×~b kai orÐzetai san to di�nusma me mètro Ðso me to ginìmenotwn mètrwn twn dÔo dianusm�twn epÐ to hmÐtono thc gwnÐac touc θ. Epomènwcto mètro tou exwterikoÔ ginomènou dÔo dianusm�twn eÐnai Ðso me to embadìntou parallhlogr�mmou pou sqhmatÐzoun. H dieÔjunsh tou dianÔsmatoc eÐnaik�jeth kai sta dÔo dianÔsmata kai h for� tou orÐzetai apì ton kanìna thcdexiìstrofhc bÐdac. Dhlad  tètoia ¸ste ta dianÔsmata ~a, ~b kai ~a ×~b naapoteloÔn èna dexiìstrofo sÔsthma.

....................................................................................................................................~a

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

~b

........

........

........

........

........

........

........

........

........

........................ ~a×~b

................................................................................................~b× ~a

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

........

........

........

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

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

θ

To exwterikì ginìmeno.

An all�xoume thn for� twn axìnwn èna di�nusma all�zei prìshmo. TadianÔsmata ~a kai ~b gia par�deigma me mia tètoia allag  twn axìnwn gÐnontai−~a kai −~b . Antijètwc to exwterikì ginìmeno paramènei analloÐwto. 'Eqoume

~a×~b → (−~a)× (−~b) = ~a×~b

15

Autì sumbaÐnei diìti to exwterikì ginìmeno den èqei saf¸c kajorismènh for�all� aut  orÐzetai sumbatik�. 'Ena tètoio di�nusma onom�zetai sun jwc yeu-dodi�nusma   axonikì di�nusma.

An dÔo mh mhdenik� dianÔsmata ~a kai ~b èqoun exwterikì ginìmeno Ðsome to mhdenikì di�nusma toÔto shmaÐnei ìti ta dianÔsmata eÐnai par�llhla.Gr�foume sumbolik� ~a ‖ ~b.

To exwterikì ginìmeno orÐzetai apì ton tÔpo 3

~a×~b = (a2b3 − a3b2, a3b1 − a1b3, a2b3 − a3b2)

To exwterikì ginìmeno orÐzetai epÐshc kai apì ton tÔpo 4

~a×~b =

∣∣∣∣a2 a3

b2 b3

∣∣∣∣~e1 +

∣∣∣∣a3 a1

b3 b1

∣∣∣∣~e2 +

∣∣∣∣a1 a2

b1 b2

∣∣∣∣~e3 =

∣∣∣∣∣∣

~e1 ~e2 ~e3

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣Ta parap�nw sÔmbola |A| parist�noun tic orÐzousec twn pin�kwn. H teleu-taÐa isìthta thc parap�nw sqèshc eÐnai kai o orismìc miac orÐzousac enìcpÐnaka me di�stash 3×3. H orÐzousa enìc pÐnaka me di�stash 2×2 orÐzetaiapì thn sqèsh ∣∣∣∣

a11 a12

a21 a22

∣∣∣∣ = a11a22 − a12a21

3Το εξωτερικό γινόμενο῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[1] := a = {a1, a2, a3}b = {b1, b2, b3}Cross[a, b]Out[1] = {a1, a2, a3}Out[2] = {b1, b2, b3}Out[3] = {−a3 b2 + a2 b3, a3 b1− a1 b3,−a2 b1 + a1 b2}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

4 ΄Ενας πίνακας στο Mathematica ορίζεται με την εντολή:῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[15] := A = {{a11, a12, a13}, {a21, a22, a23}, {a31, a32, a33}}Out[15] = {{a11, a12, a13}, {a21, a22, a23}, {a31, a32, a33}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Για να βρούμε την ορίζουσα ενός τετραγωνικού πίνακα, εκτελούμε την εντολή Det[A].

Για να αναπτύξουμε την ορίζουσα αυτή ως προς τα στοιχεία της πρώτης γραμμής γράφουμε:῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[18] := Collect[Det[t], {a11, a12, a13}, Simplify]Out[18] = a13 (−a22 a31 + a21 a32) + a12 (a23 a31− a21 a33)+

a11 (−a23 a32 + a22 a33)῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

16

Gia ta basik� monadiaÐa dianÔsmata isqÔoun oi sqèseic

~e1 × ~e2 = ~e3 ~e2 × ~e3 = ~e1 ~e3 × ~e1 = ~e2

MporoÔme na gr�youme ìlec tic parap�nw sqèseic wc ex c

~ei × ~ej =3∑

k

εijk~ek = εijk~ek

Sthn parap�nw sqèsh up�rqei �jroish wc proc ton deÐkth k pou den èqeigrafeÐ analutik�. Sto ex c ja paraleÐpoume to sÔmbolo thc �jroishc kai jaennoeÐtai ìti up�rqei �jroish wc proc k�je deÐkth pou epanalamb�netai dÔo  perissìterec forèc sthn Ðdia pleur� miac isìthtac kai sto Ðdio par�gonta(o deÐkthc k sthn prokeimènh perÐptwsh). O tanust c tou LebÔ - Tsibit�(tanust c trÐthc t�xhc) εijk orÐzetai apì thn sqèsh

εijk =

1 αν (i, j, k) = (1, 2, 3) η (2, 3, 1) η (3, 1, 2)−1 αν (i, j, k) = (3, 2, 1) η (2, 1, 3) η (1, 3, 2)0 gia ìlec tic upìloipec peript¸seic

  alli¸c

ε123 = ε231 = ε312 = 1 ε321 = ε213 = ε132 = −1 (1.3)

Me thn bo jeia tou tanust  autoÔ mporoÔme na gr�youme ton orismì touexwterikoÔ ginomènou sunoptik� wc ex c

~a×~b = εijkaibj~ek

kai ed¸ ennoeÐtai ìti up�rqei �jroish wc proc touc deÐktec i, j kai k. Tos ma thc �jroishc den gr�fetai analutik�.

To mètro tou exwterikoÔ ginomènou dÔo dianusm�twn me gwnÐa θ brÐsketaiapì thn tautìthta

(~a×~b

)·(~a×~b

)= a2b2 −

(~a ·~b

)2

Gia thn apìdeixh, parathroÔme ìti to pr¸to mèloc thc tautìthtac aut -c eÐnai Ðso me (ab sin θ) · (ab sin θ) = a2b2 sin2 θ kai to deÔtero Ðso mea2b2 − (ab cos θ)2 = a2b2 sin2 θ.

To exwterikì ginìmeno den ikanopoieÐ thn antimetajetik  idiìthta. EÐnaiantisummetrikì dhlad 

~a×~b = −~b× ~a

17

MporoÔme eÔkola na apodeÐxoume ìti to exwterikì ginìmeno eÐnai epimeristikìwc proc thn prìsjesh.

~a× (~b + ~c) = ~a×~b + ~a× ~c

Orismìc: Miktì ginìmeno tri¸n dianusm�twn ~a, ~b kai ~c orÐzetai o ar-ijmìc ~a ·(~b×~c). O arijmìc autìc eÐnai Ðsoc me ton ìgko tou parallhlepipèdoupou sqhmatÐzetai me akmèc ta trÐa aut� dianÔsmata.

ApodeiknÔetai eÔkola ìti isqÔei.

~a · (~b× ~c) =

∣∣∣∣∣∣

a1 a2 a3

b1 b2 b3

c1 c2 c3

∣∣∣∣∣∣

Apì ton orismì tou triploÔ autoÔ ginomènou san orÐzousa faÐnetai amèswcìti isqÔoun oi sqèseic

~a · (~b× ~c) = ~b · (~c× ~a) = ~c · (~a×~b)

An dÔo apì ta trÐa aut� dianÔsmata eÐnai Ðsa tìte to triplì ginìmeno eÐnaimhdèn diìti eÐnai Ðso me mia orÐzousa pou èqei dÔo grammèc Ðsec. Apì aut thn idiìthta apodeiknÔetai eÔkola ìti an to di�nusma ~d eÐnai ènac grammikìcsunduasmìc tri¸n dianusm�twn pou den brÐskontai sto Ðdio epÐpedo dhlad 

~d = k1~a + k2~b + k3~c

tìte oi suntelestèc an�ptuxhc eÐnai

k1 =~d ·~b× ~c

~a ·~b× ~ck2 =

~d · ~c× ~a

~b · ~c× ~ak3 =

~d · ~a×~b

~c · ~a×~b

Orismìc: 'Ena akìma triplì ginìmeno eÐnai to ginìmeno

~a× (~b× ~c)

pou eÐnai èna di�nusma. Gia to di�nusma autì isqÔei h sqèsh

~a× (~b× ~c) = (~a · ~c)~b− (~a ·~b)~c (1.4)

EÐnai fanerì apì thn parap�nw sqèsh ìti to di�nusma autì brÐsketai p�nwsto epÐpedo twn dianusm�twn ~b kai ~c kai ìti

~a× (~b× ~c) 6= (~a×~b)× ~c

18

MporoÔme epÐshc na apodeÐxoume thn tautìthta 5

~a× (~b× ~c) +~b× (~c× ~a) + ~c× (~a×~b) = 0

gnwst  san tautìthta tou Giakìmpi.Parat rhsh: Oi parenjèseic tou triploÔ autoÔ ginomènou eÐnai a-

paraÐthtec. Pr�gmati gia par�deigma to di�nusma ~b× (~a×~a) eÐnai mhdèn en¸to di�nusma (~b× ~a)× ~a den eÐnai. Antijètwc mporoÔme na paraleÐyoume ticparenjèseic tou triploÔ ginomènou ~a · (~b×~c) giatÐ h èkfrash (~a ·~b)×~c denorÐzetai.

TrÐa dianÔsmata pou ikanopoioÔn thn sqèsh ~a ·~b× ~c 6= 0 apoteloÔn mÐab�sh tou q¸rou R3 diìti eÐnai trÐa sto pl joc kai grammik� anex�rthta.Pr�gmati h sqèsh

k1~a + k2~b + k3~c = ~0

eÐnai isodÔnamh me tic akìloujec treic isìthtec

k1a1 + k2b1 + k3c1 = 0 k1a2 + k2b2 + k3c2 = 0 k1a3 + k2b3 + k3c3 = 0

H lÔsh tou parap�nw sust matoc wc proc k1, k2, k3 eÐnai mÐa kai monadik an h orÐzousa twn suntelest¸n twn agn¸stwn eÐnai diaforetik  apì to mhdèn,dhlad  ∣∣∣∣∣∣

a1 b1 c1

a2 b2 c2

a3 b3 c3

∣∣∣∣∣∣= ~a ·~b× ~c 6= 0

H monadik  aut  lÔsh eÐnai h mhdenik . 'Ara

k1 = k2 = k3 = 0

Epomènwc ta trÐa dianÔsmata eÐnai grammik� anex�rthta.5Ορίζουμε πρώτα τα τρία διανύσματα βάζοντας και τους τρεις ορισμούς στην ίδια εντολή

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[52] := a = {a1, a2, a3}b = {b1, b2, b3}c = {c1, c2, c3}Out[52] = {a1, a2, a3}Out[53] = {b1, b2, b3}Out[54] = {c1, c2, c3}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Ακολούθως για την απόδειξη της ταυτότητας του Γιακόμπι εκτελούμε την εντολή

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[55] := Cross[a,Cross[b, c]] + Cross[b, Cross[c, a]] + Cross[c, Cross[a, b]]Out[55] = {0, 0, 0}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

19

In[82]:= << Graphics‘Arrow‘

Show@Graphics@8Line@880, 0<, 80, 3<<D,

Line@880, 0<, 83, 0<<D, Line@880, 0<, 8-1, -1<<D,

Arrow@80, 0<, 80, 1<, HeadScaling ® RelativeD,

Arrow@80, 0<, 8.6, 0<, HeadScaling ® RelativeD,

Arrow@80, 0<, 8-.5, -.5<, HeadScaling ® RelativeD,

Arrow@80, 0<, 81, 3<D, Arrow@80, 0<, 81, 1<D,

Arrow@80, 0<, 83, 1<D, Text@"e1", 8-0.1, 1.1<D,

Text@"e2", 8-.5, -.7<D, Text@"e3", 8.6, -0.15<D<DD;

e1

e2

e3

Sq ma 1.1: TrÐa dianÔsmata grammik� anex�rthta.

1.5 Ask seic

'Askhsh 1.Na apodeiqjeÐ ìti to exwterikì ginìmeno èqei thn epimeristik  idiìthta wc

proc thn prìsjesh. Dhlad  isqÔei h sqèsh

~u× (~v + ~w) = ~u× ~v + ~u× ~w

LÔsh: Gia thn apìdeixh tètoiwn ask sewn sun jwc brÐskoume analu-tik� thn èkfrash tou pr¸tou mèlouc kai met� tou deÔterou mèlouc kai blè-poume an eÐnai Ðsa. Ja broÔme tic sunist¸sec tou dianÔsmatoc ~u× (~v + ~w)tou pr¸tou mèlouc thc isìthtac. 'Eqoume

~u× (~v + ~w) = (u1, u2, u3)× (v1 + w1, v2 + w2, v3 + w3) =

=

∣∣∣∣∣∣

~e1 ~e2 ~e3

u1 u2 u3

v1 + w1 v2 + w2 v3 + w3

∣∣∣∣∣∣= (u2(v3 + w3)− u3(v2 + w2))~e1 +

(u3(v1 + w1)− u1(v3 + w3))~e2 + (u1(v2 + w2)− u2(v1 + w1))~e3 =

20

= (u2v3 + u2w3 − u3v2 − u3w2)~e1 + (u3v1 + u3w1 − u1v3 − u1w3)~e2 +

+ (u1v2 + u1w2 − u2v1 − u2w1)~e3 (1.5)

Ja broÔme akoloÔjwc tic sunist¸sec tou dianÔsmatoc ~u×~v +~u× ~w toudeÔterou mèlouc thc isìthtac. 'Eqoume

~u× ~v + ~u× ~w =

∣∣∣∣∣∣

~e1 ~e2 ~e3

u1 u2 u3

v1 v2 v3

∣∣∣∣∣∣+

∣∣∣∣∣∣

~e1 ~e2 ~e3

u1 u2 u3

w1 w2 w3

∣∣∣∣∣∣=

(u2v3 − u3v2)~e1 + (u3v1 − u1v3)~e2 + (u1v2 − u2v1)~e3 +

(u2w3 − u3w2)~e1 + (u3w1 − u1w3)~e2 + (u1w2 − u2w1)~e3 =

(u2v3 − u3v2 + u2w3 − u3w2)~e1 + (u3v1 − u1v3 + u3w1 − u1w3)~e2 +

(u1v2 − u2v1 + u1w2 − u2w1)~e3

Ta dÔo dianÔsmata eÐnai profan¸c Ðsa kai epomènwc h �skhsh apodeÐqjhke.Ja apodeÐxoume t¸ra thn �skhsh me thn bo jeia tou tanust  eijk. Apì

ton orismì tou exwterikoÔ ginomènou èqoume

~u× (~v + ~w) = eijkui(vj + wj)~ek = eijkuivj~ek + eijkuiwj~ek = ~u× ~v + ~u× ~w

kai h tautìthta apodeÐqjhke. 6

'Askhsh 2.Na apodeiqjoÔn oi qr simec tautìthtec

(~a×~b) · (~c× ~d) = (~a · ~c)(~b · ~d)− (~a · ~d)(~b · ~c)

(~a×~b) · (~c× ~d) + (~b× ~c) · (~a× ~d) + (~c× ~a) · (~b× ~d) = 0

LÔsh: Gia na apodeÐxoume thn sqèsh ja qrhsimopoi soume thn tautìth-ta

~b× (~c× ~d) = (~b · ~d)~c− (~b · ~c)~dPollaplasi�zoume thn sqèsh aut  eswterik� me to di�nusma ~a kai brÐsk-

oume~a ·~b× (~c× ~d) = (~a×~b) · (~c× ~d) = ~a · (~b · ~d)~c− ~a · (~b · ~c)~d

6Ο υπολογιστής απλοποιεί ακόμα περισσότερο την απόδειξη της άσκησης. Αφού ορίσουμεπρώτα τα τρία διανύσματα ~v, ~u και ~w εκτελούμε την εντολή῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[59] := Cross[u, v + w]− (Cross[u, v] + Cross[u,w])Out[59] = {0, 0, 0}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

21

Apì thn opoÐa èpetai profan¸c h proc apìdeixh. 7

'Askhsh 3.Na apodeiqjeÐ h anisìthta twn KwsÔ - Sbartc

|~a ·~b| ≤ ‖a‖‖b‖Pìte isqÔei h isìthta.

LÔsh: Upojètoume ìti èna toul�qiston apì ta dianÔsmata ~a kai ~beÐnai mh mhdenikì diìti �llwc h sqèsh eÐnai profan c. Upojètoume ìti

~b 6= ~0 =⇒ ‖~b‖ 6= 0

JewroÔme thn profan  anisìthta

‖~a + λ~b‖ ≥ 0

pou isqÔei gia k�je λ . 'Eqoume

‖~a + λ~b‖2 = (~a + λ~b) · (~a + λ~b) = ~a · ~a + λ~a ·~b + λ~b · ~a + λ2~b ·~b ≥ 0

AntikajistoÔme to λ me ton arijmì −~a ·~b/~b ·~b kai paÐrnoume

‖~a + λ~b‖2 = ~a · ~a− (~a ·~b)2

~b ·~b=‖~a‖2‖~b‖2 − (~a ·~b)2

‖~b‖2≥ 0 ⇐⇒

|~a ·~b| ≤ ‖a‖‖b‖ (1.6)

EÐnai profanèc ìti h anisìthta isqÔei me to Ðson an h arqik  anisìthtaisqÔei me to Ðson dhlad  an

‖~a + λ~b‖ = 0

7Στον υπολογιστή χρειάζεται επί πλέον η εντολή Simplify για να εκτελεστεί ηαπλοποίηση της σχέσης. Είναι δυνατόν η εντολή αυτή να μην αρκεί για να επιτύχουμε τηνεπιθυμητή απλοποίηση. Το Mathematica έχει πολλές άλλες εντολές για τον σκοπό αυτόπου μπορούμε να τις χρησιμοποιήσουμε μαζύ την μία μετά την άλλη.῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[66] := Simplify[Dot[Cross[a, b], Cross[c, d]]−Dot[a, c]Dot[b, d]+

Dot[a, d]Dot[b, c]]Out[66] = 0

In[16] := Simplify[Dot[Cross[a, b], Cross[c, d]]+Dot[Cross[b, c], Cross[a, d]] + Dot[Cross[c, a], Cross[b, d]]]

Out[16] = 0῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

Φυσικά έχουμε πρώτα ορίσει τα διανύσματα με τον γνωστό τρόπο.

22

H sqèsh ìmwc aut  sunep�getai ìti

~a + λ~b = 0

pou shmaÐnei ìti ta dianÔsmata eÐnai sugrammik�. 8

'Askhsh 4.DÔo b�seic ~e1, ~e2, ~e3 kai ~e ′1, ~e

′2, ~e

′3 lègontai amoibaÐec b�seic ìtan

ikanopoioÔn tic akìloujec ennèa sqèseic

~ei · ~e ′j = δij i, j = 1, 2, 3

Na apodeÐxete ìti ta dianÔsmata thc miac b�shc prokÔptoun apì ta dianÔsmatathc �llhc.

Apì tic parap�nw sqèseic gr�foume tic treic sqèseic pou perièqoun todi�nusma ~e ′1. Autèc eÐnai

~e1 · ~e ′1 = 1 ~e2 · ~e ′1 = 0 ~e3 · ~e ′1 = 0 (1.7)

Oi dÔo teleutaÐec shmaÐnoun ìti to di�nusma ~e ′1 eÐnai k�jeto kai proc todi�nusma ~e2 kai proc to di�nusma ~e3. 'Ara to di�nusma autì eÐnai par�llhloproc to di�nusma ~e2×~e3 kai epomènwc ta dÔo aut� dianÔsmata eÐnai an�loga.'Eqoume

~e ′1 = λ~e2 × ~e3

Gia na broÔme thn stajer� analogÐac antikajistoÔme thn sqèsh aut  sthnpr¸th twn (1.7) kai brÐskoume

~e1 · ~e ′1 = λ~e1 · ~e2 × ~e3 = 1 =⇒ ~e1 · ~e2 × ~e3 = 1/λ

Dhlad  h stajer� λ eÐnai to antÐstrofo tou miktoÔ ginomènou twn tri¸ndianusm�twn. Me kuklik  enallag  twn deikt¸n mporoÔme na gr�youme t¸rakai tic treic zhtoÔmenec sqèseic.

~e ′1 =~e2 × ~e3

~e1 · ~e2 × ~e3

~e ′2 =~e3 × ~e1

~e1 · ~e2 × ~e3

~e ′3 =~e1 × ~e2

~e1 · ~e2 × ~e3

8Για τα δύο συγκεκριμένα διανύσματα ~a = (2,−1, 1) και ~b = (2,−1, 1) για παράδειγμαο υπολογιστής εκτελεί την εντολή:῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[23] := Abs[Dot[a, b]] <= Norm[a]Norm[b]Out[23] = True

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟και μας απαντάει ότι η σχέση είναι σωστή.

23

SumbolÐzoume me µ to miktì ginìmeno twn tri¸n dianusm�twn thc tonoÔ-menhc b�shc. ApodeiknÔetai ìti o arijmìc autìc eÐnai to antÐstrofo tou λ.Pr�gmati

µ = ~e ′1 · ~e ′2 × ~e ′3 = ~e ′1 · (λ−1~e1) = 1/λ

Apì mia b�sh tou trisdi�statou q¸rou mporoÔme na kataskeu�soume mia�llh b�sh amoibaÐa proc thn pr¸th. Mia b�sh amoibaÐa proc ton eautìn thconom�zetai autoamoibaÐa b�sh. Gia thn b�sh aut  profan¸c isqÔei

µ = 1/µ =⇒ µ = ±1

An µ = 1 h b�sh aut  eÐnai dexiìstrofh �llwc h b�sh onom�zetai aris-terìstrofh. Mia tètoia dexiìstrofh b�sh eÐnai kai h orjokanonik  b�sh(~e1, ~e2, ~e3) pou qrhsimopoioÔme sun jwc gia ton q¸ro R3 .

'Askhsh 5.DÐnetai to di�nusma ~v = (5, 3,−1) kai ta trÐa basik� dianÔsmata ~e1 =

(1, 0, 0), ~e2 = (1, 1, 0) kai ~e3 = (1, 1, 1). Na brejoÔn oi suntelestèc an�p-tuxhc a, b kai c ¸ste

~v = a~e1 + b~e2 + c~e3

Na breÐte touc antÐstoiqouc suntelestèc ìtan ta trÐa basik� dianÔsmata eÐnaita ~e ′1 = (1, 2, 0), ~e ′2 = (1, 0, 1) kai ~e ′3 = (2, 2, 1).

LÔsh: Pollaplasi�zoume eswterik� thn sqèsh (1.7) diadoqik� me tadianÔsmata ~e1, ~e2 kai ~e3. BrÐskoume

~e1 · ~v = ~e1 · (a~e1 + b~e2 + c~e3) = a~e1 · ~e1 + b~e1 · ~e2 + c~e1 · ~e3

~e2 · ~v = ~e2 · (a~e1 + b~e2 + c~e3) = a~e2 · ~e1 + b~e2 · ~e2 + c~e2 · ~e3

~e3 · ~v = ~e3 · (a~e1 + b~e2 + c~e3) = a~e3 · ~e1 + b~e3 · ~e2 + c~e3 · ~e3

Met� apì aplèc pr�xeic oi parap�nw treic sqèseic gia thn pr¸th tri�da twndianusm�twn ~ei, i = 1, 2, 3 gÐnontai

5 = a + b + c

8 = a + 2b + 2c

7 = a + 2b + 3c

Oi parap�nw exis¸seic eÐnai èna algebrikì sÔsthma tri¸n grammik¸n ex-is¸sewn me treic �gnwstouc touc suntelestèc a, b kai c . An afairèsoumekat� mèlh tic dÔo teleutaÐec brÐskoume amèswc c = −1. To sÔsthma gÐnetai

6 = a + b

10 = a + 2b

−1 = c

24

AkoloÔjwc afairoÔme tic dÔo pr¸tec kai brÐskoume b = 4 kai �ra telik�apì thn pr¸th a = 2.

To sÔsthma mporeÐ na lujeÐ me thn gnwst  mèjodo twn orizous¸n. 9 HlÔsh eÐnai

a = 2 b = 4 c = −1

Gia thn deÔterh tri�da twn dianusm�twn e ′i parathroÔme ìti ta trÐadianÔsmata den eÐnai grammik� anex�rthta kai �ra to sÔnolo {~e ′1, ~e ′2, ~e ′3} deneÐnai b�sh tou q¸rou. Pr�gmati èqoume

~e ′1 + ~e ′2 − ~e ′3 = 0

Dhlad  isqÔei h isìthta c1~e′1 + c2~e

′2 + c3~e

′3 = 0 qwrÐc na isqÔoun oi sqèseic

c1 = c2 = c3 = 0 efìson eÐnai c1 = c2 = 1 kai c3 = −1. 10

'Askhsh 6.

Na apodeiqjoÔn o nìmoc tou sunhmitìnou kai o nìmoc twn hmitìnwn se ènatrÐgwno.

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

A

c..................................................................................................................................................................................................................................................................................................................................

B a...........................

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

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

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

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

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

C

b

Tuqìn trÐgwno.

c2 = a2 + b2 + 2a b cos (C)sin (A)

a=

sin (B)

b=

sin (C)

c

9Το πρόγραμμα Mathematica δίνει αμέσως την λύση του συστήματος. Βρίσκουμε῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[7] := Solve[{5 == a + b + c, 8 == a + 2b + 2c, 7 == a + 2b + 3c}, {a, b, c}]Out[7] = {a = 2, b = 4, c = −1}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Το σύμβολο του ίσον μέσα στις εντολές πρέπει να γράφεται διπλό.10Η λύση του συστήματος που δίνει ο υπολογιστής είναι: (έχουμε ορίσει πρώτα τα διανύσ-

ματα ~v και ~ei)῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Solve[v == c1 ∗ e1 + c2 ∗ e2 + c3 ∗ e3, {c1, c2, c3}]Out[27] = {}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Η απάντηση δηλώνει ότι το σύστημα δεν έχει λύση. Αντί για τους αστερίσκους για τονπολλαπλασιασμό μπορούμε να αφήσουμε ένα διάστημα μεταξύ των παραγόντων του γινομένου.

25

Apìdeixh: JewroÔme tic pleurèc tou trig¸nou san dianÔsmata prosana-tolismèna ìpwc sto sq ma. 'Eqoume

~a = ~b + ~c

a) Gia na apodeÐxoume ton nìmo twn sunhmitìnwn, pollaplasi�zoume thnparap�nw exÐswsh dÔo forèc kat� mèlh. BrÐskoume

~a · ~a = (~b + ~c) · (~b + ~c) = ~b ·~b + ~c · ~c + 2~b · ~c

a2 = b2 + c2 + 2bc cos (A)

b) Gia ton tÔpo tou hmitìnou pollaplasi�zoume thn sqèsh exwterik� meto di�nusma ~a kai met� me to di�nusma ~b. BrÐskoume

~a× ~a = ~b× ~a + ~c× ~a = 0 ~a×~b = ~b×~b + ~c×~b = ~c×~b

=⇒ ~a×~b = ~c×~b = ~c× ~a =⇒ ab sin (C) = cb sin (A) = ca sin (B)

Telik� diairoÔme me to ginìmeno abc kai brÐskoume ton tÔpo

sin (C)

c=

sin (A)

a=

sin (B)

b

26

Kef�laio 2

Dianusmatikèc sunart seic

2.1 Dianusmatikèc sunart seicUp�rqoun poll� megèjh sthn fusik  pou metab�llontai san sunart seick�poiwn paramètrwn. H taqÔthta enìc kinhtoÔ gia par�deigma eÐnai dunatìnna metab�lletai kat� thn di�rkeia tou qrìnou. H dÔnamh pou askeÐtai se ènas¸ma mèsa sto pedÐo thc barÔthtac metab�lletai an�loga me thn jèsh tou s¸-matoc. EÐnai dhlad  sun�rthsh tri¸n metablht¸n. Tètoia megèjh parist�non-tai apì dianusmatikèc sunart seic. 'Ena di�nusma pou oi sunist¸sec tou eÐnaisunart seic miac   perissotèrwn pragmatik¸n metablht¸n eÐnai mÐa dianus-matik  sun�rthsh.

EÐnai gnwstì ìti èna tuqaÐo shmeÐo M tou trisdi�stato q¸rou R3 peri-gr�fetai apì mÐa tri�da arijm¸n (x, y, z) pou eÐnai oi sunist¸sec tou dianÔs-matoc ~OM se k�poio orjog¸nio sÔsthma suntetagmènwn Oxyz, dhlad 

~OM = ~r = x~i + y~j + z~k

An oi sunist¸sec tou dianÔsmatoc exart¸ntai apì k�poia par�metro t tìteèqoume:

x = x(t), y = y(t), z = z(t)

kai se k�je tim  tou t antistoiqeÐ èna shmeÐo. To sÔnolo twn shmeÐwn aut¸napoteleÐ mÐa kampÔlh C. Oi parap�nw exis¸seic thc kampÔlhc onom�zon-tai parametrikèc exis¸seic kai h par�stash aut  thc kampÔlhc parametrik anapar�stash. H exÐswsh

~r(t) = x(t)~i + y(t)~j + z(t)~k

onom�zetai dianusmatik  anapar�stash thc kampÔlhc. To ~r(t) eÐnai miadianusmatik  sun�rthsh tou t ∈ R

27

ParametricPlot3D A96 Sin@2 tD, 4 Cos@2 tD,1����

2 t=, 8t, 0, 2 Pi< ,

Boxed ® FalseE;

-5

0

5 -4

-2

0

2

40123

-5

0

5

Sq ma 2.1: H elleiptik  èlika.

Gia par�deigma, an èna ulikì shmeÐo kineÐtai ston q¸ro tìte h jèsh toueÐnai sun�rthsh tou qrìnou kai to di�nusma pou dÐnei thn jèsh tou

~r(t) = (x(t), y(t), z(t)) = x(t)~i + y(t)~j + z(t)~k

eÐnai mia dianusmatik  sun�rthsh.H dianusmatik  aut  sun�rthsh parist�nei mia kampÔlh ston trisdi�stato

EukleÐdeio q¸ro, pou onom�zetai troqi� tou kinhtoÔ. H metablht  pragmatik par�metroc t parist�nei ton qrìno. Oi sunist¸sec sunart seic

x = x(t) y = y(t) z = z(t)

onom�zontai parametrikèc exis¸seic thc troqi�c.Gia par�deigma h sun�rthsh

~r(t) = (a sin (ωt), b cos (ωt), c t) = a sin (ωt)~i + a cos (ωt)~j + c t~k

sto q¸ro R3 parist�nei kampÔlh pou lègetai elleiptik  èlika (sq ma 2.1.).Oi parametrikèc exis¸seic thc èlikac eÐnai

x(t) = a sin (ωt) y(t) = b cos (ωt) z(t) = c t (2.1)

28

Mia dianusmatik  sun�rthsh eÐnai dunatìn na èqei pedÐo orismoÔ èna up-osÔnolo tou R3 dhlad  na eÐnai sun�rthsh tri¸n metablht¸n. Par�deigmah dÔnamh thc barÔthtac eÐnai sun�rthsh thc jèshc tou ulikoÔ shmeÐou

~F (x, y, z) = f1(x, y, z)~i + f2(x, y, z)~j + f3(x, y, z)~k

Oi pragmatikèc sunart seic fi(x, y, z), i = 1, 2, 3 onom�zontai sunist¸secsunart seic thc ~F .

Se perÐptwsh pou to pedÐo orismoÔ eÐnai èna uposÔnolo tou R2 tìtegr�foume:

~F (u, v) = f1(u, v)~i + f2(u, v)~j + f3(u, v)~k

Tèloc mporoÔn na oristoÔn dianusmatikèc sunart seic twn tri¸n sunte-tagmènwn tou q¸rou kai tou qrìnou

~F = ~F (x, y, x, t) = ~F (~r, t)

Par�deigma: An sumbolÐsoume me ~r to di�nusma jèshc enìc shmeÐou,tìte eÐnai dunatìn to di�nusma autì na eÐnai mia dianusmatik  sun�rthsh dÔopragmatik¸n metablht¸n.

~r(u, v) = (x(u, v), y(u, v), z(u, v)) u ∈ R. v ∈ R

H dianusmatik  aut  sun�rthsh parist�nei mia epif�neia ston trisdi�statoEukleÐdeio q¸ro. Oi sunist¸sec sunart seic

x = x(u, v) y = y(u, v) z = z(u, v)

onom�zontai parametrikèc exis¸seic thc epif�neiac (sq ma 2.2).An apì tic sqèseic autèc apaleÐyoume ta u, v tìte paÐrnoume thn akìlou-

jh exÐswshF (x, y, z) = 0

pou eÐnai ènac �lloc trìpoc orismoÔ miac epif�neiac. Mia sunarthsiak  sqèshtwn suntetagmènwn x, y, z orÐzei mia epif�neia ston q¸ro.

E�n oi exis¸seic x = x(u, v) kai y = y(u, v) mporoÔn na lujoÔn wcproc u, v dhlad  na èqoume

u = u(x, y) και v = v(x, y)

kai antikatastajoÔn sth exÐswsh z = z(x, y) tìte paÐrnoume:

z = z (u(x, y), v(x, y)) = z(x, y)

29

ParametricPlot3D A886 Cos@uD Cos@vD, 2 Cos@uD Sin@vD, 4 Sin@uD<,

82 - u - 6 v, u, v<<, 8u, 0, 2 Pi<, 9v, -

Pi�������

2,

Pi�������

2=,

ViewPoint ® 80, 1, 1<, Shading ® FalseE;

-10010

-20246

-4-2024

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Sq ma 2.2: H tom  enìc elleiyoeidoÔc kai enìc epipèdou eÐnai mÐa kampÔlh.

h opoÐa apoteleÐ �llh mÐa par�stash tou tm matoc epifaneÐac.'Allec parast�seic miac epif�neiac eÐnai kai oi akìloujec

x = x(y, z) y = y(z, x)

An to di�nusma jèshc ~r = ~r(x, y) enìc shmeÐou miac epifaneÐac eÐnai:

~r = x~i + y~j + f(x, y)~k

tìte h parap�nw epif�neia onom�zetai epif�neia apì peristrof  gÔrw apì ton�xona z.

2.2 To ìrio kai h sunèqeiaOrismìc: 'Estw mÐa dianusmatik  sun�rthsh orismènh se k�je shmeÐo enìcanoiktoÔ diast matoc pou perièqei to t0. H sun�rthsh den eÐnai aparaÐthtona eÐnai orismènh sto Ðdio to t0. Lème ìti to di�nusma ~L eÐnai to ìrio thc~F (t) kaj¸c to t teÐnei sto t0   to ~L eÐnai to ìrio thc ~F (t) an to m koctou dianÔsmatoc ~F (t)− ~L teÐnei sto mhdèn. Dhlad 

limt→t0

~F (t) = ~L ⇐⇒ limt→t0

‖~F (t)− ~L‖ = 0

To ìrio twn dianusm�twn eÐnai isodÔnamo me èna ìrio arijm¸n. To ìrio anup�rqei eÐnai monadikì.

30

In[98]:= Plot3DAã-x2�2-y2, 8x, -2, 2<, 8y, -2, 2<E;

-2

-1

0

1

2 -2

-1

0

1

2

0

0.25

0.5

0.75

1

-2

-1

0

1

Sq ma 2.3: H epif�neia z = e−12x2−y2 .

Je¸rhma: E�n ~F (t), ~G(t) eÐnai dianusmatikèc sunart seic kai f(t)

mÐa bajmwt  sun�rthsh kai up�rqoun ta ìria limt→t0~F (t), limt→t0

~G(t) kailimt→t0 f(t) tìte:

limt→t0

(~G(t)± ~G(t)

)= lim

t→t0

~F (t)± limt→t0

~G(t),

limt→t0

(~G(t) · ~G(t)

)= lim

t→t0

~F (t) · limt→t0

~G(t),

limt→t0

(~G(t)× ~G(t)

)= lim

t→t0

~F (t)× limt→t0

~G(t),

limt→t0

(f(t)~F (t)

)= lim

t→t0f(t) lim

t→t0

~F (t), (2.2)

Gia par�deigma apodeiknÔoume eÔkola 1 ìti to ìrio thc dianusmatik c1Το όριο μιας διανυσματικής συνάρτησης

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[18] := Limit[{Cos[a(x− x0)], Sin[2 a(x− x0)]/(x− x0), x}, x → x0]Out[18] = {1, 2 a, x0}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

31

sun�rthshc

~r(x) =

(cos (a(x− x0)),

Sin(2a(x− x0))

x− x0

, x

)

gia x → x0 eÐnai to di�nusma ~r(x0) = (1, 2a, x0).Orismìc: Mia dianusmatik  sun�rthsh ~F (t) eÐnai suneq c s�ena

shmeÐo t0 tou pedÐou orismoÔ thc e�n

limt→t0

~F (t) = ~F (t0)

Mia dianusmatik  sun�rthsh ~F (t) eÐnai suneq c s�ena shmeÐo t0 e�n kaimìnon e�n k�je sunist¸sa thc eÐnai suneq c sto shmeÐo autì.

An oi sunart seic f(t), ~F (t) kai ~G(t) eÐnai suneqeÐc tìte apì ticparap�nw idiìthtec twn orÐwn sunep�getai ìti kai oi sunart seic ~F (t)± ~G(t),~F (t) · ~G(t), ~F (t)× ~G(t) kai f(t) · ~F (t) eÐnai suneqeÐc.

Orismìc: 'Estw mÐa dianusmatik  sun�rthsh orismènh se k�je shmeÐo(x0, y0, z0) enìc anoiktoÔ uposunìlou pou perièqei to (x0, y0, z0) ektìc pijan¸csto Ðdio to (x0, y0, z0). Lème ìti to di�nusma ~L eÐnai to ìrio thc ~F (x, y, z)

kaj¸c to (x, y, z) teÐnei sto (x0, y0, z0)   to ~L eÐnai to ìrio thc ~F (x, y, z)

an to ìrio twn arijm¸n ‖~F (x, y, z)− ~L‖ eÐnai to mhdèn. Dhlad 

lim(x,y,z)→(x0,y0,z0)

~F (x, y, z) = ~L ⇐⇒ lim(x,y,z)→(x0,y0,z0)

‖~F (x, y, z)− ~L‖ = 0

To ìrio an up�rqei eÐnai monadikì.Orismìc: Mia dianusmatik  sun�rthsh ~F eÐnai suneq c s�ena shmeÐo

(x0, y0, z0) tou pedÐou orismoÔ thc e�n

lim(x,y,z)→(x0,y0,z0)

~F (x, y, z) = ~F (x0, y0, z0)

Profan¸c mia dianusmatik  sun�rthsh ~F (x, y, z) eÐnai suneq c s�ena shmeÐo(x0, y0, z0) e�n kai mìnon e�n k�je sunist¸sa thc eÐnai suneq c sto shmeÐoautì.

An�loga jewr mata me ekeÐna twn dianusmatik¸n sunart sewn miac prag-matik c metablht c isqÔoun kai sth prokeÐmenh perÐptwsh twn dianusmatik¸nsunart sewn tri¸n pragmatik¸n metablht¸n.

2.3 H par�gwgoc kai to olokl rwma

Orismìc: An ~F eÐnai mia dianusmatik  sun�rthsh kai t0 shmeÐo tou

32

pedÐou orismoÔ thc, up�rqei to parak�tw ìrio kai eÐnai peperasmèno,

limt→t0

~F (t)− ~F (t0)

t− t0

tìte to ìrio autì onom�zetai par�gwgoc thc ~F sto shmeÐo t0 kai gr�foume:

~F ′(t0) = limt→t0

~F (t)− ~F (t0)

t− t0

Lème tìte ìti h ~F èqei par�gwgo sto t0   ìti h ~F eÐnai paragwgÐsimh stot0   ìti h ~F ′(t) up�rqei.

'Omoia orÐzoume thn deÔterh par�gwgo thc dianusmatik c sun�rthshc ~F (t)

~F ′′(t) =d~F ′(t)

dt= lim

t→t0

~F ′(t)− ~F ′(t0)t− t0

H dianusmatik  sun�rthsh

~F (t) = f1(t)~i + f2(t)~j + f3(t)~k

eÐnai paragwgÐsimh sto t0 e�n kai mìnon e�n oi sunart seic f1, f2, f3 eÐnaiparagwgÐsimec sto t0. Tìte èqoume

~F ′(t) = f ′1(t)~i + f ′2(t)~j + f ′3(t)~k

Je¸rhma: 2 An ~F , ~G, f eÐnai paragwgÐsimec sunart seic sto t0tìte isqÔoun

(~F ± ~G

)′(t0) = ~F ′(t0)± ~G′(t0)

(f ~F

)′(t0) = f ′(t0)~F (t0) + f(t0)~F ′(t0)

(~F · ~G

)′(t0) = ~F ′(t0) · ~G(t0) + ~F (t0) · ~G′(t0)

(~F × ~G

)′(t0) = ~F ′(t0)× ~G(t0) + ~F (t0)× ~G′(t0) (2.3)

Par�deigma: An (x, y, z) eÐnai oi suntetagmènec enìc ulikoÔ shmeÐou,pou kineÐtai ston qrìno t dhlad  oi suntetagmènec eÐnai sunart seic touqrìnou t thc morf c x = x(t), y = y(t) kai z = z(t) tìte to di�nusmajèshc tou ulikoÔ shmeÐou eÐnai:

~r(t) = x(t)~i + y(t)~j + z(t)~k

33

H taqÔthta kai h epit�qunsh tou eÐnai:

~v(t) =d~r

dt=

dx(t)

dt~i +

dy(t)

dt~j +

dz(t)

dt~k

~a(t) =d~v

dt=

d2x(t)

dt2~i +

d2y(t)

dt2~j +

d2z(t)

dt2~k

o................................................................................................................................................................x

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

y....................................................................................................................................................................................................

z

.............................................................................................................................................................~r(t)

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

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

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

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

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

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

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

......

~r(t+dt)

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

d~r=~r(t+dt)−~r(t)

Sto ìrio dt → 0 to di�nusma d~r gÐnetai efaptomenikì thc troqi�c.

'Opwc faÐnetai kai apì to sq ma, h taqÔthta eÐnai èna di�nusma efap-tomenikì thc troqi�c tou kinhtoÔ. 2

An h dianusmatik  sun�rthsh exart�tai apì perissìterec thc miac metabl-htèc (x, y, z) dhlad  ~F = ~F (x, y, z) tìte eÐnai aparaÐthto na orÐsoume thnènnoia thc merik c parag¸gou.

Orismìc: Oi merikèc par�gwgoi miac dianusmatik c sun�rthshc orÐ-zontai apì tic sqèseic

∂ ~F

∂x= lim

δx→0

~F (x + δx, y, z)− ~F (x, y, z)

δx

∂ ~F

∂y= lim

δy→0

~F (x, y + δy, z)− ~F (x, y, z)

δy

∂ ~F

∂z= lim

δz→0

~F (x, y, z + δz)− ~F (x, y, z)

δz(2.4)

2Για παράδειγμα αν ένα υλικό σημείο κινείται στην ελλειπτική έλικα της σχέσης (2.1)~r(t) = (a sin (ωt), b cos (ωt), c t) τότε έχει την ακόλουθη ταχύτητα και επιτάχυνση:῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[101] := r = {a sin[ω t], b cos[ω t], c t}v = D[r, t]g = D[D[r, t], t]Out[101] = {a sin[ω t], b cos[ω t], c t}Out[102] = {a ω cos[ω t],−b ω sin[ω t], c}Out[103] = {−a ω2 sin[ω t],−b ω2 cos[ω t], 0}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

34

Oi merikèc par�gwgoi miac dianusmatik c sun�rthshc an�gontai stonupologismì twn merik¸n parag¸gwn twn sunistws¸n thc. Gia tic merikècparagwgÐseic qrhsimopoioÔme kai touc ex c sumbolismoÔc

∂

∂xf(~r) = ∂xf(~r) = fx(~r)

H parag¸gish sthn teleutaÐa èkfrash emfanÐzetai san deÐkthc kai denja prèpei na gÐnetai sÔgqush. MporoÔme epÐshc na orÐsoume kai tic merikècparag¸gouc an¸terhc t�xhc gia par�deigma an oi parak�tw sunart seic eÐnai<< kalèc >>, èqoume

∂2f(~r)

∂x2=

∂

∂x

(∂f(~r)

∂x

)= ∂2

xxf(~r) = fxx(~r) (2.5)

∂3f(~r)

∂y∂z2=

∂

∂y

(∂

∂z

(∂f(~r)

∂z

))= ∂3

yz2f(~r) = fyz2(~r) (2.6)

Gia par�deigma brÐskoume thn akìloujh par�gwgo 3

∂3‖~r‖∂x∂z2

=∂

∂x

(∂

∂z

(∂√

(x2 + y2 + z2)

∂z

))=

∂

∂x

(∂

∂z

z√x2 + y2 + z2

)=

∂

∂x

x2 + y2

(x2 + y2 + z2)3/2= −x(x2 + y2 − 2z2)

(x2 + y2 + z2)5/2

Orismìc: To olokl rwma miac dianusmatik c sun�nthshc orÐzetai apìto olokl rwma twn sunistws¸n thc dhlad  gia èna aplì olokl rwma isqÔei

∫~r(t)d t =

∫x(t)dt~i +

∫y(t)dt~j +

∫z(t)dt~k

Gia par�deigma upologÐzoume to orismèno olokl rwma thc elleiyoeidoÔcèlikac (2.1) wc proc t apì to t1 wc to t2. 4

3Ο υπολογισμός διαδοχικών παραγωγίσεων.῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[22] := Simplify[D[D[D[

√x2 + y2 + z2, z], z], x]

Out[22] = − x(x2+y2−2z2)(x2+y2+z2)5/2

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

4Ο υπολογισμός του ορισμένου ολοκληρώματος.῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[20] := Integrate[{a Cos[ω t], b Sin[ω t], c t}, {t, t1, t2}]Out[20] =

{ aω (−Sin[t1 ω] + Sin[t2 ω]), b

ω (Cos[t1 ω]− Cos[t2 ω]), 12c (−t12 + t22)}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

35

∫ t2

t1

~r(t)d t =

∫ t2

t1

(a cos (ωt), b sin (ωt), c t) d t =

(a

ω(sin (ωt2)− sin (ωt1)) ,

b

ω(cos (ωt1)− cos (ωt2])) ,

1

2c (t22 − t21)

)

Gia èna triplì olokl rwma gia to di�nusma ~F (x, y, z) = (F1, F2, F3)èqoume

∫∫∫~F (x, y, z)dxdydz =

~i

∫∫∫~F1dxdydz +~j

∫∫∫~F2dxdydz + ~k

∫∫∫~F3dxdydz (2.7)

Pollèc forèc gia ton stoiqei¸dh ìgko dxdydz qrhsimopoioÔme thn èk-frash d3~r.

Gia par�deigma brÐskoume to akìloujo olokl rwma∫ a

1

∫ z

1

∫ y

1

~r dxdydz =

∫ a

1

dz

∫ z

1

dy

∫ y

1

dx (x, y, z) =

∫ a

1

dz

∫ z

1

dy

(x2

2, xy, xz

)∣∣∣∣y

1

=

∫ a

1

dz

∫ z

1

dy

(y2 − 1

2, y(y − 1), z(y − 1)

)=

∫ a

1

dz

(y3

6− y

2,y3

3− y2

2,

z(y2

2− y)

)∣∣∣∣z

1

=

∫ a

1

dz

(z3 − 1

6− z − 1

2,z3 − 1

3− z2 − 1

2,

z(z2 − 1

2− (z − 1))

)=

∫ a

1

dz

(z3

6− z

2+

1

3,z3

3− z2

2+

1

6,z3

2− z2 +

z

2

)

=

(z4

24− z2

4+

z

3,z4

12− z3

6+

z

6,z4

8− z3

3+

z2

4

)∣∣∣∣a

1

=

(1

24(a− 1)3(a + 3),

1

12(a− 1)3(a + 1),

1

24(a− 1)3(3a + 1)

)

An nomÐzete ìti k�poio apì ta endi�mesa apotelèsmata eÐnai l�joc mhnanhsuqeÐte, to telikì apotèlesma eÐnai swstì diìti to dÐnei o upologist c. 5

5Βρίσκουμε῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[12] :=

Integrate[Integrate[Integrate[{x, y, z}, {x, 1, y}], {y, 1, z}], {z, 1, a}]Out[12] = { 1

24 (−1 + a)3(3 + a), 112 (−1 + a)3(1 + a), 1

24 (−1 + a)3(1 + 3a)}῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

36

2.4 Ta dianusmatik� kai ta bajmwt� pedÐaOrismèna fusik� megèjh eÐnai sunart seic, bajmwtèc   dianusmatikèc twntri¸n sunistws¸n tou dianÔsmatoc thc jèshc ~r = (x, y, z) kai endeqomènwcna perièqoun analutik� kai ton qrìno t.

Orismìc: OrÐzoume san dianusmatikì pedÐo mia perioq  tou q¸roupou se k�je shmeÐo thc antistoiqeÐ èna di�nusma. Gr�foume to di�nusma sansun�rthsh tou dianÔsmatoc jèshc kai tou qrìnou ~F (~r, t). An to di�nusmaparist�nei mÐa dÔnamh tìte to pedÐo onom�zetai dunamikì pedÐo.

Orismìc: Mia perioq  pou se k�je shmeÐo thc antistoiqeÐ mia bajmwt sun�rthsh V = V (~r, t) eÐnai èna bajmwtì pedÐo.

H taqÔthta pou èqei to k�je mìrio enìc reustoÔ pou kineÐtai se mia orismèn-h perioq  gia par�deigma eÐnai èna dianusmatikì pedÐo taqut twn. To dunamikìtou pedÐou barÔthtac eÐnai èna par�deigma enìc bajmwtoÔ pedÐou.

Gia èna dianusmatikì pedÐo   èna bajmwtì pedÐo gr�foume

~F (~r, t) = F1(~r, t)~i + F2(~r, t)~j + F3(~r, t)~k V = V (~r, t)

An upojèsoume ìti to di�nusma jèshc ~r eÐnai sun�rthsh thc metablht ct, to di�nusma ~r = ~r(t) diagr�fei mia kampÔlh ston q¸ro. To pedÐo èqei tìtethn morf .

V = V (~r(t), t) = V (~r(x(t), y(t), z(t)), t)

.H par�gwgoc enìc tètoiou pedÐou wc proc ton qrìno eÐnai

dV (~r, t)

d t=

dx(t)

d t

∂V

∂x+

dy(t)

d t

∂V

∂y+

dz(t)

d t

∂V

∂z+

∂V

∂ t

2.5 H klÐsh   apìklish kai o strobilismìcOrismìc: An f(~r) eÐnai mÐa sun�rthsh h opoÐa èqei pr¸tec merikèc

parag¸gouc tìte klÐsh thc f(~r) sto shmeÐo ~r0 onom�zetai to di�nusma

grad f(~r) = ~∇f(~r0) =∂

∂xf(~r0)~i +

∂

∂yf(~r0)~j +

∂

∂zf(~r0)~k (2.8)

Se k�je shmeÐo tou pedÐou orismoÔ thc sun�rthshc f(~r) mporoÔme naantistoiqÐsoume èna di�nusma to grad f(~r) opìte dhmiourgoÔme èna dianus-matikì pedÐo. To pedÐo autì lègetai pedÐo klÐsewn thc f(~r).

To sÔmbolo ~∇ onom�zetai an�delta kai prèpei na jewreÐtai san telest cpou metasqhmatÐzei èna bajmwtì pedÐo (mÐa sun�rthsh) se èna dianusmatikì

37

In[6]:= << Calculus‘VectorAnalysis‘SetCoordinates@Cartesian@x, y, zDD

Out[7]= Cartesian@x, y, zD

Sq ma 2.4: Entolèc gia thn eisagwg  upoprogr�mmatoc.

pedÐo to grad f(~r) pou onom�zetai klÐsh   b�jmwsh (gradient) tou bajmwtoÔpedÐou f(~r).

MporoÔme na orÐsoume ton telest  an�delta san èna di�nusma me sunist¸-sec touc diaforikoÔc telestèc. 'Ara h b�jmwsh enìc bajmwtoÔ pedÐou mporeÐna jewrhjeÐ san o exwterikìc pollaplasiasmìc tou dianÔsmatoc autoÔ me thnsun�rthsh f(~r).

~∇f(~r) =

(∂

∂x,

∂

∂y,

∂

∂z

)f(~r)

Prèpei ìmwc na prosèqoume kai na b�zoume touc diaforikoÔc telestèc mprost�apì tic sunart seic.

H b�jmwsh enìc pedÐou ikanopoieÐ tic sqèseic

~∇(cf) = c~∇f ~∇(f ± g) = ~∇f ± ~∇g

~∇(fg) = g~∇f + f ~∇g ~∇(

f

g

)=

g~∇f − f ~∇g

g2

Par�deigma: An f(x, y, z) =√

x2 + y2 + z2 breÐte thn klÐsh thcf(~r) sto shmeÐo

(2√

2, 2√

2,−3). 6

LÔsh: Apì ton orismì thc klÐshc èqoume:

~∇f(~r) = fx(~r)~i + fy(~r)~j + fz(~r)~k =

6Προκειμένου να χρησιμοποιήσουμε το Mathematica για τον υπολογισμό των εκ-φράσεων αυτών πρέπει να φορτώσουμε πρώτα ένα από τα έτοιμα πακέτα του προγράμματος.Εκτελούμε την εντολή << Calculus V ectorAnalysis και μετά το σύστημα συντεταγμένωνπου θα εργασθούμε σχήμα 2.4.

Το πρόγραμμα επιστρέφει για την επιβεβαίωση το σύστημα συντεταγμένων. Αυτό θαπαραμείνει εκτός αν αλλάξει με νεότερη εντολή. Μπορούμε τώρα να χρησιμοποιήσουμε τιςεντολές για την βάθμωση μιας συνάρτησης. Για το παράδειγμα αυτό βρίσκουμε῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[21] := ReplaceAll[Grad[

√x2 + y2 + z2], {x → 2

√2, y → 2

√2, z → −3}]

Out[21] = {2√

25 , 2

√2

5 ,− 35}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

38

x√x2 + y2 + z2

~i +y√

x2 + y2 + z2~j +

z√x2 + y2 + z2

~k

kai epomènwc sto shmeÐo pou mac dìjhke brÐskoume

~∇f(2√

2, 2√

2,−3)

=2√

2√25

~i +2√

2√25

~j +−3√25

~k =2√

2

5~i +

2√

2

5~j − 3

5~k

O telest c an�delta ~∇ ìtan dr�sei p�nw se k�poia bajmwt  sun�rthshorÐzei èna dianusmatikì pedÐo. E�n èna dianusmatikì pedÐo ~F (~r) eÐnai Ðso methn b�jmwsh k�poiac diaforÐsimhc sun�rthshc dhlad 

~F (~r) = −~∇V (~r)

tìte to pedÐo autì onom�zetai sunthrhtikì dianusmatikì pedÐo kai h bajmwt sun�rthsh V (~r) dunamik  sun�rthsh   dunamikì tou ~F (~r). Poll� dianus-matik� pedÐa poÔ sunant¸ntai sth fusik  eÐnai sunthrhtik� p.q. to hlektro-statikì pedÐo, to pedÐo barÔthtac k.l.p.

EÐnai gnwstì ìti mia exÐswsh thc morf c f(x, y, z) = 0 orÐzei mia epif�neiaS. An af soume ta (x, y, z) na exart¸ntai apì ton qrìno tìte h exÐswsh ~r =~r(t) eÐnai mia kampÔlh p�nw sthn epif�neia S. To diaforikì thc sun�rthshcf(~r) eÐnai profan¸c mhdèn. 'Ara èqoume

df =∂f

∂xdx +

∂f

∂ydy +

∂f

∂zdz = ~∇f · d~r = 0

'Omwc to di�nusma d~r eÐnai efaptomenikì thc kampÔlhc �ra to di�nusma ~∇f

eÐnai k�jeto sthn kampÔlh. Epeid  h kampÔlh eÐnai tuqoÔsa to di�nusma ~∇feÐnai telik� k�jeto sthn epif�neia f(~r) = 0.

Orismìc: Apìklish enìc dianusmatikoÔ pedÐou ~F (~r) = F1(~r)~i +

F2(~r)~j + F3(~r)~k orÐzetai h akìloujh bajmwt  sun�rthsh

div ~F = ~∇ · ~F =

(∂

∂x,

∂

∂y,

∂

∂z

)· (F1, F2, F3) =

∂F1

∂x+

∂F2

∂y+

∂F3

∂z

H apìklish enìc dianusmatikoÔ pedÐou faÐnetai san to eswterikì ginìmenotou dianÔsmatoc an�delta kai thc dianusmatik c sun�rthshc. E�n h apìklishenìc pedÐou eÐnai mhdèn, tìte to pedÐo onom�zetai swlhnoeidèc.

Parat rhsh: EÐnai fanerì ìti èna tètoio bajmwtì pedÐo up�rqei mìnoan up�rqoun oi merikèc par�gwgoi ∂F1

∂x,∂F2

∂y,∂F3

∂z. Ja lème ìti oi sunart seic

eÐnai << kalèc >> kai me ton ìro autì ja ennooÔme ìti ìlec oi parast�seic pougr�foume up�rqoun, qwrÐc na perigr�foume analutik� tic sunj kec Ôparxhc.

39

Par�deigma: DÐnetai to dianusmatikì pedÐo

~F =x~i + y~j + z~k

(x2 + y2 + z2)32

=~r

r3

Na breÐte thn apìklish tou pedÐou.LÔsh: Apì ton orismì thc sun�rthshc pou mac dìjhke èpetai ìti oi

sunist¸seic tou dianusmatikoÔ pedÐou eÐnai

F1 =x

(x2 + y2 + z2)32

F2 =y

(x2 + y2 + z2)32

F3 =x

(x2 + y2 + z2)32

Dhlad  eÐnai oi suntelestèc twn basik¸n dianusm�twn (~i, ~j, ~k) antistoÐqwc.Epomènwc brÐskoume

∂F1

∂x=

1

(x2 + y2 + z2)32

+ x∂

∂x

1

(x2 + y2 + z2)32

=1

(x2 + y2 + z2)32

+

x1

(x2 + y2 + z2)52

(−3

2)(2x) =

x2 + y2 + z2

(x2 + y2 + z2)52

+−3x2

(x2 + y2 + z2)52

=

x2 + y2 + z2 − 3x2

(x2 + y2 + z2)52

=y2 + z2 − 2x2

(x2 + y2 + z2)52

Oi �llec dÔo paragwgÐseic brÐskontai me ìmoio trìpo   mporoÔme na k�noumemia kuklik  enallag  sto telikì apotèlesma. BrÐskoume

∂F2

∂y=

z2 + x2 − 2y2

(x2 + y2 + z2)52

∂F3

∂z=

x2 + y2 − 2z2

(x2 + y2 + z2)52

Sunep¸c h apìklish tou dianusmatikoÔ pedÐou eÐnai

~∇ · ~F =∂F1

∂x+

∂F2

∂y+

∂F3

∂x=

y2+z2−2x2+z2+x2−2y2+x2+y2−2z2

(x2 + y2 + z2)52

= 0

H apìklish eÐnai mhdèn kai to pedÐo onom�zetai swlhnoeidèc. To dianusmatikìautì pedÐo pollaplasiasmèno me kat�llhlec stajerèc emfanÐzetai sthn fusik san h dÔnamh Koulìmp   san h dÔnamh tou pedÐou thc barÔthtac.

~∇ ·(

~r

r3

)= 0 (2.9)

Genikìtera jewroÔme thn sun�rthsh

~F (~r) =x~i + y~j + z~k

(x2 + y2 + z2)m=

~r

rm/2

40

ApodeiknÔoume me thn bo jeia tou upologist  7 ìti h apìklish tou pedÐouautoÔ eÐnai

~∇ · ~F (~r) = (3− 2m)(x2 + y2 + z2)−m

Epomènwc h apìklish tou pedÐou eÐnai mhdèn mìno gia thn tim  m = 3/2.H apìklish enìc dianusmatikoÔ pedÐou ikanopoieÐ tic sqèseic

~∇ · (~F + ~G) = ~∇ · ~F + ~∇ · ~G ~∇ · (V ~F ) = ~∇V · ~F + V ~∇ · ~F

O diaforikìc telest c

∆ = ~∇ · ~∇ =∂2

∂x2+

∂2

∂y2+

∂2

∂z2

lègetai telest c tou Lapl�c kai h posìthta

∇2f(~r) =∂2f

∂x2+

∂2f

∂y2+

∂2f

∂z2

Laplasian . H sun�rthsh f(~r) pou ikanopoieÐ thn exÐswsh ∇2f = 0 touLapl�c lègetai armonik .

ApodeiknÔetai ìti

~∇(

1

r

)= ~∇

(1√

x2 + y2 + z2

)= − ~r

r3

kai apì thn sqèsh (2.9) sumperaÐnoume ìti

~∇2

(1

r

)= 0

Dhlad  h bajmwt  sun�rthsh V (~r) = 1/r eÐnai lÔsh thc exÐswshc Lapl�c.H sun�rthsh aut  eÐnai to dunamikì tou dianusmatikoÔ pedÐou ~r/r3 kai hexÐswsh

~∇2V (~r) = 0 (2.10)

onom�zetai exÐswsh dunamikoÔ.Se poll� probl mata thc fusik c ta di�fora pedÐa pou emfanÐzontai èqoun

analutik  qronik  ex�rthsh kai stic diaforikèc exis¸seic emfanÐzontai kai7Η απόκλιση του διανυσματικού πεδίου.

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[61] := Simplify[Div[{x, y, z}/(x2 + y2 + z2)m]]Out[61] = (3− 2m)(x2 + y2 + z2)−m

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟φυσικά έχουμε εισάγει πρώτα το κατάλληλο υποπρόγραμμα.

41

oi par�gwgoi wc proc ton qrìno. Gia par�deigma gr�foume tic exis¸seicdi�qushc kai kÔmatoc

~∇2T (~r, t) =1

a2

∂

∂tT (~r, t) ~∇2U(~r, t) =

1

c2

∂2

∂t2U(~r, t) (2.11)

Oi parap�nw diaforikèc exis¸seic èqoun kai k�poiec oriakèc   arqikèc sun-j kec pou prèpei na ikanopoioÔn oi lÔseic touc.

Orismìc: E�n ~F = F1~i + F2

~j + F3~k eÐnai èna dianusmatikì pedÐo

tètoio ¸ste oi sunart seic Fj, j = 1, 2, 3 na eÐnai << kalèc >>, tìte orÐzoumesan strobilismì tou ~F kai ton sumbolÐzoume me Cur ~F   rot ~F   ~∇× ~Fto di�nusma:

curl ~F = rot ~F = ~∇× ~F =

∣∣∣∣∣∣

~i ~j ~k∂∂x

∂∂y

∂∂z

F1 F2 F3

∣∣∣∣∣∣= (2.12)

=

(∂

∂yF3 − ∂

∂zF2

)~i +

(∂

∂zF1 − ∂

∂xF3

)~j +

(∂

∂xF2 − ∂

∂yF1

)~k

An o strobilismìc enìc dianusmatikoÔ pedÐou eÐnai mhdèn to pedÐo onom�ze-tai astrìbilo.

Par�deigma: An èna stereì s¸ma peristrèfetai gÔro apì ton stajerì�xona Oz me stajer  gwniak  taqÔthta ~ω = ω~k tìte èna shmeÐo toudiagr�fei thn kampÔlh me di�nusma jèshc

~r = (x, y, z) = (ρ cos (ωt), ρ sin (ωt), z)

ja broÔme ton strobilismì thc taqÔthtac tou. BrÐskoume pr¸ta thn taqÔthta

~v = (x, y, z) = (−ρω sin (ωt), ρω cos (ωt), 0) = ω(−y, x, 0)

Oi parap�nw telÐtsec shmaÐnoun parag¸gish wc proc ton qrìno. 'Ara o stro-bilismìc thc taqÔthtac eÐnai

~∇× ~v =

∣∣∣∣∣∣

~i ~j ~k∂x ∂y ∂z

−ωy ωx 0

∣∣∣∣∣∣= 2ω~k = 2 ~ω

O strobilismìc dÐnei èna mètro thc peristrof c tou s¸matoc kai gi autìpollèc forèc onom�zetai kai peristrof .

O strobilismìc enìc dianusmatikoÔ pedÐou ikanopoieÐ tic idiìthtec:

~∇× (~F + ~G) = ~∇× ~F + ~∇× ~G ~∇× (V ~F ) = (~∇V )× ~F + V (~∇× ~F )

42

ApodeiknÔontai eÔkola oi sqèseic 8

~∇× (~∇V ) = 0 ~∇ · (~∇× ~F ) = 0 (2.13)~∇ · (~F × ~G) = ~G · (~∇× ~F )− ~F · (~∇× ~G) (2.14)

Parat rhsh: An gia èna dianusmatikì pedÐo up�rqei k�poia bajmwt sun�rthsh dunamikoÔ tètoia ¸ste ~F = −~∇φ tìte apì tic parap�nw idiìthtecèpetai ìti to pedÐo eÐnai astrìbilo. Dhlad  ~∇ × ~F = 0 . EpÐshc an gia topedÐo ~F up�rqei k�poio dianusmatikì dunamikì ~G tètoio ¸ste ~F = ~∇× ~G

tìte to pedÐo eÐnai swlhnoeidèc dhlad  ~∇ · ~F = 0.Sthn fusik  antimetwpÐzoume to eujÔ prìblhma dhlad  na brejeÐ to di-

anusmatikì pedÐo an dojeÐ to dunamikì, all� kai to antÐstrofo prìblhmadhlad  na brejeÐ to bajmwtì   to dianusmatikì dunamikì an dojeÐ to dianus-matikì pedÐo ~F . To hlektromagnhtikì pedÐo ( ~E, ~B) gia par�deigma èqeikai dianusmatikì dunamikì kai bajmwtì dunamikì pou to gr�foume se morf tetradianÔsmatoc (φ, ~A) = (φ,A1, A2, A3). To tetradi�nusma autì to sum-bolÐzoume sun jwc me to Aν ìpou o deÐkthc me ta Ellhnik� gr�mmata paÐrneitèsseric timèc ν = 0, 1, 2, 3 kai A0 = φ.

Ta bajmwt� dunamik� pou diafèroun kat� mÐa stajer� dÐnoun to Ðdio pedÐo~F . En¸ ta dianusmatik� dunamik� pou dÐnoun to Ðdio pedÐo eÐnai dunatìn nadiafèroun kat� mia olìklhrh sun�rthsh thc morf c ~∇χ(~r). Pr�gmati an ~A

kai ~A′ = ~A + ~∇χ dÔo tètoia dunamik� tìte

~F ′ = ~∇× ~A′ = ~∇× ( ~A + ~∇χ) = ~∇× ~A + ~∇× ~∇χ = ~∇× ~A = ~F

2.6 To epikampÔlio kai to epifaneiakìolokl rwma

Orismìc: DÐnetai mia kampÔlh c pou perigr�fetai apì to di�nusma jèshc~r = ~r(t), ìpou t ∈ [a, b] ⊂ R . OrÐzoume san epikampÔlio olokl rwma kai to

8Η απόδειξη της τελευταίας δίνεται με την βοήθεια του υπολογιστή μην ξεχάσετε ναεισάγετε πρώτα το κατάλληλο υποπρόγραμμα και τις καρτεσιανές συντεταγμένες και ναορίσετε τα διανυσματικά πεδία. Μεταφέρουμε όλους του όρους στο πρώτο μέλος και βρίσκ-ουμε῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[9] := Simplify[Div[Cross[f, g]]−Dot[g, Curl[f ]] + Dot[f, Curl[g]]]Out[9] = 0

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

43

sumbolÐzoume me∫

c~F · d~r to akìloujo aplì olokl rwma

I =

∫

c

~F · d~r =

∫ b

a

~F (~r(t)) · d~r

dtdt

Orismìc: DÐnetai mia epif�neia S pou perigr�fetai apì to di�nusmajèshc ~r = ~r(u, v), ìpou (u, v) ∈ [a, b] × [c, d] ⊂ R2 . OrÐzoume san epi-faneiakì olokl rwma kai to sumbolÐzoume me

∫∫S

~F · d~S to akìloujo diplìolokl rwma

I =

∫∫

S

~F · d~S =

∫ b

a

∫ d

c

~F (~r(u, v)) · ∂~r

∂u× ∂~r

∂ududv

EpikampÔlia kai epifaneiak� oloklhr¸mata orÐzontai kai gia bajmwtècsunart seic. OrÐzoume antÐstoiqa to epikampÔlio kai to epifaneiakì olok-l rwma

∫

c

V ds =

∫ b

a

V (~r(t))

∣∣∣∣d~r

dt

∣∣∣∣ dt

∫∫

S

V dS =

∫ b

a

∫ d

c

V (~r(u, v))

∣∣∣∣∂~r

∂u× ∂~r

∂u

∣∣∣∣ dudv (2.15)

gia tic bajmwtèc sunart seic V (~r(t)) kai V (~r(u, v)).Gia V = 1 oi parap�nw tÔpoi dÐnoun to m koc thc kampÔlhc kai to embadìn

thc epif�neiac antistoÐqwc.Gia ta oloklhr¸mata aut� isqÔoun ta akìlouja oloklhrwtik� jewr mata

thc apìklishc kai tou strobilismoÔ.∫∫∫

V

~∇ · ~A d 3~r = O

∫∫

S1

~A · d~S

ìpou S1 eÐnai h kleist  epif�neia pou perikleÐei ton ìgko V . SumbolÐzoumepollèc forèc thn kleist  aut  epifaneia me ∂V , dhlad  S1 = ∂V .

∫∫

S

~∇× ~A · d ~S = O

∫

c

~A · d~r

ìpou c eÐnai h kleist  kampÔlh, to sÔnoro thc epif�neiac S. Gr�foume pol-lèc forèc c = ∂S. Oi sunart seic èqoun suneqeÐc pr¸tec merikèc parag¸gouckai oi epif�neiec, eÐnai << kalèc >> dhlad  den èqoun anwmalÐec.

44

2.7 Kampulìgrammec suntetagmènecJewroÔme tic akìloujec sunarthsiakèc sqèseic pou sundèoun tic kartesianècsuntetagmènec (x, y, z) me treic �llec metablhtèc (u1, u2, u3).

x = x(u1, u2, u3) y = y(u1, u2, u3) z = z(u1, u2, u3) (2.16)

An oi sunart seic autèc eÐnai << kalèc >> ètsi ¸ste na mporoÔn na lujoÔn wcproc (u1, u2, u3) dhlad 

u1 = u1(x, y, z) u2 = u2(x, y, z) u3 = u3(x, y, z)

tìte oi sqèseic autèc orÐzoun ènan metasqhmatismì suntetagmènwn.'Ena shmeÐo P tou q¸rou orÐzetai apì tic treic kartesianèc suntetagmènec

(x, y, z) all� kai apì tic treic kampulìgrammec suntetagmènec (u1, u2, u3)pou sundèontai me tic parap�nw sunarthsiakèc sqèseic. To di�nusma jèshctou shmeÐo P gr�fetai

~r = ~r(u1, u2, u3) = x(u1, u2, u3)~i + y(u1, u2, u3)~j + z(u1, u2, u3)~k (2.17)

Oi epif�neiec

u1 = c1 u2 = c2 u3 = c3

orÐzoun treic epif�neiec pou onom�zontai suntetagmènec epif�neiec. Stickartesianèc suntetagmènec gia par�deigma h epif�neia x = c1 eÐnai to epÐpedoyOz. Oi kampÔlec pou eÐnai tomèc aut¸n twn epifanei¸n an� dÔo onom�zontaisuntetagmènec kampÔlec. Gia par�deigma h kampÔlh pou eÐnai h tom  twn dÔoepifanei¸n x = c1 kai y = c2 eÐnai o �xonac Oz. SumbolÐzoume me

(~e1, ~e2, ~e2)

ta monadiaÐa efaptomenik� dianÔsmata stic kampÔlec autèc. An ta dianÔsmataaut� tèmnontai k�jeta tìte to sÔsthma twn suntetagmènwn eÐnai orjog¸nio.

An krat soume stajerèc tic metablhtèc u2 kai u3 kai af soume thn u1

na paÐrnei ìlec tic epitrepìmenec timèc tìte èqoume thn suntetagmènh kampÔlh~r = ~r(u1, u2 = c2, u3 = c3). To di�nusma

∂~r

∂u1

=

(∂x

∂u1

,∂y

∂u1

,∂z

∂u1

)

eÐnai wc gnwstìn efaptomenikì thc kampÔlhc aut c. SumbolÐzoume me h1 tomètro autoÔ tou dianÔsmatoc. AkoloujoÔme thn Ðdia diadikasÐa kai gia thc�llec dÔo suntetagmènec kampÔlec kai orÐzoume tic posìthtec

h1 = ‖ ∂~r

∂u1

‖ h2 = ‖ ∂~r

∂u2

‖ h3 = ‖ ∂~r

∂u3

‖

45

'Ara mporoÔme na gr�youme tic sqèseic

∂~r

∂u1

= h1~e1∂~r

∂u2

= h2~e2∂~r

∂u3

= h3~e3

PaÐrnoume to diaforikì tou dianÔsmatoc jèshc ~r kai brÐskoume

d~r =∂~r

∂u1

du1 +∂~r

∂u2

du2 +∂~r

∂u3

du3 = h1du1~e1 + h2du2~e2 + h3du3~e3

Sthn perÐptwsh twn orjogwnÐwn suntetagmènwn, to stoiqei¸dec m koc dÐnetaiapì tic sqèseic

ds = d~r · d~r = dx2 + dy2 + dz2 = h21du2

1 + h22du2

2 + h23du2

3

'Eqoume deqjeÐ ìti ta dianÔsmata (~e1, ~e2, ~e3) tèmnontai k�jeta kai epomènwc

~ej · ~ek = δjk j, k = 1, 2, 3

H apìstash dÔo geitonik¸n shmeÐwn dÐnetai genikìtera apì ton tÔpo

ds =3∑

j,k

hjkdujduk

Oi suntelestèc hjk exart¸ntai apì tic suntetagmènec uj. KajorÐzounthn fÔsh twn kampulìgrammwn suntetagmènwn kai dÐnoun thn metrik  touc.Gia thn EukleÐdeia metrik  twn kartesian¸n suntetagmènwn gia par�deigma oimetrikoÐ autoÐ suntelestèc eÐnai hjk = δjk.

O stoiqei¸dhc ìgkoc dÐnetai apì thn sqèsh

dV = dxdydz = (h1du1~e1) · (h2du2~e2)× (h3du3~e3) = h1h2h3du1du2du3 =⇒

dxdydz =∂~r

∂u1

· ∂~r

∂u2

× ∂~r

∂u3

du1du2du3

H par�stash aut  onom�zetai Iakwbian  tou metasqhmatismoÔ kai sumbolÐze-tai me

J =∂(x, y, z)

∂(u1, u2, u3)=

∂~r

∂u1

· ∂~r

∂u2

× ∂~r

∂u3

> 0

H Iakwbian  tou metasqhmatismoÔ prèpei na mhn eÐnai mhdèn ¸ste oi sunart -seic (exis¸seic 2.16) na eÐnai anex�rthtec. Deqìmaste ìti eÐnai jetik  alli¸call�zoume metaxÔ touc p.q. tic u1 kai u2.

Ja gr�youme t¸ra touc diaforikoÔc telestèc pou orÐsame se autèc ticorjog¸niec kampulìgrammec suntetagmènec. Upojètoume ìti h bajmwt  φ(~r)

46

kai h dianusmatik  sun�rthsh ~A(~r) eÐnai << kalèc >> dhlad  ìlec oi parast�seicpou emfanÐzontai stic sqèseic up�rqoun. Oi tÔpoi eÐnai

~∇φ =1

h1

∂φ

∂u1

~e1 +1

h2

∂φ

∂u2

~e2 +1

h3

∂φ

∂u3

~e3

~∇ · ~A =1

h1h2h3

[∂

∂u1

(h2h3A1) +∂

∂u2

(h3h1A2) +∂

∂u3

(h1h2A3)

]

~∇× ~A =1

h1h2h3

∣∣∣∣∣∣

h1~e1 h2~e2 h3~e3

∂u1 ∂u2 ∂u3

h1A1 h2A2 h3A3

∣∣∣∣∣∣

~∇2φ =1

h1h2h3

[∂

∂u1

(h2h3

h1

∂φ

∂u1

)+

∂

∂u2

(h3h1

h2

∂φ

∂u2

)+

∂

∂u3

(h1h2

h3

∂φ

∂u3

)]

H an�ptuxh thc parap�nw orÐzousec gÐnetai wc proc ta stoiqeÐa thc pr¸thcgramm c kai sthn an�ptuxh twn upoorizous¸n prosèqoume ¸ste oi diaforikoÐtelestèc na prohgoÔntai twn sunart sewn. Mhn xeqn�te ìti den prìkeitaigia kanonik  orÐzousa all� m�llon gia ènan mnhmonikì kanìna.

Orismìc: OrÐzoume tic kulindrikèc suntetagmènec (ρ, φ, z) apì ticparak�tw sqèseic

x = ρ cos φ y = ρ sin φ z = z

Oi metablhtèc autèc paÐrnoun tic akìloujec timèc

0 ≤ ρ < ∞ 0 ≤ φ < 2π ∞ < z < ∞

o ................................................................................................................................................................

x

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

y.............................................................................................................................................................................................................................z

....................................................................................................................................... ~r

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

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

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

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

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

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

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

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

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

φ

ρ

ρ

z

r

x

y

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

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

...........

Oi kulindrikèc suntetagmènec.

An lÔsoume tic exis¸seic autèc wc proc tic kulindrikèc suntetagmènecbrÐskoume

ρ = +√

x2 + y2 φ = arctan (y

x) z = z

47

To di�nusma jèshc enìc shmeÐou tou q¸rou eÐnai

~r = x~i + y~j + z~k = ρ cos φ~i + ρ sin φ~j + z~k

Oi suntetagmènec epif�neiec eÐnai oi ex cρ =

√x2 + y2 = c1. Par�pleurec epif�neiec kulÐndrwn me �xona par�llh-

lo proc ton �xona Oz se apìstash ρ = c1.φ = arctan (y

z) = c2 ⇒ y

z= tan (c2) = c ′2. EpÐpeda pou perièqoun ton

�xona Oz kai sqhmatÐzoun me ton �xona Ox gwnÐa Ðsh me φ = c2.z = c3. EpÐpeda par�llhla sto epÐpedo xOy k�jeta ston �xona Oz

sto shmeÐo z = c3.Oi suntetagmènec kampÔlec eÐnai oi ex c

H tom  twn dÔo pr¸twn epifanei¸n. EujeÐec par�llhlec ston �xona Oz.H tom  twn dÔo teleutaÐwn epifanei¸n. EujeÐec k�jetec ston �xona Oz.H tom  thc pr¸thc kai thc teleutaÐac. Perifèreiec kÔklwn se epÐped�

par�llhla sto xOy me kèntro ston �xona Oz.UpologÐzoume akoloÔjwc touc metrikoÔc suntelestèc. BrÐskoume

h1 = ‖ ∂

∂ρ(ρ cos φ~i + ρ sin φ~j + z~k)‖ =

‖ cos φ~i + sin φ~j‖ =

√cos2 φ + sin2 φ = 1

h2 = ‖ ∂

∂φ(ρ cos φ~i + ρ sin φ~j + z~k)‖ =

‖ − sin φ~i + cos φ~j‖ =

√ρ2 cos2 φ + ρ2 sin2 φ = ρ

h3 = ‖ ∂

∂z(ρ cos φ~i + ρ sin φ~j + z~k)‖ = ‖~k‖ = 1

SumbolÐzoume ta basik� dianÔsmata twn kulindrik¸n suntetagmènwn me

(~e1, ~e2, ~e3) = (~ρ0, ~φ0, ~k)

Gr�foume tic sqèseic pou sundèoun ta basik� aut� dianÔsmata me ta basik�kartesian� dianÔsmata se morf  pin�kwn

~ρ0~φ0~k

=

cos φ sin φ 0− sin φ cos φ 0

0 0 1

~i~j~k

O parap�nw pÐnakac metasqhmatismoÔ pou sumbolÐzetai me O(φ) eÐnai or-jog¸nioc kai èqei orÐzousa Ðsh me thn mon�da, dhlad 

O(φ)O(φ)t = 11 |O(φ)| = 1

48

EpÐshc parathroÔme ìti isqÔoun oi sqèseic

O(φ1)O(φ2) = O(φ1 + φ2) = O(φ2 + φ1) = O(φ2)O(φ1)

Gr�foume t¸ra touc diaforikoÔc telestèc pou orÐsame se kulindrikèc sun-tetagmènec

~∇ψ(ρ, φ, z) = ~ρ0∂ψ

∂ρ+ ~φ0

1

ρ

∂ψ

∂φ+ ~k

∂ψ

∂z

~∇ · ~A =1

ρ

∂

∂ρ(ρA1) +

1

ρ

∂A2

∂φ+

∂A3

∂z

~∇× ~A =1

ρ

∣∣∣∣∣∣~ρ0 ρ~φ0

~k∂ρ ∂φ ∂z

A1 ρA2 A3

∣∣∣∣∣∣

~∇2ψ =1

ρ

∂

∂ρ

(ρ∂ψ

∂ρ

)+

1

ρ2

∂2ψ

∂φ2+

∂2ψ

∂z2

Stouc parap�nw tÔpouc to di�nusma dÐnetai se kulindrikèc suntetagmènec~A = A1~ρ0 + A2

~φ0 + A3~k. H Iakobian  tou metasqhmatismoÔ twn kulindrik¸n

suntetagmènwn eÐnai J = ρ.Orismìc: OrÐzoume tic sfairikèc suntetagmènec (r, θ, φ) apì tic

parak�tw sqèseic

x = r sin θ cos φ x = r sin θ sin φ z = r cos θ

Oi metablhtèc autèc paÐrnoun tic akìloujec timèc

0 ≤ r < ∞ 0 ≤ φ < 2π 0 ≤ θ ≤ π

o ................................................................................................................................................................

x

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

y.............................................................................................................................................................................................................................z

..................................................................................................................................................~r ...................................................................................~r0

................................................................................... ..........~θ0

............................................................. .......... ~φ0

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

......................................... θ

........................................φ

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

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

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

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

r

........

........

........

........

........

........

........

........

........

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

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

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

Oi sfairikèc suntetagmènec.

49

Oi antÐstrofec lÔseic eÐnai

r = (x2+y2+z2)1/2 θ = arccos

(z

(x2 + y2 + z2)1/2

)φ = arc tan

( y

x

)

H Iakobian  tou metasqhmatismoÔ twn sfairik¸n suntetagmènwn eÐnai

J = r2 sin θ

O telest c Lapl�c eÐnai

~∇2a =1

r2 sin θ

[sin θ

∂

∂r

(r2∂a

∂r

)+

∂

∂θ

(sin θ

∂a

∂θ

)+

1

sin θ

∂2a

∂φ2

]

An mia sun�rthsh exart�tai mìno apì to r tìte o telest c Lapl�c dÐnei

~∇2a(r) =1

r2

∂

∂r

(r2∂a(r)

∂r

)=

2

r

∂a(r)

∂r+

∂2a(r)

∂r2

BrÐskoume thn lÔsh thc exÐswshc Lapl�c 9

~∇2a(r) = 0 =⇒ ∂

∂r

(r2∂a(r)

∂r

)= 0 =⇒ r2∂a(r)

∂r= C1 =⇒

∂a(r)

∂r=

C1

r2=⇒ a(r) = −C1

r+ C2

Gia par�deigma èna eleÔjero swm�tio perigr�fetai sÔmfwna me thn kban-tomhqanik  apì thn kumatosun�rthsh ψ(~r) pou ikanopoieÐ thn akìloujhdiaforik  exÐswsh

∂2ψ

∂x2+

∂2ψ

∂y2+

∂2ψ

∂z2+ k2ψ = 0 k =

√2mE

~2

An to swm�tio eÐnai egklwbismèno mèsa se èna orjog¸nio koutÐ tìte h ku-matosun�rthsh prèpei na mhdenÐzetai stic epif�neiec tou orjogwnÐou pou eÐnai

9Μπορούμε να ρωτήσουμε τον υπολογιστή να μας πει την Λαπλασιανή σε σφαιρικές συν-τεταγμένες όσο αναλυτικά γίνεται σχήμα (2.5).

Για να λύσουμε κατευθείαν την διαφορική εξίσωση του Λαπλάς γράφουμε῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[21] := DSolve[Div[Grad[a[r]]] == 0, a[r], r]Out[21] = {{a[r] → −C[1]

r + C[2]}}῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

και βρίσκουμε φυσικά το ίδιο αποτέλεσμα.

50

In[3]:= << Calculus‘VectorAnalysis‘

SetCoordinates@Spherical@r, u, fDD;

Expand@Div@Grad@a@r, u, fDDDD

Csc@uD2 aH0,0,2L@r, u, fD�����������������������������������������������������������������������

r2+

Cot@uD aH0,1,0L@r, u, fD��������������������������������������������������������������������

r2+

aH0,2,0L@r, u, fD�������������������������������������������������

r2+

2 aH1,0,0L@r, u, fD�����������������������������������������������������

r+ aH2,0,0L@r, u, fD

Sq ma 2.5: H Laplasian  se sfairikèc suntetagmènec.

epÐpeda. Autèc eÐnai oi oriakèc sunj kec kai to prìblhma lÔnetai se karte-sianèc suntetagmènec. An ìmwc h kumatosun�rthsh prèpei na mhdenÐzetai sthnepif�neia miac sfaÐrac tìte epib�lletai na ergasjoÔme se sfairikèc suntetag-mènec. H exÐswsh dèqetai mia lÔsh thc morf c ginomènou

ψ(~r) = R(r)Υ(θ, φ)

Oi sunart seic Υ(θ, φ) onom�zontai sfairikèc armonikèc. H lÔsh touprobl matoc èqei p�li thn parap�nw morf  akìma kai an k2 = f(r) + λ2.H antÐstoiqh exÐswsh eÐnai h exÐswsh tou Srèntigker gia to �tomo tou u-drogìnou.

In[53]:= Table@SphericalHarmonicY@l, m, Θ, ΦD, 8l, 8, 9<, 8m, 5, 7<D

99- 3��������64ã5 ä Φ$%%%%%%%%%%%%%%%%17017

�����������������2 Π

Cos@ΘD I-1 + 5 Cos@ΘD2M Sin@ΘD5,1�����������128

ã6 ä Φ$%%%%%%%%%%%%%%7293

��������������Π

I-1 + 15 Cos@ΘD2M Sin@ΘD6,

-3��������64ã7 ä Φ$%%%%%%%%%%%%%%%%12155

�����������������2 Π

Cos@ΘD Sin@ΘD7=,

9- 3�����������256

ã5 ä Φ$%%%%%%%%%%%%%%2717

��������������Π

I1 - 30 Cos@ΘD2 + 85 Cos@ΘD4M Sin@ΘD5,1�����������128

ã6 ä Φ$%%%%%%%%%%%%%%%%40755

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Cos@ΘD I-3 + 17 Cos@ΘD2M Sin@ΘD6,

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I-1 + 17 Cos@ΘD2M Sin@ΘD7==

Sq ma 2.6: Oi sfairikèc armonikèc gia l = 8, 9 kai m = 5, 6, 7.

51

2.8 Ask seic

'Askhsh 1.

Oi ent�seic tou hlektrikoÔ ~E kai tou magnhtikoÔ pedÐou ~H ikanopoioÔntic akìloujec tèsseric exis¸seic tou M�xgouel.

~∇ · ~E = 0 ~∇ · ~H = 0 ~∇× ~E = −1

c

∂ ~H

∂ t~∇× ~H =

1

c

∂ ~E

∂ t(2.18)

Na apodeÐxete ìti ikanopoioÔn thn exÐswsh tou kÔmatoc

~∇2 ~E − 1

c2

∂2

∂ t2~E = ~0 ~∇2 ~H − 1

c2

∂2

∂ t2~H = ~0

Dhlad  ìlec oi sunist¸sec tou hlektrikoÔ kai tou magnhtikoÔ pedÐouikanopoioÔn exis¸seic kÔmatoc. IkanopoioÔn epÐshc kai thn exÐswsh.

1

c

∂

∂ t

(1

2(E2 + H2)

)+ ~∇ · ( ~E × ~H) = 0

To di�nusma ~E × ~H onom�zetai di�nusma tou PìuntingkApìdeixh: PaÐrnoume ton strobilismì thc trÐthc sqèshc kai lìgw kai thctètarthc brÐskoume

~∇× (~∇× ~E) = −~∇× 1

c

∂ ~H

∂ t= −1

c

∂

∂ t~∇× ~H = − 1

c2

∂2 ~E

∂ t2

To pr¸to mèloc ìmwc thc parap�nw exÐswshc eÐnai apì thn exÐswsh (2.21)

~∇× (~∇× ~E) = ~∇(~∇ · ~E)− ~∇2 ~E = ~∇(~∇ · ~E)− ~∇2 ~E = −~∇2 ~E

diìti h apìklish tou hlektrikoÔ pedÐou eÐnai mhdèn apì thn pr¸th twn ex-is¸sewn tou M�xgouel. Apì tic dÔo parap�nw exis¸seic èpetai profan¸c hzhtoÔmenh. H apìdeixh gia to magnhtikì pedÐo eÐnai ìmoia.

Gia na apodeÐxoume thn deÔterh sqèsh ja qrhsimopoi soume thn tautìthta(2.14). Lìgw kai twn dÔo teleutaÐwn exis¸sewn tou M�xgouel brÐskoume

1

c

∂

∂ t

1

2(E2+H2)+ ~∇·( ~E× ~H) =

1

c

∂

∂ t

1

2(E2+H2)+ ~H ·(~∇× ~E)− ~E ·(~∇× ~H) =

1

c

∂

∂ t

1

2(E2 + H2) + ~H · (−1

c

∂ ~H

∂ t)− ~E · (1

c

∂ ~E

∂ t) = 0

52

H parap�nw sqèsh eÐnai mhdèn diìti isqÔei gia k�je di�nusma ~A

A2 = ~A · ~A =⇒ ∂

∂ tA2 =

(∂

∂ t~A

)· ~A + ~A · ∂

∂ t~A = 2 ~A · ∂

∂ t~A

'Askhsh 2.

Na apodeiqteÐ ìti k�je swlhnoeidèc di�nusma ~F mporeÐ na grafeÐ me thnmorf 

~F = ~∇× ~G (2.19)

Sth fusik  kai idiaÐtera ston hlektromagnhtismì emfanÐzontai exis¸seic tè-toiou eÐdouc kai to di�nusma ~G onom�zetai dianusmatikì dunamikì.

LÔsh: To di�nusma ~F pou dÐnetai apì thn parap�nw sqèsh eÐnaiprofan¸c èna swlhnoeidèc di�nusma. pr�gmati

~∇ · ~F = ~∇ · ~∇× ~G = 0

SumbolÐzoume me Fi kai me Gi tic sunist¸sec twn dianusm�twn ~F kai~G antistoÐqwc. Epeid  to di�nusma ~F eÐnai swlhnoeidèc isqÔei h sqèsh

~∇ · ~F = 0 =⇒ ∂

∂xF1 +

∂

∂yF2 +

∂

∂zF3 = 0 (2.20)

H sqèsh (2.19) gr�fetai analutik�

∂

∂yG3 − ∂

∂zG2 = F1

∂

∂zG1 − ∂

∂xG3 = F2

∂

∂xG2 − ∂

∂yG1 = F3

Dhlad  to prìblhma eÐnai na lujeÐ to parap�nw sÔsthma twn diaforik¸nsunart sewn. Zht�me sthn arq  mia merik  lÔsh G0 = (A,B, 0) tètoia ¸ste

~F = ~∇× ~G0

AnalÔoume thn parap�nw exÐswsh kai paÐrnoume tic exis¸seic

− ∂

∂zB = F1

∂

∂zA = F2

∂

∂xB − ∂

∂yA = F3

MporoÔme na oloklhr¸soume amèswc tic dÔo pr¸tec exis¸seic. To apotè-lesma eÐnai

A =

∫ z

z0

F2(x, y, t)d t + a(x, y) B = −∫ z

z0

F1(x, y, t)d t + b(x, y)

53

ìpou a kai b eÐnai aujaÐretec sunart seic twn metablht¸n x kai y.Epeid  zht�me mia merik  lÔsh kai èqoume mÐa akìma exÐswsh mporoÔme na

mhdenÐsoume mia apì tic dÔo autèc aujaÐretec sunart seic. Upojètoume ìtia(x, y) = 0 . AntikajistoÔme tic sunart seic A kai B sthn trÐth exÐswshtou diaforikoÔ sust matoc. BrÐskoume

∂

∂x

(−

∫ z

z0

F1(x, y, t)d t + b(x, y)

)− ∂

∂y

(∫ z

z0

F2(x, y, t)d t

)=

= −∫ z

z0

(∂

∂xF2(x, y, t) +

∂

∂yF1(x, y, t)

)d t +

∂

∂xb(x, y) = F3

To di�nusma ìmwc ~F eÐnai swlhnoeidèc kai �ra ikanopoieÐtai h sqèsh (2.20).Epomènwc h parap�nw exÐswsh gr�fetai

∫ z

z0

(∂

∂ tF3(x, y, t)

)d t +

∂

∂xb(x, y) = F3

H exÐswsh aut  met� thn olokl rwsh wc proc t gÐnetai

∂

∂xb(x, y) = F3(x, y, z0)

Apì thn sqèsh aut  brÐskoume thn �gnwsth sun�rthsh b(x, y) me mia olok-l rwsh

b(x, y) =

∫F3(x, y, z0)dx

Oi sunist¸sec A kai B tou dianÔsmatoc ~G0 eÐnai

A =

∫ z

z0

F1(x, y, t)d t B = −∫ z

z0

f2(x, y, t)d t +

∫F3(x, y, z0)dx

An ~G eÐnai mia �llh lÔsh thc exÐswshc ~F = ~∇ × ~G tìte profan¸cprèpei

~∇× ~G− ~∇× ~G0 = ~∇×(

~G− ~G0

)= ~0

Epomènwc to di�nusma ~G − ~G0 eÐnai astrìbilo kai �ra up�rqei bajmwt sun�rthsh χ(~r) tètoia ¸ste ~G− ~G0 = ~∇χ(~r) . 'Ara èqoume

~g = ~g0 + ~∇χ(~r)

ìpou χ(~r) eÐnai mia aujaÐreth sun�rthsh pou fusik� prèpei na èqei merikècparag¸gouc deÔterhc t�xhc. H sun�rthsh ~G eÐnai telik� h genik  lÔsh touprobl matoc.

54

'Askhsh 3.Na apodeiqjeÐ ìti isqÔoun oi sqèseic

~∇× (~∇× ~f) = ~∇(~∇ · ~f)− ~∇2 ~f (2.21)~∇(~f · ~g) = (~g · ~∇)~f + (~f · ~∇)~g + ~g × (~∇× ~f) + ~f × (~∇× ~g) (2.22)

~∇× (~f × ~g) = (~g · ~∇)~f − ~g(~∇ · ~f)− (~f · ~∇)~g + ~f(~∇ · ~g)

Upojètoume ìti ta dianusmatik� pedÐa èqoun sunist¸sec << kalèc >> sunart seicdhlad  ìlec oi par�gwgoi pou emfanÐzontai stic sqèseic up�rqoun.

LÔsh: Ja apodeÐxoume thn pr¸th sqèsh. Upojètoume ìti to di�nusma~f èqei sunist¸sec ~f = (f1, f2, f3, ). ja analÔsoume to di�nusma tou pr¸toumèlouc thc isìthtac. Apì ton orismì tou strobilismoÔ enìc dianÔsmatocbrÐskoume

~∇× (~∇× ~f) = ~∇×((∂yf3 − ∂zf2)~i + (∂zf1 − ∂xf3)~j + (∂xf2 − ∂yf1)~k

)

=

∣∣∣∣∣∣

~i ~j ~k∂x ∂y ∂z

∂yf3 − ∂zf2 ∂zf1 − ∂xf3 ∂xf2 − ∂yf1

∣∣∣∣∣∣= (∂y(∂xf2 − ∂yf1)−

∂z(∂zf1 − ∂xf3))~i + (∂z(∂yf3 − ∂zf2)− ∂x(∂xf2 − ∂yf1))~j + (∂x(∂zf1−∂xf3)− ∂y(∂yf3 − ∂zf2))~k = (∂2

xyf2 − ∂2yyf1 − ∂2

zzf1 + ∂2xzf3)~i+

(∂2yzf3 − ∂2

zzf2 − ∂2xxf2 + ∂2

xyf1)~j + (∂2xzf1 − ∂2

xxf3 − ∂2yyf3 + ∂2

yzf2)~k

UpologÐzoume akoloÔjwc to deÔtero mèloc thc isìthtac

~∇(~∇· ~f)− ~∇2 ~f = ~∇(∂xf1 +∂yf2 +∂zf3)− (∂xx +∂yy +∂zz)(f1~i+f2

~j +f3~k) =

∂x(∂xf1 + ∂yf2 + ∂zf3)~i + ∂y(∂xf1 + ∂yf2 + ∂zf3)~j + ∂z(∂xf1 + ∂yf2 + ∂zf3)~k−((∂xx + ∂yy + ∂zz)f1

~i + (∂xx + ∂yy + ∂zz)f2~j + (∂xx + ∂yy + ∂zz)f3)~k =

(∂2xxf1 +∂2

xyf2 +∂2xzf3)~i+(∂2

xyf1 +∂2yyf2 +∂2

yzf3)~j +(∂2xzf1 +∂2

yzf2 +∂2zzf3)~k−

(∂xx + ∂yy + ∂zz)f1~i− (∂xx + ∂yy + ∂zz)f2

~j − (∂xx + ∂yy + ∂zz)f3)~k =

(�∂2xxf1+∂2

xyf2+∂2xzf3−�∂xxf1−∂yyf1−∂zzf1)~i+(∂2

xyf1+�∂2yyf2+∂2

yzf3−∂xxf2−

�∂yyf2 − ∂zzf2)~j + (∂2xzf1 + ∂2

yzf2 +�∂2zzf3 − ∂xxf3 − ∂yyf3 −�∂zzf3)~k

55

In[31]:= << Calculus‘VectorAnalysis‘

SetCoordinates@Cartesian@x, y, zDD;

f = 8f1@x, y, zD, f2@x, y, zD, f3@x, y, zD<;

g = 8g1@x, y, zD, g2@x, y, zD, g3@x, y, zD<;

Simplify@Grad@Dot@f, gDD - Hg1@x, y, zD D@f, xD + g2@x, y, zD D@f, yD +

g3@x, y, zD D@f, zD + f1@x, y, zD D@g, xD + f2@x, y, zD D@g, yD +

f3@x, y, zD D@g, zD + Cross@g, Curl@fDD + Cross@f, Curl@gDDLD

Out[35]= 80, 0, 0<

Sq ma 2.7: H apìdeixh thc dianusmatik c sqèshc (2.22).

profan¸c ta dÔo mèlh eÐnai Ðsa kai h �ra h tautìthta apodeÐqjhke. Ed¸ me tonmh majhmatikì ìro << kalèc >> sunart seic ennooÔme ìti èqoun pr¸tec merikècparag¸gouc suneqeÐc ètsi ¸ste na mporoÔme na gr�youme p.q.

∂2

∂x∂yfj =

∂2

∂y∂xfj

pou den isqÔei p�nta.Ja apodeÐxoume thn deÔterh sqèsh me thn mèjodo tou upologist  sq ma

(2.7). To sÔmbolo << ; >> sto tèloc twn entol¸n lèei ston upologist  na mhntup¸sei thn ap�nthsh.

H apìdeixh thc trÐthc sqèshc af netai ston anagn¸sth.

'Askhsh 4.Na apodeiqjeÐ ìti oi sunart seic (exis¸seic 2.16)

x = x(u1, u2, u3) y = y(u1, u2, u3) z = z(u1, u2, u3)

eÐnai grammik� anex�rthtec an h Iakwbian  tou metasqhmatismoÔ den eÐnaimhdèn.Apìdeixh: PaÐrnoume ton akìloujo grammikì sunduasmì twn sunart sewn

c1 x(u1, u2, u3) + c2 y(u1, u2, u3) + c3 z(u1, u2, u3) = 0

Ja apodeÐxoume ìti c1 = c2 = c3 = 0.Kataskeu�zoume treic exis¸seic paragwgÐzontac thn parap�nw isìthta

diadoqik� me tic treic kampulìgrammec suntetagmènec (u1, u2, u3). BrÐskoume

c1∂x

∂u1

+ c2∂y

∂u1

+ c3∂z

∂u1

= 0

56

c1∂x

∂u2

+ c2∂y

∂u2

+ c3∂z

∂u2

= 0

c1∂x

∂u3

+ c2∂y

∂u3

+ c3∂z

∂u3

= 0

'Etsi fti�xame èna grammikì omogenèc algebrikì sÔsthma tri¸n exis¸sewn metreic �gnwstouc, tic stajerèc (c1, c2, c3). To sÔsthma èqei mia kai monadik lÔsh an h orÐzousa twn suntelest¸n twn agn¸stwn den eÐnai mhdèn, dhlad 

∣∣∣∣∣∣

∂x∂u1

∂y∂u1

∂z∂u1

∂x∂u2

∂y∂u2

∂z∂u2

∂x∂u3

∂y∂u3

∂z∂u3

∣∣∣∣∣∣=

∂~r

∂u1

· ∂~r

∂u2

× ∂~r

∂u3

= J 6= 0

H monadik  aut  lÔsh eÐnai fusik� aut  pou faÐnetai << dia gumnoÔ ofjal-moÔ >> h mhdenik . 'Ara

c1 = c2 = c3 = 0

kai h �skhsh apodeÐqjhke.An h Iakwbian  eÐnai arnhtik  mporoÔme na all�xoume metaxÔ touc tic

suntetagmènec p.q. u1 kai u2 . Me ton trìpo autì, pou fusik� den bl�ptetaih genikìthta, h orÐzousa all�zei prìshmo diìti all�zoun metaxÔ touc oi dÔopr¸tec grammèc. 'Ara telik� h anagkaÐa sunj kh eÐnai

J =∂~r

∂u1

· ∂~r

∂u2

× ∂~r

∂u3

> 0

'Askhsh 5.Mia sunarthsiak  sqèsh twn x, y, z parist�nei wc gnwstì mia epif�neia.

Mia kampÔlh ston q¸ro mporeÐ na prokÔyei apì thn tom  dÔo epifanei¸n.Upojètoume ìti oi dÔo autèc epif�neiec dÐnontai apì tic sqèseic

f1(x, y, z) = 0 f2(x, y, z) = 0

BreÐte mÐa paramètrhsh thc kampÔlhc aut c.Na efarmìsete thn lÔsh sthn perÐptwsh twn epifanei¸n.

f1(x, y, z) = 4y2 + z2− 9 = 0 f2(x, y, z) = x + 2y + 3z − 6 = 0 (2.23)

LÔsh: LÔnoume tic exis¸seic wc proc y kai z an mporoÔn na lujoÔnkai upojètoume ìti br kame tic lÔseic

y = y(x) και z = z(x)

57

tìte jètoume x = t =⇒ y = y(t) kai z = z(t) pou eÐnai oi parametrikècexis¸seic thc kampÔlhc. Epomènwc to di�nusma jèshc thc kampÔlhc eÐnai

~r(t) = (x, y, z) = (t, y(t), z(t))

Gia thn efarmog  faÐnetai apì tic exis¸seic (2.23) ìti prèpei na lÔsoumeto sÔsthma twn dÔo exis¸sewn wc proc x kai y . BrÐskoume 10

x = 6− 3z −√

9− z2, y =1

2

√9− z2 κaι

x = 6− 3z +√

9− z2 y = −1

2

√9− z2

Jètoume z = t kai �ra oi parametrikèc exis¸seic thc kampÔlhc eÐnai

x = 6− 3 t∓√

9− t2 y = ±1

2

√9− t2 z = t

Mia �llh paramètrhsh thc kampÔlhc mporeÐ na brejeÐ an jèsoume z =3 cos u. Jètoume thn lÔsh aut  sthn pr¸th apì tic exis¸seic (2.23) kai brÐsk-oume

4y2 + 9 cos2 u− 9 = 0 =⇒ y = ±3

2

√1− cos2 u = ±3

2sin u

H deÔterh exÐswsh twn (2.23) mac dÐnei kai thn metablht  x. Oi parametrikècexis¸seic thc kampÔlhc me par�metro to u eÐnai

x = ∓ 3 sin u− 9 cos u + 6 y = ±3

2sin u z = 3 cos u

Oi dÔo par�metroi sundèontai me thn sqèsh t = 3 cos u.Telik� k�noume kai mia epal jeush gia na eÐmaste sÐgouroi ìti h para-

mètrhsh ikanopoieÐ tic exis¸seic twn epifanei¸n.Gia thn pr¸th exÐswsh brÐskoume

4y2 + z2 − 9 = 4(±3

2sin u)2 + (3 cos u)2 − 9 = 9 sin2 u + 9 cos2 u− 9 = 0

10Η λύση του συστήματος῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[26] := Solve[{4y2 + z2 − 9 == 0, x + 2y + 3z − 6 == 0}, {x, y}]Out[26] = {{x → 6−3z−√9− z2, y →

√9−z2

2 }, {x → 6−3z+√

9− z2, y → − 12

√9− z2}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

58

kai gia thn deÔterh

x + 2y + 3z − 6 = ∓ 3 sin u− 9 cos u + 6 + 2(±3

2sin u) + 3(3 cos u)− 6 = 0

'Askhsh 6.Na apodeiqjeÐ ìti ikan  kai anagkaÐa sunj kh gia na eÐnai mia dianusmatik 

sun�rthsh ~F = ~F (t) k�jeth sth par�gwgì thc, eÐnai na èqei stajerì mètro.Apìdeixh: Ja apodeÐxoume pr¸ta to anagkaÐo.

'Estw ìti ‖~F‖ = C = stajer�. Tìte

~F (t) · ~F (t) = ‖~F (t)‖2 = C2

gia k�je t ∈ I

~F ′(t) · ~F (t) + ~F (t) · ~F ′(t) =⇒ 2~F ′(t) · ~F (t) = 0

sunep¸c ~F ′(t) ⊥ ~F (t)

Ja apodeÐxoume t¸ra to ikanì. 'Estw ìti ~F ′(t) ⊥ ~F (t). Tìte

~F ′(t) · ~F (t) = 0 =⇒ 1

2

d

dt

(~F ′(t)2

)= 0 =⇒ ‖~F (t)‖ = C

An èna kinhtì kineÐtai sthn perifèreia enìc kÔklou tìte to mètro toudianÔsmatoc jèshc tou eÐnai stajerì kai �ra apì thn �skhsh èpetai ìti eÐnaik�jeto sthn taqÔtht� tou.

'Askhsh 7.Na apodeiqjeÐ ìti h ikan  kai anagkaÐa sunj kh gia èqei h sun�rthsh

~F = ~F (t) stajer  dieÔjunsh, eÐnai na eÐnai par�llhlh proc thn par�gwgìthc.Apìdeixh: Ja apodeÐxoume ìti h prìtash eÐnai anagkaÐa.

Upojètoume ìti to di�nusma ~F (t) èqei stajer  dieÔjunsh tìte to di�nus-ma ~F0(t) =

~F (t)

‖~F (t)‖ èqei epÐshc stajer  dieÔjunsh kai epeid  èqei m koc thn

mon�da eÐnai stajerì. Sunep¸c ~F ′0(t) = 0. Apì tic sqèseic autèc paÐrnoume

~F (t) = ~F0(t)‖~F (t)‖ ~F ′(t) = ~F0(t)(‖~F (t)‖)′

kai epomènwc brÐskoume

~F (t)× ~F ′(t) = ~F0(t)× ~F0(t)(‖~F0(t)‖‖~F (t)‖′

)= 0 =⇒ ~F (t)× ~F ′(t) = 0

59

Ja apìdeÐxoume t¸ra ìti h sqèsh eÐnai kai ikan . 'Estw ìti isqÔei ~F (t)×~F ′(t) = 0. Apì th sqèsh ~F0(t) =

~F (t)

‖~F (t)‖ paÐrnoume

~F ′0(t) = −‖

~F (t)‖′‖~F (t)‖2

~F (t) +~F ′(t)

‖~F (t)‖=−~F (t)‖~F (t)‖′‖~F (t)‖+ ‖~F (t)‖2 ~F ′(t)

‖~F (t)‖3

all�

‖~F (t)‖2 = ~F (t) · ~F (t) =⇒ ‖~F (t)‖‖~F (t)‖′ = ~F (t) · ~F ′(t) (2.24)

opìte paÐrnoume lìgw kai thc exÐswshc (1.4)

~F ′0(t) =

−(

~F (t) · ~F ′(t))

~F (t) +(

~F (t) · ~F (t))

~F ′(t)

‖~F (t)‖3

=1

‖~F (t)‖3

[~F ′(t)×

(~F ′(t)× ~F (t)

)]= 0

diìti ~F ′(t)× ~F (t) = 0.Epomènwc ~F ′

0(t) = 0 kai �ra to di�nusma ~F0(t) eÐnai stajerì. Epeid ìmwc èqei m koc thn mon�da èqei kai stajer  dieÔjunsh.

60

Kef�laio 3

PÐnakec, OrÐzousec, Grammik�sust mata

3.1 Oi pÐnakecOrismìc: Mia di�taxh apì n ×m pragmatikoÔc   migadikoÔc arijmoÔc

eÐnai ènac pÐnakac   m tra. Gr�foume ton pÐnaka me thn morf  n gramm¸n kaim sthl¸n. Gia par�deigma ènac pÐnakac me 2 grammèc kai 3 st lec gr�fetaime thn morf 

A =

(2 3 65 3 22

)

ja lème ìti o pÐnakac autìc èqei di�stash 2×3. 'Enac pÐnakac me di�stashn×n onom�zetai tetragwnikìc pÐnakac. 'Enac tanust c me t�xh 2 parist�ne-tai apì ènan pÐnaka qwrÐc autì na shmaÐnei ìti k�je pÐnakac parist�nei ènantanust .

Sun jwc gr�foume ta stoiqeÐa enìc pÐnaka A me dÔo deÐktec (i, j),A = aij ètsi ¸ste o pr¸toc deÐkthc dhl¸nei thn gramm  kai o deÔterocthn st lh sthn opoÐa brÐsketai to stoiqeÐo. 'Enac pÐnakac me di�stash 3× 3gia par�deigma gr�fetai

A =

a11 a12 a13

a21 a22 a23

a31 a32 a33

DÔo pÐnakec A = (ajk) kai B = (bjk) me Ðdia di�stash eÐnai Ðsoi an tastoiqeÐa touc eÐnai Ðsa èna pro èna, dhlad  ajk = bjk gia k�je j kai k. 'Etsih exÐswsh metaxÔ dÔo pin�kwn A = B eÐnai sthn pragmatikìthta n × misìthtec.

61

Orismìc: Troqi�   Ðqnoc enìc tetragwnikoÔ pÐnaka eÐnai to �jroismatwn stoiqeÐwn thc kurÐac (dhlad  thc pr¸thc) diagwnÐou. Gia par�deigma htroqi� tou pÐnaka

a =

1 2 33 1 54 6 3

eÐnai Ðsh me 5 . 1

Orismìc: 'Enac tetragwnikìc pÐnakac onom�zetai diag¸nioc an ìla tastoiqeÐa tou ektìc apì ekeÐna thc kurÐac diagwnÐou tou eÐnai mhdèn. 'Enactètoioc pÐnakac eÐnai o ex c

I =

a11 0 00 a22 00 0 a33

EÐnai dunatìn ìla ta stoiqeÐa enìc diagwnÐou pÐnaka na eÐnai mon�dec. OpÐnakac autìc pou ja ton sumbolÐzoume me 11 eÐnai h mon�da tou pollaplasi-asmoÔ twn pin�kwn.

Gia par�deigma gia touc tetragwnikoÔc pÐnakec 3× 3 h mon�da eÐnai:

11 =

1 0 00 1 00 0 1

και 11A = A11 = A

Ta stoiqeÐa enìc pÐnaka mon�da ta sumbolÐzoume sun jwc me to sÔmbolo δjk

pou onom�zetai tanust c   sÔmbolo tou Krìneker.

δjk =

{1 αν j = k0 αν j 6= k

(3.1)

Orismìc: 'Enac pÐnakac A = (ajk) onom�zetai summetrikìc an ajk =akj kai antisummetrikìc an ajk = −akj. Dhlad  an ta summetrik� stoiqeÐaenìc pÐnaka wc proc thn kÔria diag¸nio eÐnai Ðsa   antÐjeta tìte o pÐnakac eÐnaisummetrikìc   antisummetrikìc antistoÐqwc. Profan¸c ta diag¸nia stoiqeÐaenìc antisummetrikoÔ pÐnaka eÐnai Ðsa me to mhdèn.

Orismìc: 'Enac pÐnakac B = (bjk) eÐnai an�strofoc enìc pÐnakaA = (ajk) an isqÔei bjk = akj. Dhlad  oi grammèc tou B eÐnai st lec tou

1Ο υπολογισμός της τροχιάς (ή ίχνος) ενός πίνακα῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[140] := Tr[{{1, 2, 3}, {3, 1, 5}, {4, 6, 3}}]Out[140] = 5

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

62

A kai oi st lec tou B eÐnai oi grammèc tou A. 'Enac tètoioc pÐnakac Bonom�zetai an�strofoc tou A kai sumbolÐzetai me At = B.

Gia par�deigma o akìloujoc pÐnakac me di�stash 3× 3

A =

a11 a12 a13

a21 a22 a23

a31 a32 a33

èqei an�strofo ton pÐnaka 2

a11 a21 a31

a12 a22 a32

a13 a23 a33

Gia thn pr�xh thc anastrof c twn pin�kwn apodeiknÔetai eÔkola ìti isqÔ-oun oi idiìthtec

(A + B)t = At + Bt (A ·B)t = Bt · At (At)t = A

MporoÔme na prosjèsoume dÔo pÐnakec A = (ajk) kai B = (bjk)an èqoun thn Ðdia di�stash n × m. To apotèlesma eÐnai o pÐnakac C =(cjk) = (ajk + bjk). MporoÔme epÐshc na touc afairèsoume me apotèlesmaD = (djk) = (ajk − bjk). OrÐzetai tèloc kai o exwterikìc pollaplasiasmìcenìc arijmoÔ pragmatikoÔ   migadikoÔ ξ me ènan pÐnaka A = (ajk). toapotèlesma eÐnai o pÐnakac ξA = (ξajk).

Gia par�deigma èqoume(

a11 a12 a13

a21 a22 a23

)+

(b11 b12 b13

b21 b22 b23

)=

(a11 + b11 a12 + b12 a13 + b13

a21 + b21 a22 + b22 a23 + b23

)

ξ

(a11 a12 a13

a21 a22 a23

)=

(ξa11 ξa12 ξa13

ξa21 ξa22 ξa23

)

'Opwc kai gia tic antÐstoiqec pr�xeic twn dianusm�twn, h pr�xh thc prìs-jeshc ikanopoieÐ thn prosetairistik  kai thn antimetajetik  idiìthta, up�rqeio mhdenikìc pÐnakac kai o antÐjetoc pÐnakac. O exwterikìc pollaplasias-mìc ikanopoieÐ epÐshc tic antÐstoiqec idiìthtec twn dianusm�twn. Tèloc oi

2Ο υπολογισμός του αναστρόφου ενός πίνακα῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[144] := Transpose[A]//MatrixForm

Out[144]//MatrixForm =

a11 a21 a31a12 a22 a32a13 a23 a33

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

63

dÔo nìmoi ikanopoioÔn kai tic epimeristikèc idiìthtec. Epomènwc to sÔnolotwn pin�kwn gÐnetai dianusmatikìc q¸roc sto s¸ma twn pragmatik¸n   twnmigadik¸n arijm¸n.

3.2 Oi orÐzousecOrismìc: OrÐzousa eÐnai mia tetragwnik  di�taxh apì stoiqeÐa ajk pou

mporoÔn na sunduastoÔn ètsi ¸ste na d¸soun thn tim 

D =

∣∣∣∣∣∣∣∣∣∣

a11 a12 a13 · · ·a21 a22 a23 · · ·a31 a32 a33 · · ·a41 a42 a43 · · ·... ... ... .........

∣∣∣∣∣∣∣∣∣∣

=∑

ijk···εijk···ai1aj2ak3 · · ·

ìpou to εijk··· eÐnai Ðso me ±1 an�loga me to an h met�jesh eÐnai �rtia  peritt . EÐnai ènac tanust c an�logoc me ton tanust  tou LebÔ - Tsibit�(sqèsh 1.3).

H orÐzousa enìc tetragwnikoÔ pÐnaka A = (ajk) mporeÐ na brejeÐ me tonakìloujo kanìna. Gia na gÐnei katanoht  h diadikasÐa paÐrnoume ton akìloujopÐnaka me di�stash 4× 4:

A =

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

Epilègoume pr¸ta mia opoiad pote gramm    st lh tou pÐnaka, èstw thndeÔterh gramm . (a21 a22 a23 a24). Pollaplasi�zoume to pr¸to stoiqeÐoa21 me thn orÐzousa tou el�ssona pÐnaka dhlad  tou pÐnaka pou prokÔptei andiagr�youme thn gramm  kai thn st lh pou brÐsketai to stoiqeÐo a21. Dhlad 

a21 |A21| = a21

∣∣∣∣∣∣∣∣

�a11 a12 a13 a14

�a21 �a22 �a23 �a24

�a31 a32 a33 a34

�a41 a42 a43 a44

∣∣∣∣∣∣∣∣= a21

∣∣∣∣∣∣

a12 a13 a14

a32 a33 a34

a42 a43 a44

∣∣∣∣∣∣

Sto parap�nw ginìmeno b�zoume mprost� to prìshmo plhn diìti to stoiqeÐobrÐsketai sthn gramm  2 kai sthn st lh 1 kai to noÔmero 2+1 eÐnai perittì,  alli¸c pollaplasi�zoume me to (−1)2+1 = −1. 'Eqoume to noÔmero

−a21 |A21|

64

Katìpin pollaplasi�zoume to deÔtero stoiqeÐo thc gramm c a22 me thnantÐstoiqh el�ssona orÐzousa. Dhlad 

a22 |A22| = a22

∣∣∣∣∣∣∣∣

a11 �a12 a13 a14

�a21 �a22 �a23 �a24

a31 �a32 a33 a34

a41 �a42 a43 a44

∣∣∣∣∣∣∣∣= a22

∣∣∣∣∣∣

a11 a13 a14

a31 a33 a34

a41 a43 a44

∣∣∣∣∣∣

Sto parap�nw ginìmeno b�zoume mprost� to prìshmo sun diìti to stoiqeÐobrÐsketai sthn gramm  2 kai sthn st lh 2 kai to noÔmero 2+2 eÐnai �rtio,  alli¸c pollaplasi�zoume me to (−1)2+2 = +1. 'Eqoume to noÔmero

a22 |A22|

SuneqÐzoume thn Ðdia diadikasÐa me to trÐto a23 kai to tètarto a24

stoiqeÐo. Tèloc prosjètoume touc tèsseric autoÔc ìrouc kai brÐskoume∣∣∣∣∣∣∣∣

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

∣∣∣∣∣∣∣∣= −a21

∣∣∣∣∣∣

a12 a13 a14

a32 a33 a34

a42 a43 a44

∣∣∣∣∣∣+ a22

∣∣∣∣∣∣

a11 a13 a14

a31 a33 a34

a41 a43 a44

∣∣∣∣∣∣−

a23

∣∣∣∣∣∣

a11 a12 a14

a31 a32 a34

a41 a42 a44

∣∣∣∣∣∣+ a24

∣∣∣∣∣∣

a11 a12 a13

a31 a32 a33

a41 a42 a43

∣∣∣∣∣∣ParathroÔme ìti an broÔme (swst�) to pr¸to prìshmo, ed¸ eÐnai to plhn, giata upìloipa b�zoume enall�x to sun kai to plhn.

Me ton trìpo autì to prìblhma tou upologismoÔ miac orÐzousac me di�s-tash 4 × 4 metafèretai sto prìblhma tou upologismoÔ 4 orizous¸n medi�stash 3× 3 dhlad  mÐa t�xh mikrìterh.

O parap�nw kanìnac profan¸c genikeÔetai gia ton upologismì orizous¸nme opoiad pote t�xh ν. O upologismìc an�getai ston upologismì ν ori-zous¸n t�xhc ν− 1, k�je mia apì autèc ston upologismìc ν − 1 orizous¸nt�xhc ν − 2 k.o.k.

Gia na upologÐsoume thn orÐzousa enìc pÐnaka me di�stash ν = 6

∣∣∣∣∣∣∣∣∣∣∣∣

5 6 7 8 9 −15 1 1 1 3 41 1 1 3 4 31 1 1 3 1 34 3 1 0 4 32 1 −2 3 −4 3

∣∣∣∣∣∣∣∣∣∣∣∣

65

prèpei na upologÐsoume sunolik� 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 6! = 720 orÐzous-ec!!!. Mhn to epiqeir sete kallÐtera na to anajèsete se k�poion �llon... tonupologist .... BrÐskoume D = 3726. 3

Gia tic orÐzousec isqÔoun oi akìloujec idiìthtec:H orÐzousa all�zei prìshmo an all�xoume metaxÔ touc dÔo grammèc   dÔo

st lec. Kat� sunèpeia mia orÐzousa pou èqei dÔo grammèc   dÔo st lec ÐseceÐnai mhdèn.

H orÐzousa den all�zei an se mÐa gramm  prosjèsoume èna pollapl�siomiac �llhc gramm c. Gia mia orÐzousa 3× 3 gia par�deigma èqoume

∣∣∣∣∣∣

a1 a2 a3

b1 b2 b3

c1 c2 c3

∣∣∣∣∣∣=

∣∣∣∣∣∣

a1 + kb1 a2 + kb2 a3 + kb3

b1 b2 b3

c1 c2 c3

∣∣∣∣∣∣To Ðdio isqÔei kai gia tic st lec. Apì thn idiìthta aut  sunep�getai ìti mÐaorÐzousa eÐnai mhdèn an dÔo grammèc   dÔo st lec eÐnai an�logec.

3.3 O Pollaplasiasmìc twn Pin�kwnOrismìc: O pollaplasiasmìc dÔo pin�kwn AB orÐzetai an o pr¸toc

pÐnakac A = (aij) tou ginomènou èqei di�stash n × m kai o deÔterocB = (bij) èqei di�stash m × k. To apotèlesma eÐnai ènac pÐnakac C medi�stash n× k kai stoiqeÐa pou dÐnontai apì thn sqèsh. 4

cij =m∑

s=1

aisbsj

3Υπολογισμός της ορίζουσας ενός πίνακα῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[3] := Det[{{5, 6, 7, 8, 9,−1}, {5, 1, 1, 1, 3, 4}, {1, 1, 1, 3, 4, 3},

{1, 1, 1, 3, 1, 3}, {4, 3, 1, 0, 4, 3}, {2, 1,−2, 3,−4, 3}}]Out[3] = 3726

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

4Ο πολλαπλασιασμός των πινάκων συμβολίζεται στο Mathematica όχι με τον αστερ-ίσκο ή το διάστημα αλλά με μια τελεία. Μπορούμε επίσης να χρησιμοποιήσουμε την εντολήDot[a, b].῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[63] := MatrixForm[{{a11, a12, a13}, {a21, a22, a23}}.

{{b11, b12}, {b21, b22}, {b31, b32}}]Out[63]//MatrixForm =(

a11 b11 + a12 b21 + a13 b31 a11 b12 + a12 b22 + a13 b32a21 b11 + a22 b21 + a23 b31 a21 b12 + a22 b22 + a23 b32

)

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

66

Gia par�deigma brÐskoume to ginìmeno enìc pÐnaka 3× 3 kai enìc pÐnaka3× 1.

1 2 34 5 67 8 9

a1

a2

a3

=

a1 + 2a2 + 3a3

4a1 + 5a2 + 6a3

7a1 + 8a2 + 9a3

'Ena �llo par�deigma eÐnai(

a11 a12

a21 a22

)·(

b11 b12

b21 b22

)=

(a11b11 + a12b21 a11b12 + a12b22

a21b11 + a22b21 a21b12 + a22b22

)

To ginìmeno dÔo pin�kwn den orÐzetai p�nta prèpei to pl joc twn sthl¸n toupr¸tou na eÐnai Ðsec me to pl joc twn gramm¸n tou deÔterou.

H pr�xh tou pollaplasiasmoÔ twn pin�kwn ikanopoieÐ thn prosetairistik idiìthta

A · (B · C) = (A ·B) · CAn dÔo pÐnakec A kai B eÐnai tetragwnikoÐ me thn Ðdia di�stash n× n

tìte orÐzontai kai ta dÔo ginìmena AB kai BA genik� ìmwc den eÐnaiÐsa AB 6= BA. H diafor� aut¸n twn dÔo pin�kwn onom�zetai metajèthc kaisumbolÐzetai me [A,B], dhlad 

[A,B] = AB −BA (3.2)

H orÐzousec twn dÔo pin�kwn AB kai BA eÐnai Ðsec. 5

|AB| = |BA|H parap�nw idiìthta twn orizous¸n isqÔei kai gia thn troqi� tou ginomènou

dÔo pin�kwn dhlad Tr(AB) = Tr(BA)

Gia touc pÐnakec isqÔei h epimeristik  idiìthta

A(B + C) = AB + AC (B + C)A = BA + CA

5Η απόδειξη για πίνακες τάξης 3× 3 δίνεται με την βοήθεια του υπολογιστή. Βρίσκουμε῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[69] := a = {{a11, a12, a13}, {a21, a22, a23}, {a31, a32, a33}}b = {{b11, b12, b13}, {b21, b22, b23}, {b31, b32, b33}}Simplify[Det[a.b]−Det[b.a]]Out[69] = {{a11, a12, a13}, {a21, a22, a23}, {a31, a32, a33}}Out[70] = {{b11, b12, b13}, {b21, b22, b23}, {b31, b32, b33}}Out[71] = 0

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

67

Prèpei na prosèqoume ston pollaplasiasmì twn pin�kwn diìti h pr�xh deneÐnai antimetajet . Gia par�deigma èqoume

(A + B)2 = (A + B)(A + B) = A2 + B2 + AB + BA

To apotèlesma den eÐnai Ðson oÔte me A2+B2+2AB oÔte me A2+B2+2BA.MporoÔme na pollaplasi�soume tou pÐnakec pollèc forèc diadoqik� gia

na upologÐsoume p.q. to ginìmeno A3 = AAA   to A2B3A k.l.p.   akìmana broÔme kai sunart seic pin�kwn f(A) arkeÐ oi sun�rthsh f na eÐnaianalutik . Gia par�deigma èqoume

eλA = 1 + λA +1

2!λ2A2 +

1

3!λ3A3 + · · ·

Orismìc: 'Enac tetragwnikìc pÐnakac B eÐnai antÐstrofoc enìc �lloupÐnaka A me tic Ðdiec diast�seic an

AB = BA = 11

sumbolÐzoume ton pÐnaka autìn me A−1 = B. O antÐstrofoc enìc pÐnaka eÐnaimonadikìc. Gia par�deigma o antÐstrofoc tou pÐnaka

A =

1 1 0 20 0 0 12 1 3 12 2 1 1

eÐnai o pÐnakac 6

5 −8 1 −3−4 6 −1 3−2 3 0 10 1 0 0

An o pÐnakac èqei mhdenik  orÐzousa |A| = 0 tìte o antÐstrofoc tou denorÐzetai.

6Ο υπολογισμός του αντίστροφου ενός πίνακα῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[169] := Inverse[{{1, 1, 0, 2}, {0, 0, 0, 1}, {2, 1, 3, 1}, {2, 2, 1, 1}}]

//MatrixForm

Out[169]//MatrixForm =

5 −8 1 −3−4 6 −1 3−2 3 0 10 1 0 0

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

68

Gia na broÔme ton antÐstrofo enìc tetragwnikoÔ pÐnaka A = (ajk) sqh-matÐzoume ton pÐnaka A∗ = (a∗jk) ìpou a∗jk = (−1)(j+k)Akj kai Akj eÐnai horÐzousa tou pÐnaka pou prokÔptei an apaleÐyoume thn k− gramm  kai thnj− st lh pou brÐsketai to stoiqeÐo akj. O antÐstrofoc tou pÐnaka A eÐnaio akìloujoc pÐnakac.

A−1 =1

|A|A∗

ìpou |A| eÐnai h orÐzousa tou pÐnaka A.H orÐzousa enìc diagwnÐou pÐnaka A = (ajδjk), eÐnai Ðsh me to ginìmeno

twn mh mhdenik¸n stoiqeÐwn thc kurÐac diagwnÐou tou. 'Enac pÐnakac pou èqeiìla tou ta stoiqeÐa pou brÐskontai p�nw   k�tw apì thn kurÐa diag¸nio Ðsame mhdèn, onom�zetai trigwnikìc kai èqei orÐzousa epÐshc Ðsh me to ginìmenotwn stoiqeÐwn thc kurÐac diagwnÐou tou.

H tim  thc orÐzousac enìc pÐnaka A eÐnai Ðsh me thn orÐzousa tou an�s-trofou pÐnaka At. Dhlad 

|A| = |At|H orÐzousa tou ginomènou dÔo pin�kwn eÐnai Ðsh me to ginìmeno twn ori-

zous¸n. H idiìthta aut  sunep�getai ìti h orÐzousa tou ginomènou dÔo  perissotèrwn pin�kwn den exart�tai apì thn di�taxh twn pin�kwn. Dhlad gia dÔo pÐnakec èqoume

|AB| = |A||B| = |BA|

O pÐnakac mon�da 11 = (δjk) èqei orÐzousa Ðsh me thn mon�da. 'Ara horÐzousa enìc pÐnaka A kai h orÐzousa tou antÐstrofou A−1 eÐnai arijmoÐantÐstrofoi.

AA−1 = 11 =⇒ |A||A−1| = 1

'Estwsan dÔo pÐnakec A kai B pou sundèontai me thn bo jeia enìc trÐtoupÐnaka S me thn sqèsh A = S−1BS. H sqèsh aut  onom�zetai metasqhma-tismìc omoiìthtac. Oi dÔo pÐnakec èqoun thn Ðdia orÐzousa. Pr�gmati

|A| = |S−1BS| = |S−1SB| = |11B| = 1|B| = |B|

An mporèsoume na broÔme ènan pÐnaka S antistrèyimo, ètsi ¸ste o parap�nwpÐnakac A na eÐnai diag¸nioc tìte profan¸c o upologismìc thc orÐzousactou B brÐsketai eÔkola, diìti o orÐzousa tou A eÐnai Ðsh me to ginìmenotwn diagwnÐwn stoiqeÐwn tou diagwnÐou pÐnaka A.

Orismìc: 'Enac pÐnakac onom�zetai orjog¸nioc an o antÐstrofoc isoÔ-tai me ton an�strofo dhlad 

A−1 = At ⇐⇒ AAt = AtA = 11

69

H orÐzousa enìc orjog¸niou pÐnaka eÐnai Ðsh me sun   plhn èna. H apìdeixhèpetai

|AAt| = |A||At| = |A||A| = |A|2 = 1 =⇒ |A| = ±1

3.4 Ta grammik� sust mataMia efarmog  thc jewrÐac twn pin�kwn eÐnai h lÔsh enìc grammikoÔ sust -matoc algebrik¸n exis¸sewn.

a11x1 + a12x2 + · · · a1nxn = b1

a21x1 + a22x2 + · · · a2nxn = b2

· · · · · · · · · · · · · · · · · · · · ·am1x1 + am2x2 + · · · amnxn = bm

me m exis¸seic kai n agn¸stouc.OrÐzoume touc pÐnakec A = (ajk) , X = (xj) kai B = (bj) . To sÔsthma

gr�fetai san mia exÐswsh kai h lÔsh tou brÐsketai eÔkola toul�qiston tupik�.BrÐskoume

AX = B =⇒ X = A−1B

Epomènwc prèpei na broÔme ton antÐstrofo tou pÐnaka A (an up�rqei) kaimet� na ton pollaplasi�soume me ton B.

An gia par�deigma jèloume na lÔsoume to akìloujo grammikì sÔsthmatwn tess�rwn exis¸sewn me tèsseric agn¸stouc.

x1 + x2 + x3 + x4 = 3x1 + 2x2 − x3 + 2x4 = 52x1 + 3x2 + x3 − x4 = 4x1 + 2x2 + x3 + x4 = 4

Gr�foume to sÔsthma me thn morf  pin�kwn AX = B ìpou oi pÐnakec A,X kai B eÐnai

A =

1 1 1 11 2 −1 22 3 1 −11 2 1 1

X =

x1

x2

x3

x5

B =

3544

To prìblhma eÐnai na brejeÐ o antÐstrofoc tou pÐnaka A an up�rqeikai na ton pollaplasi�soume me ton pÐnaka B. Gia thn lÔsh enìc tètoiousust matoc èqoun anaptuqjeÐ kai pollèc �llec mèjodoi. Fusik� h kalÔterh

70

apì ìlec eÐnai h mèjodoc tou upologist . 7 BrÐskoume ìti x1 = 1, x2 = 1,x3 = 0 kai x4 = 1.

Ja perigr�youme akoloÔjwc m�llon gia << istorikoÔc lìgouc >> thn mèjodotwn orizous¸n pou isqÔei gia èna sÔsthma n exis¸sewn me n agn¸stouc.

UpologÐzoume pr¸ta tic parak�tw orÐzousec

|A| =

∣∣∣∣∣∣∣∣∣∣

a11 a12 · · · a1n

a21 a22 · · · a2n

a31 a32 · · · a3n... ... ... .........

an1 an2 · · · ann

∣∣∣∣∣∣∣∣∣∣

|A1| =

∣∣∣∣∣∣∣∣∣∣

b1 a12 · · · a1n

b2 a22 · · · a2n

b3 a32 · · · a3n... ... ... .........bn an2 · · · ann

∣∣∣∣∣∣∣∣∣∣

|A2| =

∣∣∣∣∣∣∣∣∣∣

a11 b1 · · · a1n

a21 b2 · · · a2n

a31 b3 · · · a3n... ... ... .........

an1 bn · · · ann

∣∣∣∣∣∣∣∣∣∣

· · · |An| =

∣∣∣∣∣∣∣∣∣∣

a11 a12 · · · b1

a21 a22 · · · b2

a31 a32 · · · b3... ... ... .........

an1 an2 · · · bn

∣∣∣∣∣∣∣∣∣∣

An |A| 6= 0 tìte to sÔsthma èqei mia kai monadik  lÔsh thn akìloujh

x1 =|A1||A| , x2 =

|A2||A| , · · ·xn =

|An||A|

An ìlec oi orÐzousec eÐnai mhdèn tìte to sÔsthma eÐnai aìristo   adÔnatoen¸ an |A| = 0 kai mia toul�qiston apì tic orÐzousec |Aj| eÐnai diaforetik apì to mhdèn |Aj| 6= 0 tìte to sÔsthma eÐnai adÔnato.

Mia �llh mèjodoc eÐnai aut  twn diadoqik¸c apaloif¸n. MporoÔme na an-tikatast soume mia apì tic exis¸seic me ènan grammikì sunduasmì thc exÐsw-shc aut c me opoiad pote �llh. Epidi¸koume fusik� aut  h exÐswsh na eÐnaipio apl .

7Το Mathematica έχει εντολή για την λύση ενός τέτοιου συστήματος. Αφού ορίσουμεπρώτα τους πίνακες A και B βρίσκουμε την λύση με την εντολή:῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[170] := LinearSolve[A, B]Out[170] = {{1}, {1}, {0}, {1}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Η απάντηση σημαίνει ότι x1 = 1, x2 = 1, x3 = 0 και x4 = 1.

71

In[20]:= LinearSolve@A, BD

LinearSolve::nosol : Linear equation encountered

which has no solution. More�

LinearSolve@881, 1, 1, 1<, 81, 1, 1, 1<, 82, 3, 1, -1<,

81, 2, 1, 1<<, 881<, 82<, 84<, 84<<D

Sq ma 3.1: H perÐptwsh enìc sust matoc pou den èqei lÔsh

Gia par�deigma ja lÔsoume to sÔsthma 8

x + y + 2z = 1 2x− y + 2z = −4 4x + y + 4z = −2

AntikajistoÔme thn deÔterh me ton grammikì sunduasmì ε2 − 2ε1 kai thntrÐth me ton grammikì sunduasmì ε3 − 4ε1. Me to sÔmbolo εj sumbolÐzoumethn j exÐswsh. 'Eqoume t¸ra to sÔsthma

x + y + z = 1 − 3y − 2z = −2 − 3y − 4z = 2

AfairoÔme tèloc tic dÔo teleutaÐec kai me to apotèlesma antikajistoÔme thnteleutaÐa. BrÐskoume

x + y + z = 1 − 3y − 2z = −2 − 2z = 4

H teleutaÐa dÐnei z = −2. To apotèlesma autì to antikajistoÔme sthndeÔterh kai brÐskoume y = 2. Tèloc h pr¸th dÐnei x = 1.

H ap�nthsh tou upologist  se mia perÐptwsh pou to sÔsthma den èqeilÔsh faÐnetai sto sq ma (3.1).

3.5 Idiotimèc kai idiodianÔsmata enìc pÐna-ka

Se pollèc efarmogèc katal goume sto akìloujo grammikì omogenèc alge-brikì sÔsthma

AX = λX (3.3)8Ο υπολογιστής δίνει την λύση

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[4] := LinearSolve[{{1, 1, 2}, {2,−1, 2}, {4, 1, 4}}, {−1,−4,−2}]Out[4] = {1, 2,−2}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

72

ìpou A eÐnai ènac tetragwnikìc pÐnakac bajmoÔ n× n.Se autèc tic peript¸seic anazhtoÔme tic timèc tou λ gia tic opoÐec to

sÔsthma èqei mh mhdenik  lÔsh. Oi timèc autèc tou λ onom�zontai idiotimèckai oi antÐstoiqec lÔseic X onom�zontai idiosunart seic.

Gia par�deigma ja broÔme tic idiotimèc kai tic idiosunart seic tou pÐnaka 9

1 −1 11 2 1−1 1 2

BrÐskoume me thn mèjodo tou upologist  ìti oi idiotimèc tou pÐnaka eÐnai

λ1 = 3 λ2 = 1 + i√

2 λ3 = 1− i√

2

kai ta antÐstoiqa idiodianÔsmata

~e1 = (0, 1, 1) ~e2 = (−i√

2,−1, 1) ~e3 = (i√

2,−1, 1)

EÐnai dunatìn to ginìmeno AB dÔo pin�kwn A kai B na eÐnai mhdènqwrÐc na eÐnai mhdèn oÔte o pÐnakac A oÔte o B. An ìmwc ènac apì toucdÔo pÐnakec èqei antÐstrofo tìte aparaÐthta o �lloc ja eÐnai Ðsoc me mhdèn.EÐnai gnwstì ìti sto sÔnolo twn pragmatik¸n arijm¸n isqÔei h sunepagwg ab = ac ⇒ b = c mìno an a 6= 0. Gia touc pÐnakec h sqèsh

AB = AC η A(B − C) = 0

sunep�getai thn sqèsh B = C mìno an |A| 6= 0 dhlad  an o pÐnakac Aèqei antÐstrofo. Kat� sunèpeia up�rqei mh mhdenik  lÔsh X thc exÐswshcidiotim¸n

AX = λX η (A− λ11)X = 0

gia ekeÐnec tic timèc tou λ pou o pÐnakac A − λ11 den èqei antÐstrofo eÐnaidhlad  lÔseic thc exÐswshc

|A− λ11| = 0 (3.4)

pou onom�zetai qarakthristik  exÐswsh tou pÐnaka A.9Ο υπολογισμός των ιδιοτιμών και των αντίστοιχων ιδιοδιανυσμάτων ενός πίνακα

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[12] := Eigensystem[{{1,−1, 1}, {1, 2, 1}, {−1, 1, 2}}]Out[12] = {{3, 1 + i

√2, 1− i

√2}, {{0, 1, 1}, {−i

√2,−1, 1}, {i√2,−1, 1}}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

73

Parat rhsh: EÐnai fanerì ìti an antikatast soume sthn parap�nwexÐswsh to λ me ton pÐnaka A h exÐswsh ikanopoieÐtai. Upojètoume ìti hqarakthristik  exÐswsh eÐnai

P (λ) = |A− λ11| = anλn + an−1λn−1 + · · ·+ a1λ + a0 (3.5)

Jètoume λ → A kai brÐskoume P (A) = anAn+an−1A

n−1+ · · ·+a1A+a0 = 0H exÐswsh aut  dÐnei ènan trìpo na upologÐsoume ton antÐstrofo pÐnaka

tou A. BrÐskoume

A−1 = −(an/a0)An−1 − (an−1/a0)A

n−2 + · · · − (a1/a0)

ParathroÔme ìti an jèsoume λ = 0 sthn qarakthristik  exÐswsh (3.5)brÐskoume a0 = |A|.

Gia na lÔsoume tètoia probl mata idiotim¸n brÐskoume pr¸ta tic idio-timèc λj lÔnontac thn qarakthristik  exÐswsh idiotim¸n (3.4) tou pÐnakaA. Katìpin antikajistoÔme k�je idiotim  sto omogenèc sÔsthma (3.3) kai lÔ-noume to sÔsthma pou èqei profan¸c mia toul�qiston mh mhdenik  lÔsh. Anse mia idiotim  up�rqoun poll� grammik� anex�rthta idiodianÔsmata h idiotim aut  onom�zetai ekfulismènh.

H qarakthristik  exÐswsh idiotim¸n gia ènan pÐnaka t�xhc n×n eÐnai ènapolu¸numo bajmoÔ n. 'Ara èqei n sto pl joc lÔseic. EÐnai ìmwc dunatìnk�poiec apì autèc na eÐnai pollaplèc lÔseic me pollaplìthta èstw aj, tìteto pl joc twn grammik� anexart twn idiodianusm�twn thc eÐnai ≤ aj. K�jegrammikìc sunduasmìc twn dianusm�twn aut¸n eÐnai epÐshc idiodi�nusma thcÐdiac idiotim c.

An oi idiotimèc λj enìc pÐnaka eÐnai diaforetikèc metaxÔ touc an� dÔotìte ta antÐstoiqa idiodianÔsmata Xj eÐnai grammik� anex�rthta. An k�poiaidiotim  èqei pollaplìthta megalÔterh apì èna, tìte ta grammik� anex�rthtaidiodianÔsmata thc idiotim c aut c kai ta grammik� anex�rthta idiodianÔsmatapou antistoiqoÔn stic �llec idiotimèc eÐnai grammik� anex�rthta.

An ènac pÐnaka eÐnai diag¸nioc tìte ta mh mhdenik� stoiqeÐa thc kurÐacdiagwnÐou tou eÐnai oi idiotimèc touc. H prìtash aut  dÐnei kai mÐa mèjodo giana broÔme tic idiotimèc enìc pÐnaka.

DÔo tetragwnikoÐ pÐnakec A kai B eÐnai ìmoioi an sundèontai me tonakìloujo metasqhmatismì omoiìthtac

B = S−1AS

DÔo ìmoioi pÐnakec èqoun Ðsec orÐzousec kai tic Ðdiec idiotimèc. Epomènwc anbroÔme ènan antistrèyimo pÐnaka S ètsi ¸ste o pÐnakac B na eÐnai diag¸nioctìte brÐskoume polÔ eÔkola tic idiotimèc tou pÐnaka A. Ja lème ìti o pÐnakacA eÐnai diagwnopoi simoc.

74

Gia thn apìdeixh pollaplasi�zoume thn exÐswsh idiotim¸n apì ta arister�me ton pÐnaka S−1 kai apì dexi� me ton pÐnaka S. BrÐskoume

AX = λX =⇒ S−1AXS = λS−1XS =⇒ (S−1AS)(S−1XS) = λ(S−1XS)

H teleutaÐa sqèsh mac lèei ìti o pÐnakac B = S−1AS èqei idiotim  to λkai idiodi�nusma to Y = S−1XS.

Gia na eÐnai ènac pÐnakac bajmoÔ n × n diagwnopoi simoc prèpei na è-qei n− idiodianÔsmata grammik� anex�rthta. ApodeiknÔetai ìti o pÐnakacS èqei st lec ta idiodianÔsmata aut�. An èqei sunolik� ligìtera apì n−idiodianÔsmata tìte den diagwnopoieÐtai.

Gia par�deigma ja broÔme tic idiotimèc kai ta antÐstoiqa idiodianÔsmata toupÐnaka

A =

(−3 −23 4

)

H qarakthristik  exÐswsh tou pÐnaka eÐnai

|A− λ11| =∣∣∣∣−3− λ −2

3 4− λ

∣∣∣∣ = (−3− λ)(4− λ) + 2 ∗ 3 = λ2 − λ− 6 = 0

pou dÐnei tic lÔseicλ1 = −2 λ2 = 3

Gia thn pr¸th idiotim    exÐswsh idiotim¸n gÐnetai(−1 −2

3 6

)(x1

x2

)=

(00

)=⇒ x1 = −2x2

Epilègoume thn aploÔsterh tim  x2 = 1 kai brÐskoume to idiodi�nusma

X1 =

(−21

)

Gia thn deÔterh idiotim  brÐskoume(−6 −2

3 1

)(x1

x2

)=

(00

)=⇒ x2 = −3x1

Epilègoume p�li thn aploÔsterh tim  x1 = 1 kai brÐskoume to idiodi�nusma

X2 =

(1−3

)

SqhmatÐzoume ton akìloujo pÐnaka S pou èqei st lec ta parap�nw idio-dianÔsmata

S =

(−2 11 −3

)

75

H orÐzousa tou pÐnaka autoÔ eÐnai |S| = 5 kai o antÐstrofoc pÐnakac eÐnai

S−1 =1

5

(−3 −1−1 −2

)

Tèloc brÐskoume ton ìmoio pÐnaka

B = S−1AS =1

5

(−3 −1−1 −2

) (−3 −23 4

)(−2 11 −3

)=

1

5

(−3 −1−1 −2

)(4 3−2 −9

)=

(−2 00 3

)

pou eÐnai ènac diag¸nioc pÐnakac me diag¸nia stoiqeÐa tic idiotimèc tou dedomè-nou pÐnaka A.

76

3.6 Ask seic

'Askhsh 1.Gia na broÔme ton antÐstrofo enìc tetragwnikoÔ pÐnaka A = (ajk) sqh-

matÐzoume ton pÐnaka A∗ = (a∗jk) ìpou a∗jk = (−1)(j+k)Akj kai Akj eÐnai horÐzousa tou pÐnaka pou prokÔptei an apaleÐyoume thn k− gramm  kai thnj− st lh pou brÐsketai to stoiqeÐo akj. O antÐstrofoc tou pÐnaka A eÐnai:

A−1 =1

|A|A∗

Na apodeÐxete me èna par�deigma ton isqurismì thc �skhshc.Apìdeixh: Ja apodeÐxoume thn �skhsh me ton pÐnaka:

A =

1 2 31 5 67 8 9

BrÐskoume pr¸ta thn orÐzousa tou pÐnaka A. AnaptÔssoume thn orÐzousawc proc thn pr¸th gramm .

∣∣∣∣∣∣

1 2 31 5 67 8 9

∣∣∣∣∣∣= 1 ∗

∣∣∣∣5 68 9

∣∣∣∣− 2 ∗∣∣∣∣1 67 9

∣∣∣∣ + 3 ∗∣∣∣∣1 57 8

∣∣∣∣ =

1 ∗ (5 ∗ 9− 6 ∗ 8)− 2 ∗ (1 ∗ 9− 6 ∗ 7) + 3 ∗ (1 ∗ 8− 5 ∗ 7) = −18

BrÐskoume met� ta stoiqeÐa tou pÐnaka A∗. 10 'Eqoume

a∗11 = A11 =

∣∣∣∣5 68 9

∣∣∣∣ = −3 a∗12 = −A21 =

∣∣∣∣2 38 9

∣∣∣∣ = 6

a∗13 = A31 =

∣∣∣∣2 35 6

∣∣∣∣ = −3 a∗21 = −A12 =

∣∣∣∣1 67 9

∣∣∣∣ = 33

a∗22 = A22 =

∣∣∣∣1 37 9

∣∣∣∣ = −12 a∗23 = −A32 =

∣∣∣∣1 31 6

∣∣∣∣ = −3

10Μπορούμε να βρούμε τα στοιχεία a∗jk όλα μαζί. Η εντολή για τις ελάσσονες ορίζουσεςείναι Minors[A][[j, k]] . Βρίσκουμε:῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[32] := Do[Print[(−1)j+kMinors[A][[3− k + 1, 3− j + 1]]], {j, 1, 3}, {k, 1, 3}]FromIn[32] := −3 6 − 3 33 − 12 − 3 − 27 6 3

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

77

a∗31 = A13 =

∣∣∣∣1 57 8

∣∣∣∣ = −27 a∗32 = −A23 =

∣∣∣∣1 27 8

∣∣∣∣ = 6

a∗33 = −A33 =

∣∣∣∣1 21 5

∣∣∣∣ = 3

Tèloc o antÐstrofoc pÐnakac eÐnai o ex c 11

A−1 = − 1

18

−3 6 −333 −12 −3−27 6 3

=

1

6

1 −2 1−11 4 19 −2 −1

'Askhsh 2.

Na brejoÔn oi idiotimèc kai oi idiosunart seic tou pÐnaka 12

A =

1 −2 01 5 11 0 2

LÔsh: Oi idiotimèc kai oi idiosunart seic enìc pÐnaka eÐnai oi lÔseic thc

11Ο υπολογιστής δίνει τον αντίστροφο ενός πίνακα απλά και γρήγορα.῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[37] := Inverse[{{1, 2, 3}, {1, 5, 6}, {7, 8, 9}}]//MatrixForm

Out[37]//MatrixForm =

16 − 1

316

− 116

23

16

32 − 1

3 − 16

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Η επαλήθευση έπεται

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[8] := Dot[{{1, 2, 3}, {1, 5, 6}, {7, 8, 9}},{{1/6,−1/3, 1/6}, {−11/6, 2/3, 1/6}, {3/2,−1/3,−1/6}}]//MatrixForm

Out[8]//MatrixForm =

1 0 00 1 00 0 1

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Σημειώστε ότι οι πράξεις των πινάκων δεν εκτελούνται στο Mathematica όταν οι πίνακες

ορισθούν σε MatrixForm.12Ο υπολογιστής δίνει με μια εντολή τις ιδιοτιμές και τις ιδιοσυναρτήσεις.

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[2] := Eigenvalues[{{1,−2, 0}, {1, 5, 1}, {1, 0, 2}}]Eigenvectors[{{1,−2, 0}, {1, 5, 1}, {1, 0, 2}}]Out[2] = {4, 3, 1}Out[3] = {{2,−3, 1}, {1,−1, 1}, {−1, 0, 1}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

78

akìloujhc exÐswshc idiotim¸n.

AX = λX η

1 −2 01 5 11 0 2

xyz

= λ

xyz

=⇒

(A− λ11)X = 0 η

1− λ −2 01 5− λ 11 0 2− λ

xyz

=

000

H akìloujh exÐswsh onom�zetai qarakthristik  exÐswsh tou pÐnaka A

|A− λ11| = 0 η

∣∣∣∣∣∣

1− λ −2 01 5− λ 11 0 2− λ

∣∣∣∣∣∣= 0

LÔnoume thn exÐswsh wc proc thn par�metro λ. AnaptÔssoume thn orÐzousawc proc thn pr¸th gramm 

|A− λ11| = (1− λ)

∣∣∣∣5− λ 1

0 2− λ

∣∣∣∣ + 2

∣∣∣∣1 11 2− λ

∣∣∣∣ = (1− λ)(5− λ)(2− λ)+

(2− λ− 1) = (1− λ)(5− λ)(2− λ) + 2(1− λ) = (1− λ)[(5− λ)(2− λ) + 2]

= (1− λ)(10− 7λ + λ2 + 2) = (1− λ)(λ2 − 7λ + 12) = (1− λ)(λ− 3)(λ− 4)

Epomènwc h lÔsh thc qarakthristik c exÐswshc eÐnai 13

λ = 1 λ = 3 λ = 4

Gia thn idiotim  λ = 1 h exÐswsh idiotim¸n gÐnetai

0 −2 01 4 11 0 1

xyz

=

000

=⇒

−2y = 0x + 4y + z = 0

x + z = 0=⇒

y = 0x = −z

'Ara to antÐstoiqo idiodi�nusma eÐnai

~e1 =

−z0z

= z

−101

13Η λύση της χαρακτηριστικής εξίσωσης῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[15] := Solve[Det[{{1,−2, 0}, {1, 5, 1}, {1, 0, 2}}−

λ{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}] == 0, λ]Out[15] = {{λ → 1}, {λ → 3}, {λ → 4}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

79

EÐnai profanèc apì thn exÐswsh idiotim¸n ìti an to di�nusma ~ej eÐnai lÔshthc exÐswshc idiotim¸n tìte kai to di�nusma ξ~ej eÐnai epÐshc lÔsh. 'Ara giato parap�nw idiodi�nusma mporoÔme na epilèxoume to z ¸ste to antÐstoiqoidiodi�nusma na èqei apl  èkfrash. Epilègoume z = 1 kai to idiodi�nusmagÐnetai

~e1 =

−101

Gia thn idiotim  λ = 3 h exÐswsh idiotim¸n gÐnetai−2 −2 01 2 11 0 −1

xyz

=

000

=⇒

−2x− 2y = 0x + 2y + z = 0

x− z = 0=⇒

y = −xx + 2y + z = 0

z = x=⇒

y = −xx− 2x + x = 0

z = x=⇒

y = −x

z = x

'Ara to antÐstoiqo idiodi�nusma gia x = 1 eÐnai

~e2 =

1−11

Tèloc gia thn idiotim  λ = 4 h exÐswsh idiotim¸n gÐnetai−3 −2 01 1 11 0 −2

xyz

=

000

=⇒

−3x− 2y = 0x + y + z = 0x− 2z = 0

=⇒

y = −32x

x + y + z = 0z = 1

2x

=⇒y = −3

2x

x− 32x + 1

2x = 0

z = 12x

=⇒y = −3

2x

z = 12x

'Ara to antÐstoiqo idiodi�nusma gia x = 2 eÐnai

~e3 =

2−31

ParathroÔme ìti ta trÐa parap�nw dianÔsmata eÐnai grammik� anex�rthta. 14

14Η απόδειξη ότι τρία διανύσματα είναι γραμμικά ανεξάρτητα῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[12] := Solve[c1{2,−3, 1}+ c2{1,−1, 1}+ c3{−1, 0, 1} == {0, 0, 0}

, {c1, c2, c3}]Out[12] = {{c1 → 0, c2 → 0, c3 → 0}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

80

Kataskeu�soume ton pÐnaka S pou èqei st lec ta idiodianÔsmata aut�

2−31

1−11

−101

−→ S =

2 1 −1−3 −1 01 1 1

O ìmoioc diag¸nioc pÐnakac tou parap�nw pÐnaka eÐnai ènac diag¸nioc pÐ-nakac me diag¸nia stoiqeÐa tic idiotimèc.

B =

2 1 −1−3 −1 01 1 1

−1

1 −2 01 5 11 0 2

2 1 −1−3 −1 01 1 1

=

4 0 00 3 00 0 1

Oi parap�nw pr�xeic èginan me ton upologist . 15

'Askhsh 3.Na upologÐsete thn orÐzousa tou pÐnaka

A =

1 a a2

1 b b2

1 c c2

LÔsh: EÐnai gnwstì ìti h orÐzousa enìc pÐnaka den all�zei an an-tikatast soume mia gramm  me ènan grammikì sunduasmì thc gramm c aut cme mia �llh. Efarmìzoume thn idiìthta aut  kai afairoÔme apì thn deÔterhkai thn trÐth gramm , thn pr¸th kai me ta apotèlesmata antikajistoÔme thndeÔterh kai thn trÐth gramm  antistoÐqwc. BrÐskoume

Det(A) =

∣∣∣∣∣∣

1 a a2

1 b b2

1 c c2

∣∣∣∣∣∣=

∣∣∣∣∣∣

1 a a2

0 b− a b2 − a2

0 c− a c2 − a2

∣∣∣∣∣∣=

∣∣∣∣∣∣

1 a a2

0 b− a (b− a)(b + a)0 c− a (c− a)(c + a)

∣∣∣∣∣∣AkoloÔjwc bg�zoume koinì par�gonta to b− a apì thn deÔterh gramm  kaiton par�gonta c−a apì thn trÐth. Met� afairoÔme apì thn trÐth gramm  thndeÔterh kai antikajistoÔme me to apotèlesma thn trÐth gramm . BrÐskoume

Det(A) = (b− a)(c− a)

∣∣∣∣∣∣

1 a a2

0 1 b + a0 1 c + a

∣∣∣∣∣∣= (b− a)(c− a)

∣∣∣∣∣∣

1 a a2

0 1 b + a0 0 c− b

∣∣∣∣∣∣15Ο πολλαπλασιασμος των πινάκων S−1BS.

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[24] := Dot[(Inverse[S]), B, S]Out[24] = {{4, 0, 0}, {0, 3, 0}, {0, 0, 1}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

81

In[53]:= A =

i

k

jjjjjjjjjjjjj

1 a a2 a3

1 b b2 b3

1 c c2 c3

1 d d2 d3

y

{

zzzzzzzzzzzzz

;

In[54]:= Simplify@Det@ADD

Out[54]= Ha - bL Ha - cL Hb - cL Ha - dL Hb - dL Hc - dL

Sq ma 3.2: H orÐzousa enìc pÐnaka 4× 4 tou Ðdiou tÔpou me thn �skhsh.

H orÐzousa tou teleutaÐou pÐnaka upologÐzetai eÔkola kai telik� brÐskoume

Det(A) = (b− a)(c− a)(c− b)

H orÐzousa upologÐzetai pio eÔkola me thn mèjodo tou upologist . Gia miamegalÔterh orÐzousa tou idÐou tÔpou brÐskoume sto sq ma (3.2) thn orÐzousatou pÐnaka.

'Askhsh 4.DÐnontai oi akìloujoi pÐnakec tÔpou 2× 2

11 =

(1 00 1

)s1 =

(0 −11 0

)

Na brejoÔn oi perittèc kai oi �rtiec dun�meic tou pÐnaka s2n+11 kai s2n

1 giak�je fusikì arijmì n kai na upologisjeÐ h ekjetik  sun�rthsh tou pÐnakaapì to akìloujo an�ptugma kata Tèulor.

O(φ) = e−φs1 = 11 +∞∑

n=1

(−1)n

n!φnsn

1

Na apodeÐxete tèloc ìti O(φ)O(θ) = O(φ + θ).LÔsh: UpologÐzoume tic dun�meic tou pÐnaka. BrÐskoume

s21 = s1 ∗ s1 =

(0 −11 0

)(0 −11 0

)=

(−1 00 −1

)= −11

s31 = s1 ∗ s2

1 = −s1 s41 = s2

1 ∗ s21 = (−11)(−11) = 11

Apo tic parap�nw sqèseic sumperaÐnoume ìti oi perittèc dun�meic tou pÐ-naka eÐnai an�logec tou pÐnaka s1 kai oi �rtiec dun�meic eÐnai an�logec toupÐnaka 11. 'Eqoume

s2n+11 = (−1)ns1 s2n

1 = (−1)n11

82

Epomènwc brÐskoume

e−φs1 = 11 +∞∑

n=1

(−1)n

n!φnsn

1 = 11∞∑

n=1

(−1)n

(2n)!φ(2n) − s1

∞∑n=0

(−1)n

(2n + 1)!φ(2n+1)

Ta parap�nw ajroÐsmata eÐnai ta gnwst� anaptÔgmata kat� Tèulor tousunhmitìnou kai tou hmitìnou. 'Ara

O(φ) = 11 cos φ− s1 sin φ =

(cos φ sin φ− sin φ cos φ

)

O zhtoÔmenoc pÐnakac eÐnai o pÐnakac pou perigr�fei mia peristrof  ston q¸rotwn dÔo diast�sewn.

ApodeiknÔoume t¸ra thn deÔterh prìtash

O(φ)O(θ) =

(cos φ sin φ− sin φ cos φ

)(cos θ sin θ− sin θ cos θ

)=

(cos φ cos θ − sin φ sin θ cos φ sin θ + sin φ cos θ− sin φ cos θ − cos φ sin θ − sin φ sin θ + cos φ cos θ

)=

(cos (φ + θ) sin (φ + θ)− sin (φ + θ) cos (φ + θ)

)= O(φ + θ)

ìpou qrhsimopoi same gnwstoÔc tÔpouc thc trigwnometrÐac.To apotèlesma faÐnetai profanèc ìmwc den eÐnai diìti genik� ta dÔo ek-

jetik� den eÐnai Ðsa. Gia dÔo tuqìntec pÐnakec A kai B isqÔei o tÔpoc

eAeB = eA+B+ 12[A,B]+ 1

12[A−B,[A,B]]+···

ìpou o metajèthc twn pin�kwn A, B orÐzetai apì thn sqèsh

[A,B] = AB −BA

83

84

Kef�laio 4

Seirèc Fouriè

4.1 O dianusmatikìc q¸rocOrismìc: 'Ena sÔnolo V onom�zetai dianusmatikìc   grammikìc q¸roc

epÐ enìc s¸matoc F (= R η C) , ìtan eÐnai dunatìn na oristoÔn dÔo pr�xeicsÔnjeshc, h prìsjesh kai o (exwterikìc) pollaplasiasmìc. An to s¸ma FeÐnai oi pragmatikoÐ   oi migadikoÐ arijmoÐ, o dianusmatikìc q¸roc onom�zetaiantÐstoiqa pragmatikìc   migadikìc dianusmatikìc q¸roc.

Orismìc: Ta dianÔsmata {φ1, φ2, · · · , φn} onom�zontai grammik�anex�rthta, an h isqÔei h sqèsh

n∑i=1

aiφi = 0 ⇐⇒ ai = 0 ∀i = 1, 2, · · · , n

Orismìc: 'Ena �peiro sÔnolo dianusm�twn eÐnai grammik¸c anex�rthto,an k�je peperasmèno uposÔnolo tou eÐnai grammik¸c anex�rthto.

Orismìc: To sÔnolo twn dianusm�twn {φ1, φ2, · · · φn, · · · }eÐnai èna sÔnolo gennhtìrwn, an k�je di�nusma tou q¸rou mporeÐ na grafeÐsan grammikìc sunduasmìc twn φi

x =∞∑i=1

aiφi

Orismìc: 'Ena sÔnolo B onom�zetai b�sh tou q¸rou, an eÐnai gram-mik� anex�rthto kai sÔnolo gennhtìrwn. To pl joc twn stoiqeÐwn tou Bonom�zetai di�stash tou q¸rou. 'Enac grammikìc q¸roc pou den èqei peperas-mènh di�stash, onom�zetai apeÐrwn diast�sewn.

Par�deigma: 1. To sÔnolo twn diatetagmènwn n− �dwn apì arijmoÔc:

V = {x / x = (x1, x2, · · · , xn), xi ∈ R}

85

Oi pr�xeic sÔnjeshc orÐzontai apì tic sqèseic:

x + φ = (x1 + φ1, x2 + φ2, · · · , xn + φn) και ax = (ax1, a2x, · · · , axn)

O q¸roc autìc onom�zetai n− di�statoc EukleÐdeioc q¸roc. 'Eqoume  dhmelet sei ton 3− di�stato dianusmatikì q¸ro R3.

Par�deigma: 2. To sÔnolo twn �peirwn akolouji¸n apì arijmoÔc

`2(∞) =

{ξ / ξ = (ξ1, ξ2, · · · , ξn, · · · ) ξi ∈ R oπoυ

∞∑

k=1

|ξk|2 < ∞}

H prìsjesh kai o pollaplasiasmìc orÐzontai ìpwc kai sto prohgoÔmenopar�deigma. O q¸roc autìc eÐnai apeÐrwn diast�sewn.

Par�deigma: 3. To sÔnolo twn suneq¸n sunart sewn miac prag-matik c metablht c t, me pr�xeic orismènec apì tic sqèseic

1) (f + g)(t) = f(t) + g(t) ∀f , g ∈ C(R)

2) (af)(t) = af(t) ∀f ∈ C(R) ∀a ∈ CTo sÔnolo autì sumbolÐzetai me C(R).

Par�deigma: 4. To sÔnolo twn sunart sewn miac metablht c t,gia tic opoÐec to olokl rwma kat� Lempègk

∫R |f(t)|2dt up�rqei kai eÐnai

peperasmèno. H prìsjesh kai o pollaplasiasmìc orÐzontai ìpwc kai sto pro-hgoÔmeno par�deigma. O q¸roc autìc sumbolÐzetai me L2(R) kai anafèretaisun jwc san o q¸roc twn tetragwnik� oloklhr¸simwn sunart sewn.

4.2 Eswterikì ginìmenoOrismìc: 'Enac migadikìc dianusmatikìc q¸roc V onom�zetai q¸roc

eswterikoÔ ginomènou, tìte kai mìno tìte ìtan eÐnai efodiasmènoc me miaapeikìnish

(, ) : V × V 7−→ (C)

me tic idiìthtec

1) (f, g) = (g, f)∗ ∀f , g ∈ V

2) (af + bg, h) = a∗(f, h) + b∗(g, h) ∀f, g, h ∈ V ∀a , b ∈ C3) (f, f) ≥ 0 ∀f ∈ V και (f, f) = 0 ⇐⇒ f = 0

Me asterÐsko sumbolÐzoume ton suzug  tou migadikoÔ arijmoÔ. Gia ton mi-gadikì arijmì z ∈ C èqoume

z = ρeiφ = a + ib z∗ = ρe−iφ = a− ib ρ, φ, a, b ∈ R

86

To eswterikì ginìmeno orÐzetai kai gia pragmatikoÔc dianusmatikoÔc q¸rouc,ìpou isqÔoun oi Ðdiec idiìthtec qwrÐc touc asterÐskouc.

Je¸rhma: H st�jmh   norm enìc q¸rou me eswterikì ginìmenoorÐzetai apì th sqèsh

‖χ‖ = (χ, χ)1/2

Parat rhsh: Se èna dianusmatikì q¸ro mporeÐ na orÐsoume mia normanex�rthta apì to eswterikì ginìmeno. 'Enac q¸roc eswterikoÔ ginomènou eÐ-nai plousiìteroc se plhroforÐec apì ènan stajmhtì dianusmatikì q¸ro, diìtiektìc apì to m koc enìc dianÔsmatoc, mac dÐnei kai thn gwnÐa dÔo dianusm�twn.

H gwnÐa metaxÔ dÔo dianusm�twn dÐnetai apì thn sqèsh:

cos θ =|(χ, ψ)|‖χ‖‖ψ‖ ≤ 1

H parap�nw anisìthta eÐnai gnwst  san anisìthta tou Sbartc.Par�deigma: 1. O n− di�statoc EukleÐdeioc dianusmatikìc q¸roc

gÐnetai q¸roc eswterikoÔ ginomènou kai to eswterikì ginìmeno orÐzetai apìth sqèsh

(χ, ψ) =n∑

j=1

χ∗jψj

EÐnai mia epèktash tou gnwstoÔ eswterikoÔ ginomènou tou trisdi�statouq¸rou R3.

Par�deigma: 2. Ston q¸ro l2(∞) to eswterikì ginìmeno orÐzetaiomoÐwc apì th sqèsh

(χ, ψ) =∞∑

j=1

χ∗jψj

Sthn perÐptwsh ìmwc aut  prèpei na epib�lloume thn sunj kh

‖χ‖2 =∞∑

j=1

|χj|2 < ∞

pou eÐnai anagkaÐa ¸ste h st�jmh tou q¸rou na eÐnai peperasmènh.Par�deigma: 3. Ston q¸ro L2(R) to eswterikì ginìmeno orÐzetai

apì thn sqèsh

(χ, ψ) =

∫

Rχ∗(t)ψ(t)dt

H sunj kh pou prèpei na isqÔei ed¸, ¸ste na orÐzetai to m koc enìc dianÔs-matoc, eÐnai

‖χ‖2 =

∫

R|χ(t)|2dt < ∞

87

O q¸roc autìc eÐnai o q¸roc twn tetragwnik� oloklhr¸simwn (kat� Lempègk)sunart sewn. An to pedÐo orismoÔ twn sunart sewn eÐnai to di�sthma [−π, π]tìte gr�foume L2([−π, π]). O q¸roc autìc eÐnai ènac q¸roc QÐlmpert dhlad ènac pl rhc dianusmatikìc q¸roc me eswterikì ginìmeno. O q¸roc èqei �peirhdi�stash.

4.3 Seirèc FourièOrismìc: 'Ena sÔnolo dianusm�twn φ1, φ2, · · · , φn onom�zetai orjokanon-ikì sÔnolo, tìte kai mìno tìte ìtan:

(φi, φj) =

∫ b

a

φ∗i (x)φj(x)dx = δij =

{0 gia i 6= j

1 gia i = j

Dhlad  ta dianÔsmata aut� eÐnai an� dÔo orjog¸nia kai èqoun m koc thn mon�-da. Mia orjokanonik  b�sh enìc dianusmatikoÔ q¸rou eswterikoÔ ginomènou,eÐnai mia b�sh pou perièqei orjokanonik� dianÔsmata.

Parat rhsh: Se k�je n− di�stato dianusmatikì q¸ro mporoÔmep�nta na kataskeu�soume mia orjokanonik  b�sh me thn mèjodo Gkram - Smit.

Je¸rhma: An èna di�nusma f analÔetai san grammikìc sunduasmìctwn orjokanonik¸n dianusm�twn φ1, φ2, · · · , φn, dhlad 

f =n∑

j=1

aiφi

tìte oi suntelestèc aj dÐnontai apì thn sqèsh

aj = (φj, f)

kai onom�zontai suntelestèc Fouriè. Pr�gmati

(φi, f) = (φi,

n∑j=1

ajφj) =n∑

j=1

aj(φi, φj) =n∑

j=1

ajδij = ai

Dhlad  èqoume thn an�ptuxh

f =n∑

j=1

(φj, f)φj

Parat rhsh: H prìblhma thc an�ptuxhc miac sun�rthshc se seir�Fouriè eÐnai sthn pragmatikìthta tautìshmo me to prìblhma thc an�ptuxhcenìc dianÔsmatoc wc proc k�poia orjokanonik  b�sh.

88

Stic epìmenec paragr�fouc ja epekteÐnoume thn jewrÐa twn seir¸n Fourièse q¸rouc me �peirh di�stash. H diafor� apì touc sunhjismènouc dianus-matikoÔc q¸rouc eÐnai ìti ed¸ to pl joc twn dianusm�twn thc b�shc eÐnai(arijm simo) �peiro. 'Ara gia na gr�youme tic an�logec sqèseic gia tètoioucq¸rouc eÐnai fanerì ìti qreiazìmaste k�poia krit ria gia thn sÔgklish.

Orismìc: To sÔnolo V twn sunart sewn lème ìti apoteloÔn ènasunarthsiakì q¸ro. SumbolÐzoume me L2(a, b) to sÔnolo ìlwn twn fragmèn-wn migadik¸n sunart sewn f(x) pou eÐnai tetragwnik� oloklhr¸simec se ènadi�sthma a ≤ x ≤ b. To eswterikì ginìmeno kai to m koc twn dianusm�twnorÐzontai antistoÐqwc apì tic sqèseic

(f(x), g(x)) =

∫ b

a

f ∗(x)g(x)dx ‖f‖2 =

∫ b

a

|f(x)|2dx < 0

H teleutaÐa anisìthta shmaÐnei ìti to olokl rwma sugklÐnei. O q¸roc autìceÐnai apeÐrwn diast�sewn.

To sÔnolo twn apeÐrwn sunart sewn {φ1(x), φ2(x), · · · } onom�zetaiorjokanonikì ìtan

(φi, φj) = δij (4.1)Je¸rhma: jewroÔme thn an�ptuxh thc sun�rthshc f ∈ V sto

parak�tw �peiro �jroisma

f(x) = c1φ1(x) + c2φ2(x) + · · · cnφn(x) + · · · =∞∑

j=1

cjφj(x) (4.2)

An h parap�nw seir� sugklÐnei omoiìmorfa sth sun�rthsh f(x) sto di�sthma[a, b] , tìte oi stajerèc an�ptuxhc cn dÐnontai apì thn sqèsh

cn = (φn, f) =

∫ b

a

f(x)φ∗n(x)dx

Apìdeixh: Pollaplasi�zoume thn seir� (4.2) me thn fragmènh sun�rthshφ∗n(x). H seir� pou prokÔptei sugklÐnei epÐshc omoiìmorfa kai epomènwcmporoÔme na oloklhr¸soume ìro proc ìro apì to a wc to b. BrÐskoume

∫ b

a

f(x)φ∗n(x)dx =∞∑i=1

ci

∫ b

a

φi(x)φ∗n(x)dx =∞∑i=1

ciδin = cn

Oi arijmoÐ cn onom�zontai stajerèc Fouriè thc f(x) wc proc to orjokanon-ikì sÔnolo {φn(x)} . H seir�

∑cnφn(x) lègetai seir� Fouriè antÐstoiqh thc

f(x) wc proc to sÔnolo autì kai gr�foume:

f(x) =∞∑

j=1

cjφj(x) a ≤ x ≤ b

89

anex�rthta an h seir� sugklÐnei h ìqi proc thn f(x).Je¸rhma: An φ1, φ2, · · · , φn, · · · eÐnai èna orjokanonikì sÔnolo tìte

isqÔei h anisìthta tou Mpèsel∞∑

j=1

|(φj, f)|2 =∞∑

j=1

|cj|2 ≤ ‖f‖2 =

∫ b

a

|f(x)|2dx

H seir� twn tetrag¸nwn twn stajer¸n Fouriè k�poiac sun�rthshc f(x)sugklÐnoun se èna �jroisma pou den uperbaÐnei to m koc thc f(x).

Epeid  to �peiro �jroisma thc anisìthtac tou Mpèsel sugklÐnei, oi ìroi|cn|2 sqhmatÐzoun profan¸c mia mhdenik  akoloujÐa. 'Ara èqoume thn sqèsh

limn→∞

cn = 0

pou onom�zetai je¸rhma RÐman.Orismìc: 'Ena orjokanonikì sÔnolo onom�zetai pl rec, an den up�rqei

�llh sun�rthsh tou F orjog¸nia proc ìlec tic sunart seic tou sunìlou.Je¸rhma: 'Ena orjokanonikì sÔnolo φ1, φ2, · · · eÐnai pl rec, tìte

kai mìno tìte, ìtan isqÔoun oi akìloujec isodÔnamec prot�seic

(φk, x) = 0 ∀k = 1, 2, · · · =⇒ x = 0

limn→∞

‖f −n∑

k=1

φk(φk, f)‖ = 0 ∀ f ∈ V

MporoÔme dhlad  na gr�youme

f =∞∑

k=1

φk(φk, f) ∀f ∈ V

ìpou to �peiro �jroisma sugklÐnei wc proc to norm tou q¸rou.

(f, g) =∞∑

k=1

(f, φk)(φk, g) ∀f , g ∈ V

pou onom�zetai tautìthta tou P�rsebal. Gia f = g h tautìthta tou P�rse-bal gr�fetai

‖f‖ =∞∑

k=1

|(φk, f)|2 ∀f ∈ V

Par�deigma: 1. Gia to q¸ro L2[−π, π] to sÔnolo twn suneq¸npragmatik¸n sunart sewn

φ0 =1√2π

, φn =1√π

cos nx, ψn =1√π

sin nx, n = 1, 2, · · ·

90

eÐnai mÐa orjokanonik  b�sh tou q¸rou. k�je sun�rthsh tou q¸rou eÐnai èna�peiro �jroisma suneq¸n sunart sewn thc morf c:

f(x) =a0

2+

∞∑n=0

an cos nx +∞∑

n=1

bn sin nx

Oi suntelestèc Fouriè an kai bn dÐnontai apì tic sqèseic:

an =1

π

∫ π

−π

f(x) cos nxdx, bn =1

π

∫ π

−π

f(x) sin nxdx

H anisìthta tou Mpèsel gr�fetai

a0

2+

∞∑n=1

(a2n + b2

n) ≤ 1

π

∫ π

−π

|f(x)|2dx

h opoÐa sthn ousÐa eÐnai isìthta, h tautìthta tou P�rsebal. To je¸rhma touRÐman gr�fetai

limn→∞

∫ ∞

−∞f(x) cos nx = lim

n→∞

∫ ∞

−∞f(x) sin nx = 0

Par�deigma: 2. Gia ton Ðdio q¸ro, to sÔnolo twn suneq¸n migadik¸nsunart sewn

φk(x) =1√2π

eikx

eÐnai mÐa epÐshc mia orjokanonik  b�sh tou q¸rou. k�je sun�rthsh tou q¸roueÐnai èna �peiro �jroisma suneq¸n sunart sewn thc morf c:

f(x) =∞∑

k=−∞cke

ikx (4.3)

Oi suntelestèc Fouriè ck dÐnontai apì tic sqèseic:

ck = (f(x), φk(x)) =1

π

∫ π

−π

f(x)φ∗k(x)dx =1

π

∫ π

−π

f(x)e−ikxdx

Parat rhsh: H migadik  aut  èkfrash thc f(x) prokÔptei eÔkolakai apì tic tautìthtec tou 'Ouler.

eikx = cos kx + i sin kx e−ikx = cos kx− i sin kx

cos kx =1

2

(eikx + e−ikx

)sin kx =

1

2i

(eikx − e−ikx

)

91

H tautìthta tou P�rsebal sthn perÐptwsh aut  gr�fetai∞∑

k=−∞|ck|2 =

1

π

∫ π

−π

|f(x)|2dx (4.4)

Par�deigma: 3. MÐa sun�rthsh orismènh sto di�sthma [−L,L]mporeÐ epÐshc na analujeÐ se seir� Fouriè. Ston q¸ro L2[−L,L] h seir�Fouriè miac sun�rthshc f(x) eÐnai:

f(x) =a0

2+

∞∑n=0

an cosnπx

L+

∞∑n=1

bn sinnπx

L

Oi suntelestèc Fouriè an kai bn dÐnontai apì tic sqèseic:

an =1

L

∫ L

−L

f(x) cosnπx

Ldx, bn =

1

L

∫ L

−L

f(x) sinnπx

Ldx

Mia seir� Fouriè sugklÐnei p�nta me thn ènnoia thc st�jmhc tou q¸rou.Dhlad 

limn→∞

∫

R‖

n∑m

cmφm(x)− f(x)‖2dx = 0

Sto er¸thma gia thn shmeiak  sÔgklish thc seir�c Fouriè kai m�lista stashmeÐa asuneqeÐac thn ap�nthsh dÐnei to parak�tw je¸rhma.

Je¸rhma: To je¸rhma tou Ntiriklè. Upojètoume ìti:a) H f(x) orÐzetai sto di�sthma −L ≤ x ≤ L , ektìc Ðswc apì peperas-

mèno pl joc shmeÐwn. Ektìc tou diast matoc autoÔ orÐzetai ètsi ¸ste naeÐnai periodik , periìdou 2L dhlad  f(x) = f(x + 2L).

b) Oi sunart seic f(x) kai f ′(x) eÐnai kat� tm mata suneqeÐc. dhlad to pedÐo orismoÔ mporeÐ na diairejeÐ se peperasmèno arijmì upodiasthm�twnìpou oi sunart seic f(x) kai f ′(x) eÐnai suneqeÐc, en¸ sta �kra teÐnoun sepeperasmèna ìria.

Tìte h seir� Fouriè sugklÐnei sthn tim  f(x), an to shmeÐo x eÐnai shmeÐosuneqeÐac kai sthn tim :

f(x+) + f(x−)

2

an to shmeÐo x eÐnai shmeÐo asuneqeÐac, ìpou

f(x+) = limε→0

f(x + ε) f(x−) = limε→0

f(x− ε)

Oi sunj kec tou jewr matoc eÐnai ikanèc all� ìqi anagkaÐec. EÐnai dhlad dunatìn na mhn ikanopoioÔntai all� ìmwc h seir� na sugklÐnei.

92

Parat rhsh: 'Otan h f(x) eÐnai mia �rtia sun�rthsh f(x) = f(−x) ,tìte h seir� Fouriè eÐnai èna �jroisma sunhmitìnwn

f(x) =∞∑

n=0

an cos nx

kai oi suntelestèc Fouriè an kai bn dÐnontai apì tic sqèseic

an =2

π

∫ π

0

f(x) cos nxdx bn = 0

pr�gmati an h sun�rthsh f(x) eÐnai �rtia tìte

an =

∫ π

−π

f(x) cos nxdx =

∫ 0

−π

f(x) cos nxdx +

∫ π

0

f(x) cos nxdx =

−∫ 0

π

f(−x) cos nxdx +

∫ π

0

f(x) cos nxdx = 2

∫ π

0

f(x) cos nxdx

kai

bn =

∫ π

−π

f(x) sin nxdx =

∫ 0

−π

f(x) sin nxdx +

∫ π

0

f(x) sin nxdx =

∫ 0

π

f(−x) sin nxdx +

∫ π

0

f(x) sin nxdx = 0

'Otan h f(x) eÐnai mia peritt  sun�rthsh f(x) = −f(−x), tìte h seir�Fouriè eÐnai èna �jroisma hmitìnwn

f(x) =∞∑

n=1

bn sin nx

Oi suntelestèc Fouriè an kai bn dÐnontai apì tic sqèseic:

an = 0, bn =2

π

∫ π

0

f(x) sin nxdx

H apìdeixh eÐnai ìmoia me thn perÐptwsh thc �rtiac sun�rthshc. Bèbaia up-�rqoun kai sunart seic pou den eÐnai oÔte �rtiec oÔte perittèc.

Parat rhsh: 'Otan h sun�rthsh f(x) eÐnai orismènh sto di�sthma[0, π] mporoÔme na epekteÐnoume thn sun�rthsh kai sto di�sthma [−π, 0], ¸stena eÐnai eÐte �rtia eÐte peritt . 'Etsi mia sun�rthsh orismènh sto [0, π] mporeÐna analujeÐ se �jroisma sunhmitìnwn   se �jroisma hmitìnwn.

93

OrÐzoume thn sun�rthsh

F1(x) =

{f(x) 0 < x < π

f(−x) −π < x < 0

H sun�rthsh aut  eÐnai �rtia, diìti F1(x) = F1(−x) kai onom�zetai �rtiaepèktash thc f(x). H antÐstoiqh seir� Fouriè eÐnai mia seir� hmitìnwn.

Gia thn peritt  epèktash thc f(x), orÐzoume thn sun�rthsh

F2(x) =

{f(x) 0 < x < π

−f(−x) −π < x < 0

H sun�rthsh aut  eÐnai peritt , diìti F2(x) = −F2(−x) kai analÔetai seseir� hmitìnwn.

4.4 OloklhrwtikoÐ MetasqhmatismoÐJewroÔme dÔo sunart seic f(x) kai F (x) pou sundèontai me ton akìloujhsqèsh

F (k) =

∫ b

a

f(x)G(k, x)dx

H sun�rthsh F (k) onom�zetai oloklhrwtikìc metasqhmatismìc thc f(x)me pur na thn sun�rthsh G(k, x).

'Enac qr simoc oloklhrwtikìc metasqhmatismìc eÐnai o metasqhmatismìcLapl�c

F (ω) =

∫ ∞

0

f(t)e−ωtdt

pou èqei pur na thn sun�rthsh e−ωt. ParadeÐgmata sto parak�tw sq ma.Orismìc: An mia sun�rthsh f(x) ikanopoieÐ tic sunj kec tou jewr -

matoc tou Ntiriklè kai eÐnai apolÔtwc oloklhr¸simh sto R dhlad ∫ ∞

−∞|f(x)|dx < ∞

tìte orÐzoume san metasqhmatismì Fouriè thc f(x) thn sun�rthsh F (k)ìpou

F (k) =1√2π

∫ ∞

−∞f(x)eikxdx f(x) =

1√2π

∫ ∞

−∞F (k)e−ikxdk (4.5)

Oi tÔpoi autoÐ eÐnai mia epèktash twn tÔpwn (4.3) twn seir¸n Fouriè. Hsun�rthsh eikx onom�zetai pur nac tou oloklhrwtikoÔ metasqhmatismoÔ.

94

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kEE

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k4-

12�������

k2

Sq ma 4.1: Oi metasqhmatismoÐ Lapl�c tou sin (x + 1) kai tou H3(x)poluwnÔmou tou ErmÐt.

An antikajistoÔme thn sun�rthsh F (k) tou deÔterou tÔpou apì tonpr¸to kai diat�xoume kat�llhla tic dÔo oloklhr¸seic, brÐskoume

f(x) =1

2π

∫ ∞

−∞

∫ ∞

−∞f(y)eikydye−ikxdk =

∫ ∞

−∞

[1

2π

∫ ∞

−∞e−ik(x−y)dk

]f(y)dy

H parap�nw par�stash mèsa stic agkÔlec onom�zetai dèlta sunarthsiakìtou Ntir�k   dèlta sun�rthsh kai eÐnai mia epèktash thc sqèshc (4.1) pouekfr�zei thn orjokanonikìthta twn sunart sewn. 'Eqoume

δ(x− y) =1

2π

∫ ∞

−∞e−ik(x−y)dk f(x) =

∫ ∞

−∞δ(x− y)f(y)dy

An F (k) kai J(k) eÐnai oi metasqhmatismoÐ Fouriè twn sunart sewn f(x)kai j(x) tìte isqÔei kai ed¸ mia antÐstoiqh sqèsh tou P�rsebal pou t¸ragr�fetai

∫ ∞

−∞F (k)J∗(k)dk =

∫ ∞

−∞f(x)j∗(x)dx

Par�deigma: Gia par�deigma ja broÔme thn metasqhmatismènh Fourièthc sun�rthshc f(x) = e−x2/2. 'Eqoume 1

f(k) =1√2π

∫ ∞

−∞e−x2/2eikxdx =

1√2π

∫ ∞

−∞e−(x−ik)2/2e−k2/2dx =

1Βεβαίως και εδώ η μέθοδος του υπολογιστή είναι καταλυτική. Βρίσκουμε῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[4] := FourierTransform[Exp[−x2/2], x, k]Out[4] = e−

k22

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

95

e−k2/2 1√2π

∫ ∞

−∞e−y2/2dy = e−k2/2

ìpou qrhsimopoi same to gnwstì olokl rwma∫ ∞

−∞e−ay2

dy =

√π

a

96

4.5 Ask seic

'Askhsh 1.Na analujeÐ se seir� Fouriè h peritt  sun�rthsh 2

f(x) =

{−h2

−π < x < 0

h2

0 < x < π

LÔsh: H sun�rthsh eÐnai peritt  kai antistoiqeÐ se mia seir� hmitìnwn.

bn =2

π

∫ π

0

h

2sin nxdx =

[h

2

2

nπcos nx

]π

0

=h

2

2

nπ(1− cos nπ)

bn =

{2hnπ

gia n perittì0 gia n �rtio

opìte sto di�sthma −π < x < π h seir� Fouriè eÐnai

f(x) =2h

π

∞∑

k=1

sin (2k + 1)x

2k + 1=

2h

π

(sin x

1+

sin 3x

3+

sin 5x

5+ · · · · · ·

)

'Askhsh 2.Na brejeÐ h seir� Fouriè thc peritt c sun�rthshc:

f(x) = x − π < x < π

2Θα λυθεί με την μέθοδο του υπολογιστή. Ορίζουμε την συνάρτηση῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[1] := f [x−] = If [x < 0,−h/2, h/2]Out[1] = If [x < 0,−h/2, h/2]

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Υπολογίζουμε το ολοκλήρωμα

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[2] := a[n−] = Integrate[ 2

π Sin[n x]f [x], {x, 0, π}], Assumptions → n ∈ Integers]Out[2] = (−1+(−1)n) h

n π῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

Τέλος ορίζουμε το άθροισμα Φουριέ και βρίσκουμε την γραφική παράσταση στο σχήμα.Παρατηρούμε ότι η προσέγγιση είναι καλλίτερη στη μέση των διαστημάτων. Στην αρχή

και στο τέλος των διαστημάτων δηλαδή στα σημεία ασυνεχείας 0, π και −π υπάρχειμια μεγάλη απόκλιση από την συνάρτηση. Αυτό συμβαίνει σε όλες τις σειρές Φουριέ καιονομάζεται φαινόμενο Γκιμπς.

Τα σύμβολα που γράφουμε βγαίνουν σε μορφή παραθύρου στο File − Palettes −BasicTypesetting ή με κατάλληλη πληκτρολόγηση. Παράδειγμα για να τυπώσουμε τοσύμβολο ∈ πατάμε κατά σειρά τα πλήκτρα Esc e l Esc.

97

In[77]:= h = 1;

m = 10;

g1@x_D = ân =1

ma@nD Sin@n xD;

m = 100;

g2@x_D = ân =1

ma@nD Sin@n xD;

Plot@8f@xD, g1@xD, g2@xD<, 8x, -Π, Π<D;

-3 -2 -1 1 2 3

-0.6

-0.4

-0.2

0.2

0.4

0.6

Sq ma 4.2: H seir� Fouriè me 10 ìrouc kai me 100 ìrouc.

LÔsh: H sun�rthsh f(x) orÐzetai ∀x ∈ [−π, π]. EpÐshc oi sunart seicf(x) kai f ′(x) eÐnai suneqeÐc. 'Ara h seir� Fouriè

f(x) =a0

2+

∞∑n=1

(an cos nx + bn sin nx)

sugklÐnei. Gia touc suntelestèc an�ptuxhc brÐskoume an = 0 kai

bn =2

π

∫ π

0

x sin nxdx =2

π

(−

[x cos nx

n

]π

0− 1

n

∫ π

0

cos nxdx

)

=2

π

(−π

ncos nπ −

[1

n2sin nx

]π

0

)=

2

n(−1)n+1

opìte sto di�sthma −π < x < π èqoume

x

2=

∞∑n=1

1

n(−1)n+1 sin nx = sin x− 1

2sin2x +

1

3sin 3x− · · ·

98

'Askhsh 3.Na brejeÐ h seir� Fouriè thc �rtiac sun�rthshc

f(x) = |x| − π < x < π

Na efarmìsete thn tautìthta tou P�rsebal gia na upologÐsete to �jroismathc seir�c

S =∞∑

n=1

1

n4

LÔsh: UpologÐzoume touc suntelestèc an�ptuxhc. BrÐskoume bn = 0kai

a0 =2

π

∫ π

0

xdx =2

π

[x2

2

]π

0

= π

an =2

π

∫ π

0

x cos nxdx =2

π

(−

[x sin nx

n

]π

0

− 1

n

∫ π

0

sin nxdx

)

=2

π

[1

n2cos nx

]π

0

=2

n2π(cos nπ − 1) =

2

n2π((−1)n − 1)

an =

{− 4

n2πn perittì,

0 n �rtio.Epomènwc, sto di�sthma −π < x < π èqoume,

|x| = π

2−

∞∑

k=0

4 cos (2k + 1)x

π(2k + 1)2=

π

2− 4

π

(cos x +

cos 3x

32+

cos 5x

52+ · · ·

)

Parat rhsh: Efarmìzoume thn tautìthta tou P�rsebal sthn seir�aut .

1

π

∫ π

−π

(|x|)2 dx =1

π

∫ π

−π

x2dx =2π2

3=

a20

2+

∞∑n=1

a2n =

π2

2+

∞∑

k=0

16

(2k + 1)4π2

Epomènwc èqoume∞∑

k=0

1

(2k + 1)4=

π4

96

All� ìmwc èqoume

S =∞∑

n=1

1

n4=

∞∑

k=0

1

(2k + 1)4+

∞∑

k=1

1

(2k)4=

∞∑

k=0

1

(2k + 1)4+

1

16S =⇒

99

S =16

15

∞∑

k=0

1

(2k + 1)4=

16

15

π4

96=

π4

90

'Askhsh 4.Na analujeÐ se seir� Fouriè h sun�rthsh

f(x) = x2 − π ≤ x ≤ π

LÔsh: H sun�rthsh eÐnai �rtia kai epomènwc oi suntelestèc bn = 0.BrÐskoume touc suntelestèc an.

a0 =2

π

∫ π

0

f(x)dx =2

π

∫ π

0

x2dx =2

π

[x3

3

]π

0

=2π2

3

an =2

π

∫ π

0

f(x) cos nxdx =2

π

∫ π

0

x2 cos nxdx =

=2

π

[1

nx2 sin nx +

2

n2x cos nx− 2

n3sin nx

]π

0

=4

πn2π cos nπ =

4

n2(−1)n

'Ara h seir� Fouriè thc sun�rthshc aut c eÐnai

x2 =π2

3+

∞∑n=1

4(−1)n

n2cos nx

Gia x = π h seir� Fouriè dÐnei

π2 =π2

3+

∞∑n=1

4(−1)n

n2cos nπ =⇒

∞∑n=1

1

n2=

π2

6

H seir� sugklÐnei omoiìmorfa gia k�je x ∈ (−π, π). pr�gmati èqoume∣∣∣∣4(−1)n

n2cos nx

∣∣∣∣ ≤4

n2= Mn

Epeid  h seir�

S =π2

3+

∞∑n=1

Mn

sugklÐnei, apì to je¸rhma tou B�ierstrac, èpetai ìti h seir� Fouriè sugklÐneiomoiìmorfa.

An oloklhr¸soume mia seir� Fouriè prostÐjetai ènac akìma par�gontacn ston paranomast  k�je ìrou thc seir�c. Autì shmaÐnei ìti h seir� pou

100

ja prokÔyei sugklÐnei akìma grhgorìtera apì thn arqik  seir�. MporoÔmeepomènwc na oloklhr¸soume p�nta mia seir� Fouriè ìro proc ìro kai m�l-ista, h prokÔptousa seir� sugklÐnei omoiìmorfa sto olokl rwma thc arqik c.sun�rthshc.

Oloklhr¸noume thn seir� thc �skhshc kai èqoume∫ x

−π

x2dx =

∫ x

−π

π2

3dx +

∞∑n=1

∫ x

−π

4(−1)n

n2cos nxdx =⇒

[x3

3

]x

−π

=

[π2

3x

]x

−π

+∞∑

n=1

[4(−1)n

n3sin nx

]x

−π

=⇒

x3

3+

π3

3=

π2

3x +

π3

3+

∞∑n=1

4(−1)n

n3sin nx =⇒

1

12x

(x2 − π

)=

∞∑n=1

(−1)n+1

n3sin nx

Ja efarmìsoume t¸ra sthn seir� aut  to je¸rhma tou P�rsebal. 'Eqoume

1

π

∫ π

−π

[x

(x2 − π2

)]2dx =

1

π

∫ π

−π

[x2

(x4 − 2x2π2 + π4

)]dx =

1

π

[x7

7− 2x5

5π2 +

x3

3π4

]π

−π

= 2π6

(1

7− 2

5+

1

3

)=

8π6

105=⇒

∞∑n=1

1

n2=

π6

945

'Otan paragwgÐzoume mia seir� ìro proc ìro prostÐjetai èna n stonarijmht  kai eÐnai dunatìn h prokÔptousa seir� na mhn sugklÐnei. Prèpeina eÐmaste polÔ prosektikoÐ ìtan paragwgÐzoume mia seir� Fouriè ìro procìro. Gia na paragwgÐsoume mia seir� Fouriè mporoÔme na qrhsimopoi soumeto genikì je¸rhma pou isqÔei gia k�je seir�. 'Omwc to je¸rhma autì perièqeianagkaÐec sunj kec pou den eÐnai aparaÐthtec gia thn parag¸gish twn seir¸nFouriè.

ParagwgÐzoume thn seir� Fouriè thc sun�rthshc x2 ìro proc ìro kaièqoume

2x =∞∑

n=1

4(−1)n

n2(−n sin nx) =⇒ x

2=

∞∑n=1

(−1)n+1

nsin nx

To apotèlesma eÐnai  dh gnwstì apì prohgoÔmenh �skhsh.

101

An paragwgÐsoume akìma mia for� thn seir� aut  ìro proc ìro brÐskoume

1

2=

∞∑n=1

(−1)n+1 cos nx

H seir� aut  profan¸c den sugklÐnei efìson o teleutaÐoc ìroc thc den mh-denÐzetai.

'Askhsh 5.Na apodeiqjeÐ ìti gia −π < x < π h seir� Fouriè twn sunart sewn

x cos x kai x sin x eÐnai

x cos x = −1

2sin x + 2

∞∑n=2

(−1)nn

(n− 1)(n + 1)sin nx

x sin x = 1− 1

2cos x− 2

∞∑n=2

(−1)n

(n− 1)(n + 1)cos nx

LÔsh: H sun�rthsh f(x) = x sin x eÐnai �rtia diìti

f(−x) = (−x) sin (−x) = x sin x = f(x)

'Ara bn = 0 kai oi suntelestèc an dÐnontai apì thn sqèsh

an =2

π

∫ π

0

f(x) cos nxdx =2

π

∫ π

0

x sin x cos nxdx

Gia n = 0 to olokl rwma gÐnetai

a0 =2

π

∫ π

0

x sin xdx =2

π[−x cos x + sin x]π0 = − 2

ππ cos π = 2

Gia n = 1 to olokl rwma gÐnetai

a1 =2

π

∫ π

0

x sin x cos xdx =1

π

∫ π

0

x sin 2xdx =

=1

π

[−1

2x cos 2x +

1

22sin 2x

]π

0

= − 1

π

1

2π cos 2π = −1

2

Ja upologÐsoume t¸ra to olokl rwma ìtan n 6= 1.

an =2

π

∫ π

0

x sin x cos nxdx =

102

1

π

∫ π

0

x sin (n + 1)xdx +1

π

∫ π

0

x cos (n− 1)xdx =

1

π

[− 1

n + 1x cos (n + 1)x +

1

(n + 1)2sin (n + 1)x

]π

0

+

1

π

[1

n− 1x cos (n− 1)x− 1

(n− 1)2sin (n− 1)x

]π

0

=

− 1

n + 1cos (n + 1)π +

1

n− 1cos (n− 1)π = −(−1)n+1

n + 1+

(−1)n−1

n− 1=

= (−1)n+1

[− 1

n + 1+

1

n− 1

]=

2(−1)n+1

(n + 1)(n− 1)=

2(−1)n+1

n2 − 1

Epomènwc to an�ptugma se seir� Fouriè eÐnai

x sin x = 1− 1

2cos x +

∞∑n=2

2(−1)n+1

(n + 1)(n− 1)cos nx

H seir� aut  lìgw tou jewr matoc tou B�ierstrac sugklÐnei omoiìmorfa diìti∣∣∣∣

2(−1)n+1

(n + 1)(n− 1)cos nx

∣∣∣∣ <1

n2= Mn

kai h seir�∑∞

n=2 Mn sugklÐnei.ParagwgÐzoume thn seir� kai brÐskoume

x cos x + sin x =1

2sin x +

∞∑n=2

2n(−1)n+1

(n + 1)(n− 1)sin nx =⇒

x cos x = −1

2sin x +

∞∑n=2

2n(−1)n+1

(n + 1)(n− 1)sin nx

'Askhsh 6.Na analujeÐ se seir� Fouriè h sun�rthsh

f(x) =

{2 −2 ≤ x ≤ 0

x 0 < x ≤ 2

sto di�sthma −2 ≤ x ≤ 2.LÔsh: BrÐskoume

a0 =1

2

∫ 2

−2

f(x)dx =1

2

(∫ 0

−2

2dx +1

2

∫ 2

0

xdx

)=

103

=1

2

([2x]0−2 +

[x2

2

]2

0

)=

1

2(4 + 2) = 3

an =1

2

∫ 2

−2

f(x) cosnπx

2dx =

1

2

∫ 0

−2

2 cosnπx

2dx +

1

2

∫ 2

0

x cosnπx

2dx =

[2

nπsin

nπx

2

]0

−2

+

[1

nπx sin

nπx

2+

2

n2π2cos

nπx

2

]2

0

=2 ((−1)n − 1)

n2π2

an =

{0 an n = 2k �rtio

− 4(2k+1)2π2 an n = 2k + 1 perittì

bn =1

2

∫ 2

−2

f(x) sinnπx

2dx =

1

2

∫ 0

−2

2 sinnπx

2dx +

1

2

∫ 2

0

x sinnπx

2dx =

[− 2

nπcos

nπx

2

]0

−2

+

[− 1

nπx cos

nπx

2+

2

n2π2sin

nπx

2

]2

0

= − 2

nπ

Gia x 6= 0 h sun�rthsh eÐnai suneq c kai �ra h seir� Fouriè sugklÐnei sthntim  f(x).

f(x) =3

2− 2

∞∑n=1

[1− (−1)n

n2π2cos

nπx

2+

1

nπsin

nπx

2

](4.6)

To shmeÐo x = 0 eÐnai shmeÐo asuneqeÐac kai h seir� Fouriè sugklÐneisthn tim 

1

2[f(x + 0) + f(x− 0)]x=0 =

1

2

[lim

ε→0+f(ε) + lim

ε→0+f(−ε)

]=

1

2

[lim

ε→0+ε + lim

ε→0+2

]= 1

Gia x = 0 h seir� Fouriè dÐnei thn sqèsh

1 =3

2−

∞∑

k=1

4

(2k + 1)2π2=⇒ π2

8=

∞∑

k=1

1

(2k + 1)2

Sto Ðdio apotèlesma katal goume an jèsoume x = 2 sthn seir� Fouriè ìpouh sun�rthsh eÐnai suneq c.

H seir� èqei sqediasjeÐ sto sq ma (4.3). ParathroÔme to fainìmeno Gkimp-c sto shmeÐo asuneqeÐac x = 0±. Sto shmeÐo x = 0 pr�gmati h sun�rthshsugklÐnei sthn tim  1.

104

In[125]:= f@x_D = If@x < 0, 2, xD;

g@x_D =

3�����

2- 2 ã

n=1

30 ikjjjj1 - H-1Ln��������������������������

n2 Π2

CosAn Π

x�����

2E +

1���������

n Π

SinAn Π

x�����

2Ey{zzzz;

Plot@8f@xD, g@xD<, 8x, -2, 2<D;

-2 -1 1 2

0.5

1

1.5

2

Sq ma 4.3: H seir� Fouriè thc exÐswshc (4.6).

'Askhsh 7.Na analujeÐ se seir� Fouriè hmitìnwn h sun�rthsh

f(x) =

{x 0 ≤ x ≤ π

2

π − x π2≤ x ≤ π

kai na apodeiqjeÐ ìti∞∑

k=1

1

(2n + 1)4=

π4

96

LÔsh: EpekteÐnoume thn sun�rthsh sto di�sthma [−π, 0] ètsi ¸ste hsun�rthsh na eÐnai peritt 

F (x) =

{f(x) 0 ≤ x ≤ π

−f(−x) 0 ≥ x ≥ −π

BrÐskoume touc suntelestèc an�ptuxhc an kai bn. 'Eqoume an = 0 kai

bn =2

π

∫ π

0

f(x) sin nxdx =2

π

∫ π2

0

f(x) sin nxdx +2

π

∫ π

π2

f(x) sin nxdx =

105

2

π

∫ π2

0

x sin nxdx +2

π

∫ π

π2

(π − x) sin nxdx =2

π

[−x

ncos nx +

1

n2sin nx

]π2

0

− 2

π

[−x

ncos nx +

1

n2sin nx

]π

π2

− 2

π

[π

ncos nx

]π

π2

=

2

π

(− π

2ncos

nπ

2+

1

n2sin

nπ

2

)− 2

π

(−π

ncos nπ +

π

2ncos

nπ

2− 1

n2sin

nπ

2

)

− 2

n

(cos nπ − cos

nπ

2

)=

4

πn2sin

nπ

2

Epomènwc 3

b2k =4

π(2k)2sin kπ = 0

kaib2k+1 =

4

π(2k + 1)2sin

((2k + 1)

π

2

)=

4(−1)k

π(2k + 1)2

kai h seir� Fouriè eÐnai

f(x) =4

π

∞∑

k=0

(−1)k

(2k + 1)2sin ((2k + 1)x) (4.7)

H seir� èqei sqediasjeÐ sto sq ma (4.4). H prosèggish eÐnai polÔ kal mìno me touc treic pr¸touc ìrouc diìti h sun�rthsh eÐnai suneq c.

Efarmìzoume thn tautìthta tou P�rsebal sthn seir� Fouriè kai èqoume

1

π

∫ π

−π

|F (x)|2 dx =2

π

∫ π

0

(f(x))2 dx =2

π

(∫ π2

0

x2dx +

∫ π

π2

(π − x)2dx

)=

=2

π

[[x3

3

]π2

0

−[(π − x)3

3

]π

π2

]=

2

π

[π3

24+

π3

24

]=

π2

6

kai h tautìthta tou P�rsebal gr�fetai

π2

6=

16

π2

∞∑

k=1

1

(2k + 1)4=⇒ π4

96=

∞∑

k=1

1

(2k + 1)4

3Ο υπολογισμός του ολοκληρώματος῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[101] := f [x ] = Which[x < Pi/2, x, x ≥ Pi/2, P i− x]Integrate[(2/P i)f [x] Sin[n x], {x, 0, P i}, Assumptions → {n ∈ Integers}]Out[101] = Which[x < π

2 , x, x ≥ π2 , π − x]

Out[102] = 4 Sin[ nπ2 ]

n2 π῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

106

In[53]:= f@x_D = If@x < Pi�2, x, Pi - xD;F@x_D = If@x < 0, -f@-xD, f@xDD;

PlotA9F@xD, 4��������

Pi ãk=0

3 H-1L^k������������������������������H2 k + 1L^2 Sin@H2 k + 1L xD=, 8x, 0, Π<E;

0.5 1 1.5 2 2.5 3

0.25

0.5

0.75

1

1.25

1.5

Sq ma 4.4: H seir� Fouriè thc exÐswshc (4.7).

H diafor� twn dÔo teleutaÐwn arijm¸n gia mia prosèggish me 100 ìrouc,brÐskoume ìti eÐnai 4 d = 2× 10−8 perÐpou.

4Για n = 100 βρίσκουμε῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[11] := N [Limit[π4

96 −∑n

k=01

(2 k+1)4 , n → 100]]Out[11] = 2.0219636631679805× 10−8

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟Το άθροισμα εισάγεται από το παράθυρο File − Parettes − BasicInput ή με την εντολήSum[ 1

(2 k+1)4 , {k, 0, 100}].

107

108

Kef�laio 5

'Alutec Ask seic

'Askhsh 1.Na apodeiqjeÐ h sqèsh

~a× (~b× ~c) = (~a · ~c)~b− (~a ·~b)~c

kai h tautìthta tou Giakìmpi

~a× (~b× ~c) +~b× (~c× ~a) + ~c× (~a×~b) = 0

'Askhsh 2.Na apodeiqjeÐ ìti h exÐswsh tou epipèdou pou dièrqetai apì trÐa shmeÐa me

di�nusma jèshc ~r1, ~r2 kai ~r3, eÐnai

(~r − ~r1) · (~r2 − ~r1)× (~r3 − ~r1) =

∣∣∣∣∣∣

x− x1 y − y1 z − z1

x2 − x1 y2 − y1 z2 − z1

x3 − x1 y3 − y1 z3 − z1

∣∣∣∣∣∣= 0

'Askhsh 3.DÐnontai trÐa dianÔsmata

~a =~i +~j ~b = ~j + ~k ~c =~i−~j

Na upologÐsete to triplì bajmwtì ginìmeno ~a ·~b×~c. UpologÐste epÐshcta dianÔsmata

~a× (~b× ~c) ~c× (~a×~b) ~b× (~c× ~a)

109

'Askhsh 4.Na apodeiqjeÐ ìti to eswterikì ginìmeno ìpwc orÐzetai ston q¸ro R3

ikanopoieÐ tic idiìthtec tou eswterikoÔ ginomènou thc paragr�fou (4.2) . NaapodeiqjeÐ ìti h sqèsh

(f, g) =

∫f ∗(x)g(x)dx

ikanopoieÐ epÐshc tic Ðdiec idiìthtec tou eswterikoÔ ginomènou ektìc apì thnsunepagwg .

‖f‖2 = (f, f) = 0 =⇒ f = 0

'Askhsh 5.

Na apodeiqjeÐ ìti to di�nusma ~c = ‖a‖~b + ‖b‖~a diqotomeÐ thn gwnÐa pousqhmatÐzoun ta mh mhdenik� dianÔsmata ~a kai ~b.

'Askhsh 6.

Na gr�yete to ginìmeno ~R = (~a × ~b) × (~c × ~d) sthn morf  k1~c + k2~d

ìpou k1 kai k2 bajmwt� megèjh. An ta dianÔsmata ~a, ~b, ~c kai ~d eÐnaiomoepÐpeda deÐxte ìti ~R = ~0.

'Askhsh 7.Na apodeiqjeÐ ìti oi di�mesoi enìc trig¸nou ABC tèmnontai se èna shmeÐo

D pou diaireÐ k�je di�meso se lìgo 2/1. An ~a, ~b, ~c, ~d eÐnai ta dianÔsmatajèshc twn antistoÐqwn shmeÐwn tìte

3~d = ~a +~b + ~c

'Askhsh 8.

An ~T , ~N , ~B eÐnai to efaptìmeno di�nusma, h pr¸th k�jetoc kai h deÔterhk�jetoc miac kampÔlhc na apodeÐxete ìti ta dianÔsmata aut� ikanopoioÔn ticakìloujec sqèseic

d~T

ds= κ ~N,

d ~B

ds= −τ ~N,

d ~N

ds= τ ~B − κ~T

To κ onom�zetai kampulìthc kai to τ strèyh thc kampÔlhc en¸ ta an-tÐstrofa touc R = 1/κ kai σ = 1/τ onom�zontai aktÐna kampulìthtac kaiaktÐna strèyhc antistoÐqwc.

110

'Askhsh 9.Na brejeÐ h taqÔthta kai h epit�qunsh enìc kinhtoÔ se sfairikèc sunte-

tagmènec

x = r sin θ cos φ y = r sin θ sin φ z = r cos θ

Ap�nthsh:~v = r~r0 + rθ~θ0 + rφ sin θ~φ0

~γ =(r − rθ2 − rφ2 sin2 θ

)~r0 +

(2rθ + rθ − rφ2 sin θ cos θ

)~θ0+

+(2rθφ + 2rφ sin θ + rφ sin θ

)~φ0

'Askhsh 10.To di�nusma jèshc enìc swmatÐou me m�za m pou kineÐtai sto epÐpedo

eÐnai xOy.~r = a cos ωt~i + b cos ωt~j

Na deÐxte ìti h troqi� eÐnai èlleiyh kai ìti h dÔnamh pou askeÐtai sto swm�tiokateujÔnetai p�ntote proc thn arq  twn axìnwn. Na deÐxte epÐshc ìti toèrgo thc dÔnamhc kat� thn kÐnhsh mia for� gÔro apì thn èlleiyh eÐnai mhdèn.BreÐte tèloc thn rop  thc dÔnamhc kai thn stroform  wc proc thn arq .

Ap�nthsh:

~F = −mω2~r, ~M = ~0, ~L = 2mabω~k

'Askhsh 11.Na breÐte tic stajerèc a, b, c ¸ste to pedÐo twn dun�mewn

~F = (x + 2y + az)~i + (bx− 3y − z)~j + (4x + vy + 2z)~k

na eÐnai sunthrhtikì. Na breÐte to dunamikì pou par�gei autì to dunamikìpedÐo.

Ap�nthsh:

a = 4, b = 2, c = −1 V = −1

2x2 +

3

2y2 − z2 − 2xy − 4xz + yz

111

In[159]:= f@1D = x y z;

f@2D = x2 y2 z2;

f@3D =

�!!!!!!!!!!!x y z ;

Do@Print@PowerExpand@8D@f@nD, xD, D@f@nD, yD, D@f@nD, zD<DD,8n, 1, 3<DDo@Print@PowerExpand@D@D@f@nD, xD, xD + D@D@f@nD, yD, yD +

D@D@f@nD, zD, zDDD, 8n, 1, 3<D

8y z, x z, x y<

92 x y2 z2, 2 x2 y z2, 2 x2 y2 z=

9�!!!!y�!!!!z

���������������������

2�!!!!x

,�!!!!x�!!!!z

���������������������

2�!!!!y

,�!!!!x�!!!!y

���������������������

2�!!!!z=

0

2 x2 y2 + 2 x2 z2 + 2 y2 z2

-

�!!!!x�!!!!y

���������������������

4 z3�2-

�!!!!x�!!!!z

���������������������

4 y3�2-

�!!!!y�!!!!z

���������������������

4 x3�2

Sq ma 5.1: H ap�nthsh thc �skhshc (12).

'Askhsh 12.

Na brejeÐ h b�jmwsh twn parak�tw bajmwt¸n sunart sewn

f1(~r) = xyz f2(~r) = x2y2z2 f3(~r) =√

xyz

BreÐte epÐshc thn Laplasian  ~∇2fj twn sunart sewn aut¸n.Ap�nthsh: H ap�nthsh sto sq ma (5.1) ìpou èqoume apofÔgei thn

eisagwg  tou kat�llhlou upoprogr�mmatoc kai qrhsimopoi same ton orismìtwn zhtoumènwn telest¸n.

'Askhsh 13.

Na brejeÐ h apìklish kai o strobilismìc twn parak�tw dianusmatik¸nsunart sewn

~F1 = (x2 + yz, y2 + zx, z2 + zy) ~F2 = (x2, y2, z2) ~F3 =1

‖~r‖(yz, zx, xy)

112

'Askhsh 14.

An ~F = φ(r)~r na apodeÐxete ìti ~∇ · ~F = rφ′(r) + 3φ(r) Na breÐteepÐshc thn sun�rthsh φ(r) pou ikanopoieÐ thn exÐswsh ~∇ · ~F = 0 dhlad to di�nusma ~F na eÐnai swlhnoeidèc.

'Askhsh 15.Na brejoÔn oi timèc twn κ, λ kai µ ¸ste ta parak�tw dianusmatik�

pedÐa na eÐnai astrìbila.

~F = x(κy − λzµ)~i + (κ− 1)x2~j + (2 + λ)x2z~k

~F = (x + 2y + κz)~i + (λx− 3y − z)~j + (4x + µy + 2z)~k

'Askhsh 16.Na apodeÐxete thn tautìthta

~A× (~∇× ~A) =1

2~∇(A2)− ( ~A · ~∇) ~A

An ta dianÔsmata ~A kai ~B eÐnai stajer� na apodeÐxete ìti

~∇( ~A · ~B × ~r) = ~A× ~B

kai na brejeÐ h klÐsh thc sun�rthshc ( ~A× ~r) · ( ~B × ~r).

'Askhsh 17.

Na deÐxete ìti to pedÐo twn dun�mewn ~F = −kr3~r eÐnai sunthrhtikì.Gr�yte thn dunamik  enèrgeia kai thn olik  stajer  kinhtik  enèrgeia.

Ap�nthsh:

E =1

2m

(dr

d t

)2

+1

5kr5

'Askhsh 18.DÐnontai oi pÐnakec

A =

3 5 −12 0 −1−1 2 0

B =

1 0 11 3 4−1 1 3

113

Na ektelestoÔn oi parak�tw pr�xeic

AB −BA A2B + 5A AB − 2A

'Askhsh 19.Na lujeÐ me thn mèjodo twn orizous¸n to sÔsthma

x + y + z = 2 9x + y − 2z = 6, 3x− y − z = 6

Ap�nthsh: 1 x = 2, y = −4, z = 4.

'Askhsh 20.Na brejoÔn ta idiodianÔsmata kai oi idiotimèc twn pin�kwn

1 0 10 1 01 0 1

1 1 01 0 10 1 1

5 0 20 1 02 0 2

3 0 00 2 −50 1 −2

'Askhsh 21.DÐnontai oi akìloujoi pÐnakec pou onom�zontai m trec tou P�ouli.

σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

)

Na upologÐsete tic parast�seic

σ1σ2 + σ2σ1 σ2σ3 − σ3σ2 σ21

DÐnetai o pÐnakac S(a) = a1σ1 + a2σ2 + a3σ3. Na upologÐsete tic akìloujecdun�meic kai thn ekjetik  sun�rthsh. (H LÔsh sto parak�tw sq ma).

S2, S3, S2k, S2k+1 B = eiS = 1 + iS +1

2!(iS)2 +

1

3!(iS)3 + · · ·

1Η λύση του συστήματος῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟In[124] := Solve[{x + y + z == 2, 9x + y − 2z == 6, 3x− y − z == 6}, {x, y, z}]Out[124] = {{x → 2, y → −4, z → 4}}

῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟῟

114

In[217]:=

B = Cos@AD J 1 0

0 1N + ä S

Sin@AD�������������������

A;

vv = 8-a12 - a22 - a32 ® -A2<;S = a1 J 0 1

1 0N + a2 J 0 -ä

ä 0N + a3 J 1 0

0 -1N;

Simplify@PowerExpand@ReplaceAll@MatrixExp@ä SD, vvD - BDDOut[220]=

880, 0<, 80, 0<<

Sq ma 5.2: ApodeiknÔoume ìti eiS = B ìpou A =√

a21 + a2

2 + a23.

'Askhsh 22.Na deÐxete ìti oi sunart seic

1√2L

,1√2L

sin nπx

L

1√2L

cos nπx

Ln = 0, 1, 2, · · ·

eÐnai èna orjokanonikì sÔnolo sunart sewn ston q¸ro L2(−L,L).Na apodeÐxete epÐshc ìti kai oi sunart seic

1√2π

eimφ m = 0,±1,±2,±3, · · ·

ston q¸ro L2(−π, π).

'Askhsh 23.Na anaptÔxete thn sun�rthsh

x(x− π) 0 < x < π

se seir� hmitìnwn kai se seir� sunhmitìnwn. Qrhsimopoi ste tic anaptÔxeicautèc gia na apodeÐxete tic sqèseic

∞∑n=1

1

n2=

π2

6,

∞∑n=1

(−1)n−1

n2=

π2

12,

∞∑n=1

(−1)n−1

(2n− 1)3=

π3

32,

Tèloc me thn bo jeia thc tautìthtac tou P�rsebal na apodeÐxete ìti∞∑

n=1

1

n4=

π4

90,

∞∑n=1

1

n6=

π6

945.

115

Ap�nthsh:

x(π − x) =8

π

(1

13sin x +

1

33sin 3x +

1

53sin 5x + · · · · · ·

)

x(π − x) =π2

6−

(1

12cos 2x +

1

22cos 4x +

1

32cos 6x · · · · · ·

)

'Askhsh 24.Na anaptuqjoÔn se seir� Fouriè

a) h sun�rthsh

f1(t) =

{0 −π ≤ ω t ≤ 0

sin (ω t) 0 ≤ ω t ≤ π

kai b) h �rtia sun�rthsh

f2(t) =

{− sin (ω t) −π ≤ ω t ≤ 0

sin (ω t) 0 ≤ ω t ≤ π

Oi sunart seic èqoun perÐodo L = 2π/ω. Na breÐte thn an�ptuxh thc pr¸thcapì thn an�ptuxh thc deÔterhc.

Ap�nthsh:

f1(t) =1

π+

1

2sin ω t− 2

π

∞∑

k=1

1

4k2 − 1cos (2kω t)

f2(t) =2

π− 4

π

∞∑

k=1

1

4k2 − 1cos (2kω t)

f1(t) =1

2sin (ωt) +

1

2f2(t)

'Askhsh 25.Na anaptuqjeÐ h sun�rthsh

f(x) = sin x 0 ≤ x ≤ 2

a) Se seir� hmitìnwn b) Se seir� sunhmitìnwn

116

'Askhsh 26.Se poll� probl mata thc fusik c eÐnai protimìtero na analÔoume mia

sun�rthsh se seir� Fouriè thc morf c

f(x) =A0

2+

∞∑n=1

An cos (nx− θn)

Na apodeÐxete ìti h èkfrash aut  eÐnai isodÔnamh me thn gnwst  an�lush,ìpou

an = An cos θn bn = An sin θn

A2n = a2

n + b2n tan θn = bn/an

Na efarmìsete thn ap�nthsh sthn sun�rthsh thc parak�tw �skhshc.

'Askhsh 27.Na apodeiqjeÐ ìti h ekjetik  sun�rthsh eµx gia −π < x < π dèqetai

thn akìloujh an�ptuxh Fouriè

ex =2 sinh µπ

π

[1

2µ+

∞∑n=1

(−1)n

n2 + µ2(µ cos nx− n sin nx)

]

jèsate x → −x sthn parap�nw sqèsh kai apodeÐxte tic sqèseic

sinh µx =1

2

(eµx − e−µx

)= −2 sinh µπ

π

∞∑n=1

(−1)n

n2 + µ2sin nx − π < x < π

cosh µx =1

2

(eµx + e−µx

)=

2µ sinh π

π2

∞∑n=0

(−1)n

n2 + µ2cos nx − π ≤ x ≤ π

Na jèsete tèloc γ = 1 kai x = 0 kai x = π sthn teleutaÐa sqèsh gia naapodeÐxete tic tautìthtec

π

sinh π= 1 + 2

∞∑n=1

(−1)n

n2 + 1

π

tanh π= 1 + 2

∞∑n=1

1

n2 + 1

'Askhsh 28.Na apodeÐxete thn akìloujh an�ptuxh se seir� Fouriè thc sun�rthshc

cos γx ìpou h stajer� γ den eÐnai akèraioc arijmìc.

cos γx =2γ sin γπ

π

[1

2γ2− cos x

γ2 − 1+

cos 2x

γ2 − 22− cos 3x

γ2 − 32+ · · · · · ·

]γ /∈ A

117

In[37]:= m = 15;

PlotA9Π x än=1

m ikjj1 -

x�����

n

y{zz ikjj1 +

x�����

n

y{zz, Sin@Π xD=, 8x, -4, 4<E;

-4 -2 2 4

-2

-1

1

2

Sq ma 5.3: H prosèggish eÐnai kal  sthn perioq  tou mhdenìc.

Apì thn sqèsh aut , na jèsete x = π kai γ = x, gia na apodeÐxete thnsqèsh

cot xπ =2x

π

[1

2x2+

1

x2 − 12+

1

x2 − 22+

1

x2 − 32+

1

x2 − 42· · · · · ·

]

Na apodeÐxete ìti h seir� aut  sugklÐnei omoiìmorfa gia 0 ≤ x ≤ b < 1 kaina thn oloklhr¸sete ìro proc ìro gia na apodeÐxete thn sqèsh

sin πx = πx

∞∏n=1

(1− x

n

)(1 +

x

n

)

H grafik  par�stash twn dÔo parap�nw sunart sewn gia mia prosèggish mem = 15 ìrouc faÐnetai sto sq ma (5.3).

Na jèsete tèloc x = 12

gia na apodeÐxete thn isìthta

π

2=

∞∏n=1

2n

2n− 1

2n

2n + 1

118

'Askhsh 29.Na apodeiqtoÔn oi parak�tw seirèc Fouriè gia 0 ≤ x ≤ 2π.Upìdeixh: Na apodeÐxete thn pr¸th kai met� tic upìloipec me diadoqikèc

oloklhr¸seic   na apodeÐxete thn teleutaÐa kai met� tic upìloipec me diado-qikèc paragwgÐseic.

∞∑

k=1

cos kx

k2=

π2

6− πx

2+

x2

4

∞∑

k=1

sin kx

k3=

π2x

6− πx2

4+

x3

12

∞∑

k=1

cos kx

k4=

π4

90− π2x2

12+

πx3

12− x4

48

∞∑

k=1

sin kx

k5=

π4x

90− π2x3

36+

πx4

48− x5

240

'Askhsh 30.Na apodeiqjeÐ h sqèsh

∞∑

k=1

(−1)k+1 cos (2k + 1)x

(2k − 1)(2k + 1)(2k + 3)=

π

8cos2 x − 1

3cos x − π

2≤ x ≤ π

2

'Askhsh 31.Mia qord  m kouc L eÐnai demènh sta shmeÐa 0 kai L tou x−�xona. Thn

qronik  stigm  t = 0 èqei arqik  jèsh kai arqik  taqÔthta pou dÐnontai apìtic sunart seic f(x) kai g(x) antistoÐqwc. Na brejeÐ to pl�toc tal�ntwshcy(x, t) thc qord c. To pl�toc tal�ntwshc ikanopoieÐ thn diaforik  exÐswshtou kÔmatoc. To prìblhma eÐnai to akìloujo.

Na lujeÐ h diaforik  exÐswsh

∂2y(x, t)

∂t2=

∂2y(x, t)

∂x20 < x < L, t > 0

me oriakèc sunj kecy(0, t) = y(L, t) = 0

kai arqikèc sunj kec

y(x, 0) = f(x) yt(x, 0) = g(x)

Ap�nthsh:

y(x, t) =∞∑

n=1

sinnπx

L

[An cos

nπt

L+ Bn sin

nπt

L

]

119

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Out[48]= 91, t, -1����

2+

3 t2�����������

2, -

3 t��������

2+

5 t3�����������

2,

3����

8-

15 t2��������������

4+

35 t4��������������

8=

Sq ma 5.4: Ta polu¸numa Lez�ntr gia n = 0, 1, 2, 3, 4.

ìpou oi stajerèc An kai Bn eÐnai oi suntelestèc Fouriè twn sunart sewnf(x) kai g(x), dhlad 

An =2

L

∫ L

0

f(x) sinnπx

Ldx Bn =

2

L

∫ L

0

g(x) sinnπx

Ldx

'Askhsh 32.DÐnontai ta grammik¸c anex�rthta dianÔsmata χ1, χ2, · · ·. Apì ta di-

anÔsmata aut� kataskeu�zoume ta dianÔsmata

ψ1 = χ1 φ1 = ψ1/‖ψ1‖ψ2 = χ2− < φ1|χ2 > φ1 φ2 = ψ2/‖ψ2‖· · · · · · · · · · · ·

ψn = χn −∑n−1

k=1 < φk|χn > φk φn = ψn/‖ψn‖

Na apodeiqjeÐ ìti ta dianÔsmata {φ1, φ2, · · ·} eÐnai èna orjokanonikì sÔno-lo. Na apodeiqjeÐ epÐshc ìti gia k�je m ta dianÔsmata {χ1, χ2, · · · , χm}kai {φ1, φ2, · · · , φm} eÐnai dÔo sÔnola gennhtìrwn tou Ðdiou q¸rou.

H Mèjodoc aut  onom�zetai mèjodoc orjokanonikopoÐhshc Gkram - Smit.Na efarmìsete thn mèjodo gia tic sunart seic: χn(t) = tn−1, n = 1, 2, · · ·

ìpou t ∈ (−1, 1).Ap�nthsh: Ta polu¸numa tou Lez�ntr.

120

BibliografÐa

[1] Dianusmatik  An�lush. D. Sourl�c Panepist mio Pa-tr¸n 1997

[2] Mathematical methods for physicists. G. Arfken Academ-ic press 1971

[3] Seirèc Fouriè Schaum’s outline series McGraw-Hill

[4] An¸tera Majhmatik�. M. Spiegel Schaum’s outline seriesMcGraw-Hill

[5] Majhmatik  An�lush L. Brand Ellhnik  Majhmatik EtaireÐa 1984

[6] Genik� Majhmatik�. Q. ZagoÔrac PanepisthmiakècParadìseic 1999

[7] Merikèc Diaforikèc Exis¸seic. S. Traqan�c Panepisth-miakèc Ekdìseic Kr thc 2001

[8] Eisagwg  sto Mathematica . K. Papad�khc EkdìseicTziìla JessalonÐkh 2003

121

Kat�logoc onom�twn kai h apìdosh touc sta Ellhnik�.

EukleÐdhc ( 300 p.q. )Rene Descartes Ren�toc Kartèsioc (1596 - 1650)Brook Taylor MproÔk Tèulor (1685 - 1731)Leonhard Euler Lèonarnt 'Ouler (1707 - 1783)Pierre - Simon de Laplace Pièr - Simìn tou Lapl�c (1749 - 1827)Adrian - Mari Legendre Antri�n - MarÐ Lez�ntr (1752 - 1833)Marc - Antoine Parseval Mark - Antou�n P�rsebal (1755 - 1836)Jean - Batiste Fourier Zan - MpatÐst Fouriè (1768 - 1830)Karl Friedrich Gauss Karl Fr ntriq Gk�ouc (1777 - 1855)Friedrich Bessel Fr ntriq Mpèsel (1784 - 1846)Augustin - Louis Causchy Wgkustèn - LouÐ KwsÔ (1789 - 1857)Karl Jacob Karl Giakìmpi (1804 - 1851)Peter Dirichlet Pèter Ntiriklè (1805 - 1859)Karl Weierstrass Karl B�ierstrac (1815 - 1897)Charles Hermit Sarl ErmÐt (1822 - 1901)Leopold Kronecker Lèopolnt Krìneker (1823 - 1891)Georg Riemann Gkèorgk R man (1826 - 1866)James Clerk Maxwell Tzèimc Klèrk M�xgouel (1831 - 1879)Josiah Gibbs Tzos�ia Gkimpc (1839 - 1903)John Poynting Tzwn Pìuntingk (1852 - 1914)David Hilbert Nt�bit QÐlmpert (1862 - 1943)Toulio Levi - Chivita ToÔlio LebÐ - Tsibit� (1873 - 1941)Henri - Leon Lebesque AnrÐ - Leìn Lempègk (1875 - 1941)Erhard Schmidt 'Erqart SmÐt (1876 - 1959)Erwin Schrodinger 'Erbin Srèntingker (1887 - 1961)Wolfgang Pauli Bìlfgkangk P�ouli (1900 - 1958)Paul Dirac Pwl Ntir�k (1902 - 1984)Laurent Schwartz Lwr�n Sbartc (1915 - 2002)

122

Kat�logoc Sqhm�twn

1.1 TrÐa dianÔsmata grammik� anex�rthta. . . . . . . . . . . . . . 20

2.1 H elleiptik  èlika. . . . . . . . . . . . . . . . . . . . . . . . . 282.2 H tom  enìc elleiyoeidoÔc kai enìc epipèdou eÐnai mÐa kampÔlh. 292.3 H epif�neia z = e−

12x2−y2 . . . . . . . . . . . . . . . . . . . . . 29

2.4 Entolèc gia thn eisagwg  upoprogr�mmatoc. . . . . . . . . . . 372.5 H Laplasian  se sfairikèc suntetagmènec. . . . . . . . . . . . 492.6 Oi sfairikèc armonikèc gia l = 8, 9 kai m = 5, 6, 7. . . . . . . 502.7 H apìdeixh thc dianusmatik c sqèshc (2.22). . . . . . . . . . . 55

3.1 H perÐptwsh enìc sust matoc pou den èqei lÔsh . . . . . . . . 723.2 H orÐzousa enìc pÐnaka 4× 4 tou Ðdiou tÔpou me thn �skhsh. . 82

4.1 Oi metasqhmatismoÐ Lapl�c tou sin (x + 1) kai tou H3(x)poluwnÔmou tou ErmÐt. . . . . . . . . . . . . . . . . . . . . . . 94

4.2 H seir� Fouriè me 10 ìrouc kai me 100 ìrouc. . . . . . . . . . . 974.3 H seir� Fouriè thc exÐswshc (4.6). . . . . . . . . . . . . . . . 1034.4 H seir� Fouriè thc exÐswshc (4.7). . . . . . . . . . . . . . . . 104

5.1 H ap�nthsh thc �skhshc (12). . . . . . . . . . . . . . . . . . . 1105.2 ApodeiknÔoume ìti eiS = B ìpou A =

√a2

1 + a22 + a2

3. . . . . 1135.3 H prosèggish eÐnai kal  sthn perioq  tou mhdenìc. . . . . . . 1165.4 Ta polu¸numa Lez�ntr gia n = 0, 1, 2, 3, 4. . . . . . . . . . . . 118

123

124