vector calculus

DIANUSMATIKOS LOGISMOS

I. Muritz

2009

Perieqmena

1 DIANUSMATIKH ALGEBRA 5

1.1 Dianusmatik sunartsei mia metablht . . . . . . . . . . . . . . . . . . 5

1.2 Dianusmatiko qroi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Upqwroi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Grammik anexarthsa . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Distash kai bsh en dianusmatiko qrou . . . . . . . . . . . . . 12

1.2.4 Efarmog sth lsh Grammikn Susthmtwn . . . . . . . . . . . . . 14

1.2.5 Gewmetrik ermhnea twn grammikn susthmtwn . . . . . . . . . . . 16

1.3 Grammik apeikonsei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 SUNARTHSEIS POLLWN METABLHTWN 23

2.1 Pragmatik sunartsei polln metablhtn . . . . . . . . . . . . . . . . . 24

2.1.1 Grfhma kai snolo stjmh . . . . . . . . . . . . . . . . . . . . . . 24

2.1.2 Merik pargwgo . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.3 Bajmda gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.4 Pargwgo kat mko kamplh kai

kateujunmenh pargwgo . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Dianusmatik peda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.1 O telest andelta. Apklish kai strobilism . . . . . . . . . . . 35

2.2.2 Efarmog: Metdosh jermthta . . . . . . . . . . . . . . . . . . . 38

2.3 To Jerhma Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Akrtata bajmwtn pedwn . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5 To jerhma twn peplegmnwn sunartsewn . . . . . . . . . . . . . . . . . . 43

3 OLOKLHRWSH SUNARTHSEWN POLLWN METABLHTWN 47

3.1 Epikamplia oloklhrmata . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Dipl oloklhrmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Tripl oloklhrmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Allag metablhtn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 Epifaneiak oloklhrmata . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3

4 PERIEQOMENA

3.5.1 Parrthma: Upologism tou epifaneiako oloklhrmato . . . . . 62

3.6 To Jerhma Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.7 To Jerhma Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 OI EXISWSEIS THS MAJHMATIKHS FUSIKHS 77

4.1 Klassik diaforik exissei me merik paraggou . . . . . . . . . . . 77

4.2 Lsei twn MDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 H exswsh diqush . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Oi exissei Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5 Oi exissei knhsh twn reustn . . . . . . . . . . . . . . . . . . . . . . . 86

Keflaio 1

DIANUSMATIKH ALGEBRA

H nnoia tou diansmato w diatetagmnh trida tou R3enai gnwst ap to mjhma tou

Apeirostiko Logismo. To keflaio aut enai mia eisagwg sth Grammik 'Algebra kai

genikeei thn lgebra tou Rnse afhrhmnou dianusmatiko qrou. Arqzoume me thn

gnwst nnoia th parametrik anaparstash kampuln.

Sumbolism. Upenjumzoume ti sti trei diastsei me r = (x, y, z) sumbolzoume

to dinusma jsh kai r = r =x2 + y2 + z2. Gia elfrunsh tou sumbolismo sunjw

grfoume |r| ant r.

1.1 Dianusmatik sunartsei mia metablht

'Opw dh gnwrzoume, sunartsei tou tpou r : I Rn pou I enai kpoio disthmapragmatikn arijmn kai n = 2, 3 onomzontai kample sto eppedo ston tridistato

qro.

Pardeigma 1.1.1. An t R kai a,v R3 stajer diansmata, tte r(t) = a + tvqei w eikna thn eujea pou pern ap to shmeo a kai qei th diejunsh tou diansmato

v.

Pardeigma 1.1.2. H kamplh r(t) = (x(t), y(t)) me x(t) = a cos t kai y(t) = a sin t

pou a > 0 kai t [0, 2], enai parametrik parstash kklou kntrou (0, 0) kai aktnaa.

Pardeigma 1.1.3. H kamplh r(t) = (x(t), y(t)) me x(t) = a cost kai y(t) = a sint

pou a > 0, > 0 kai t [0, 2/], enai parametrik parstash tou diou kklou, all todinusma jsh kinetai me diaforetik gwniak taqthta.H kamplh r(t) = (x(t), y(t))

me x(t) = a cos (t) = a cost kai y(t) = a sin (t) = a sint pou a > 0, > 0kai t [0, 2/], enai parametrik parstash tou diou kklou, all to dinusma jshkinetai me antjeto prosanatolism.

5

6 KEFALAIO 1. DIANUSMATIKH ALGEBRA

Pardeigma 1.1.4. 'Estw f : R R, sunrthsh mia metablht. To grfhma thenai mia kamplh C pou algebrik parstatai ap thn y = f (x) , dhlad

C ={(x, y) R2 : y = f (x)} .

H C qei w parametrik anaparstash, x(t) = t kai y(t) = f (t) .

Ap ta paradegmata prokptei ti ma kamplh mpore na qei poll anaparastsei,

p.q. oi

x (t) = t, y (t) = t, z (t) = t, t [0,),x (t) = t2 1, y (t) = t2 1, z (t) = t2 1, t [1,),

pariston thn dia kamplh (eujea pou pern ap thn arq twn axnwn kai ap to shmeo

(1, 1, 1)).

Pargwgo mia dianusmatik sunrthsh r : t 7 (x(t), y(t), z(t)) sto shmeo t enaito dinusma

dr

dt= lim

t0r (t+t) r (t)

t,

arke fusik na uprqei to rio. Ekola prokptei ti sthn perptwsh aut oi pragmatik

sunartsei x(t), y(t), z(t) enai paragwgsime kai mlista

r (t) drdt

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

To dinusma r (t) enai parllhlo sthn eujea pou efptetai sthn kamplh sto shmeo r (t).

Gia pardeigma, na swmtio pou kinetai sthn elikoeid troqi r(t) = (cos t, sin t, t),

t R, qei taqthta v(t) = r (t) = ( sin t, cos t, 1) pou to mtro th enai |v(t)| = 2.Me anlogo trpo orzetai w epitqunsh h deterh pargwgo, r = d2r/dt2 =

(x(t), y(t), z(t)) (ef' son oi sunartsei x(t), y(t) kai z(t) enai do for paragwgsime).

Gia pardeigma, to dinusma jsh en swmatidou pou ektele omal kuklik knhsh

(dialgoume ssthma axnwn ste to kntro tou kklou na enai to shmeo (0, 0)), grfetai

r(t) = a cost i+ a sint j, a, > 0.

H taqthta ja enai loipn

v(t) = a sint i+ a cost j.

Parathrome ti to mtro th enai stajer, v(t) = a, kai enai diark kjeth stodinusma jsh, r(t).v(t) = 0. H epitqunsh

dv

dt= 2a cost i 2r sint j = 2r(t),

1.1. DIANUSMATIKES SUNARTHSEIS MIAS METABLHTHS 7

enai antrroph th dianusmatik aktna, kat sunpeia enai kjeth sthn taqthta.

(Genik, an mia dianusmatik sunrthsh v qei stajer norm, tte enai orjognia sthn

pargwgo th, v. Apdeixh: Paragwgzonta thn |v(t)|2 = v(t).v(t) = const, qoume2v(t).v(t) = 0, o.e.d.)

To ginmeno pragmatik sunrthsh ep dianusmatik sunrthsh r enai mia na

dianusmatik sunrthsh q = r pou orzetai w, q(t) = (t)r(t), t R. H pargwgo thq dnetai ap ton tpo q(t) = (t)r(t)+ (t)r(t). Me anlogo trpo, gia do dianusmatik

sunartsei r kai p prokptei

d

dt(r.p) = r.p+ r.p,

d

dt(r p) = r p+ r p.

Ma kamplh lgetai apl kamplh an den autotmnetai. H parametrik anaparstash

mia apl kamplh enai mia na pro na apeiknish

r : I R3,

pou I kpoio disthma kai r to dinusma jsh en shmeou th kamplh.

Ma kamplh lgetai lea n h pargwgo th den mhdenzetai poujen sto pedo orismo

th,

dr

dt= (x, y, z) 6= (0, 0, 0) t I.

(Dikaiologeste ton orism).

Se kje apl kamplh antistoiqon do prosanatolismo kateujnsei. An p.q. h

kamplh C sundei do shmeaA kai B, mporome na epilxoume ton prosanatolism ap to

A sto B. H parametrik anaparstash r = r(t), t [t1, t2], sbetai ton prosanatolismaut an, r(t1) = A, r(t2) = B (dste pardeigma).

Asksei

Epanalambnoume kpoiou tpou. H seir aut apotele epanlhyh tou sqetiko ke-

falaou ap ton Apeirostik Logism.

1. An a,b, c enai grammik exarthmna tte a. (b c) = 0 kai antstrofa.

2. Upenjumzoume ti h exswsh tou epipdou pou pern ap to shmeio a kai enai kjeto

sto dinusma n grfetai

(x a) .n = 0.'Ena eppedo qei exswsh x + 2y 2z + 7 = 0. Brete: a) na monadiao kjetodinusma b) ta shmea tom me tou xone g) thn apstash tou epipdou ap thn

arq d) thi suntetagmne tou shmeou tou epipdou pou qei thn elqisth apstash

ap th arq.


3. Upenjumzoume ti h exswsh th eujea pou pern ap to shmeio a kai qei th

diejunsh tou diansmato u grfetai

r (t) = tu+ a.

Dexte ti aut mpore en gnei na grafe w

x x0a

=y y0b

=z z0c

. (1.1.1)

Brete thn exswsh th eujea pou pern ap to shmeio (1, 1, 1) kai enai kjeth sto

eppedo me exswsh 4x 3y + z = 5.

4. Brete th dianusmatik taqthta kai to mtro th sti paraktw kample

r (t) = t2i + tj+ k, r (t) = (a cos t, b sin t, 0) , r (t) = tu+ a,

pou a, b stajer kai u, a stajer diansmata.

5. Brete parametrik anaparastsei r(t) twn paraktw kampuln: eujea pou pern

ap ta shmea (1, 1, 1) kai (2, 3, 2), {(x, y) R2 : y = ex}, lika aktna 2 kai bmato1/3.

6. Upologste ta mkh twn kampuln

r(t) =(2 cos t, 2 sin t,

3t), t [0, 3] , r(t) = (t sin t, 1 cos t, 0) , t [0, 2] .

Upmnhsh:

l (r) =

ba

|r(t)| dt = ba

x2(t) + y2(t) + z2(t) dt.

1.2 Dianusmatiko qroi

'Ena snolo V lgetai dianusmatik qro (DQ) grammik qro upernw tou R, an ta

stoiqea tou mporon na prostejon metax tou kai na pollaplasiaston me pragmatiko

arijmo kat trpo pou na ikanopoiontai oi aklouje idithte: (Me x,y, z sumbolzoume

stoiqea tou V kai me , pragmatiko arijmo. Ta stoiqea tou V lgontai diansmata

kai oi arijmo suqn lgontai bajmwt).

DQ 1 An x kai y ankoun sto V tte kai x+ y ankei epsh sto V .

DQ 2 Gia la ta stoiqea x,y tou V isqei x+ y = y + x.

DQ 3 x ankei sto V .

DQ 4 (x+ y) + z = x+ (y + z).

1.2. DIANUSMATIKOI QWROI 9

DQ 5 Uprqei na stoiqeo tou V pou sumbolzetai me 0 ttoio ste gia kje stoiqeox tou V na isqei x + 0 = x.

DQ 6 Gia kje x sto V uprqei na stoiqeo x ttoio ste x+ (x) = 0.

DQ 7 (x + y) = x + y.

DQ 8 (+ )x = x + x.

DQ 9 ()x = (x).

DQ 10 1x = x. (1 enai o arijm 1).

To prtupo ma enai fusik o R3, tou opoou oi algebrik idithte tjentai w oris-

m tou dianusmatiko qrou. Ta pio endiafronta snola sta majhmatik, kajstantai

dianusmatiko qroi an oriston katllhla oi prxei th prsjesh dianusmtwn kai o

pollaplasiasm me bajmwt.

Pardeigma 1.2.1. To snolo Rn, dhlad to snolo twn diatetagmnwn n-dwn (x1, ..., xn)

pragmatikn arijmn enai o gnwst ma n-distato Eukledeio qro. Ta stoiqea tou

ikanopoion ti idithte DQ1 w DQ10, an orzoume jroisma dianusmtwn

(x1, ..., xn) + (y1, ..., yn) = (x1 + y1, ..., xn + yn) ,

kai pollaplasiasm me bajmwt

(x1, ..., xn) = (x1, ..., xn) .

Suqn, oi perisstere idithte ikanopoiontai kat tetrimmno trpo kai gia na

dexoume ti na snolo V enai dianusmatik qro, arke na elgxoume ti:

Ta stoiqea tou mporon na prostejon metax tou kai na pollaplasiaston me prag-

matiko arijmo kat trpo pou na ikanopoiontai oi aklouje idithte:

A. An x kai y ankoun sto V tte kai x+ y ankei epsh sto V .

B. An R kai x V , tte x ankei sto V .

Pardeigma 1.2.2. To snolo twn suneqn sunartsewn f : [a, b] R (sumbolzetaime C[a, b]) me prxei thn prsjesh sunartsewn

(f + g)(x) = f(x) + g(x)

kai pollaplasiasm me bajmwt R

(f)(x) = f(x)


enai DQ diti ikanopoie ti idithte DQ1 w DQ10. H apdeixh afnetai ston anagn-

sth. Smfwna me thn prohgomenh paratrhsh, elgqoume mno ti an f kai g ankoun

sto C[a, b], tte kai f + g kai f ankoun epsh sto C[a, b]. H aplousteumnh aut

diadikasa enai efikt diti to mhdenik stoiqeo tou qrou orzetai me profan trpo

(sthn perptwsh ma enai h mhdenik sunrthsh) kai oi idithte DQ2, DQ4 kai DQ6 w

DQ10, enai sunpeie gnwstn idiottwn twn pragmatikn arijmn.

Pardeigma 1.2.3. To snolo Mmn twn m n pinkwn, me prxei thn prsjeshpinkwn kai pollaplasiasm me pragmatik arijm, enai dianusmatik qro.

1.2.1 Upqwroi

Orism 1.1. 'Ena uposnolo W en dianusmatiko qrou V lgetai dianusmatik

upqwro, an ikanopoiontai oi aklouje sunjke:

a. An x kai y ankoun sto W tte kai x + y ankei epsh sto W .

b. An R kai x W , tte x ankei sto W .g. To mhdenik dinusma 0 tou V ankei epsh sto W .

Gia pardeigma, to snolo twn stoiqewn tou R3pou qoun mhdenik z suntetagmnh,

dhlad stoiqea th morf (x, y, 0) me x, y R, enai na dianusmatik upqwro touR3. (Argtera ja dome na trpo na ton tautopoisoume me ton R

2). An V enai to

snolo lwn twn sunartsewn f : [a, b] R, mporome ekola na dexoume ti V enaidianusmatik qro kai C[a, b] enai na upqwro tou. AnMnn enai o DQ twn n npinkwn kai W to snolo twn n n summetrikn pinkwn A (dhlad aij = aji) tte Wenai upqwro tou Mnn.

H tom do upoqrwn enai upqwro. Brete do upqwrou tou R3. Poioi enai oiupqwroi tou R?

Orism 1.2. 'Estw V na dianusmatik qro kai x1, . . . ,xn stoiqea tou V . An

a1, . . . , an enai pragmatiko arijmo, tte to dinusma

a1x1 + . . .+ anxn

lgetai grammik sunduasm twn x1, . . . ,xn.

Gia pardeigma, ap ta stoiqea sin t, cos t tou d.q. C[0, ] pargetai o grammik

sunduasm a sin t+ b cos t.

Jerhma 1.2.1. To snolo W lwn twn grammikn sunduasmn twn x1, . . . ,xn, enai

dianusmatik upqwro tou V .


Apdeixh. An a1, . . . , an kai b1, . . . , bn enai pragmatiko arijmo, tte

(a1x1 + . . .+ anxn) + (b1x1 + . . .+ bnxn) = (a1 + b1)x1 + . . .+ (an + bn)xn

dhlad to jroisma do stoiqewn tou W enai pli stoiqeo tou W . An c enai arijm,

tte

c (a1x1 + . . .+ anxn) = ca1x1 + . . .+ canxn

enai epsh grammik sunduasm twn x1, . . . ,xn, ra stoiqeo tou W . Tlo, to W

periqei to 0, diti

0 = 0x1 + . . .+ 0xn.

Lme ti o upqwro W pargetai ap ta x1, . . . ,xn. Eidik an ta diansmata

x1, . . . ,xn pargoun ton V , dhlad kje stoiqeo tou V grfetai w grammik sundu-

asm tou, tte lme ti o dianusmatik qro V pargetai ap ta x1, . . . ,xn. Gia

pardeigma, o R2pargetai ap ta i, j, all pw mporete na elgxete, pargetai kai ap

ta i, i+ j, j, kai ap ta i j, i+ j, j. Tjetai loipn to erthma, an uprqei pnta ttoiosnolo pou pargei na DQ kai an nai, poio enai o elqisto arijm twn dianusmtwn

pou apaitontai ste kje stoiqeo tou V na grfetai w grammik sunduasm tou. Gia

na apantsoume sto erthma, ja qreiastome thn nnoia th grammik anexarthsa.

1.2.2 Grammik anexarthsa

Orism 1.3. Ta diansmata x1, . . . ,xn lgontai grammik anexrthta an h isthta

a1x1 + . . .+ anxn = 0

sunepgetai ti loi oi arijmo a1, . . . , an enai mhdn.

Ta diansmata x1, . . . ,xn lgontai grammik exarthmna an uprqoun arijmo a1, . . . , an,

qi loi mhdn, ttoioi ste

a1x1 + . . .+ anxn = 0.

Gia pardeigma, ta stoiqea sin t, cos t tou d.q. C[0, ] enai grammik anexrthta.

Prgmati, gia na enai na grammik sunduasm tou so me mhdn gia kje t [0, ],dhlad a sin t + b cos t = 0, t [0, ], ja prpei a = 0, b = 0, p.q. gia t = 0 prokpteib = 0 kai gia t = /2 prokptei a = 0.

Dexte ti ston R2 ta i, j, enai grammik anexrthta, to dio kai ta i, i + j. Tai j, i+ j, j, enai grammik exarthmna. 'Estw C1(a, b) o qro twn paragwgismwnsunartsewn sto disthma (a, b). Dexte ti oi sunartsei et, e2t enai grammik

anexrthte.


Jerhma 1.2.2. An ta diansmata x1, . . . ,xn enai grammik exarthmna, tte na ap

aut mpore na grafe w grammik sunduasm twn upolopwn. Antstrofa, an na ap

ta diansmata x1, . . . ,xn mpore na grafe w grammik sunduasm twn upolopwn, tte

aut enai grammik exarthmna.

Apdeixh. 'Estw ti ta diansmata x1, . . . ,xn enai grammik exarthmna. Tte uprqoun

arijmo a1, a2, . . . , an, qi loi mhdn, ttoioi ste

a1x1 + a2x2 + . . .+ anxn = 0.

'Estw p.q. ti a1 6= 0, opte

x1 = a2a1 . . . an

a1xn.

Antstrofa, an na ap ta diansmata x1, . . . ,xn, p.q. to x1 mpore na grafe w grammik

sunduasm twn upolopwn, tte

x1 = 2x2 + . . .+ nxn,

sunep

(1)x1 + 2x2 + . . .+ nxn = 0dhlad ta x1, . . . ,xn enai grammik exarthmna.

1.2.3 Distash kai bsh en dianusmatiko qrou

Orism 1.4. O mgisto arijm grammik anexarttwn dianusmtwn en dianus-

matiko qrou V lgetai distash tou V , kai sumbolzetai me dim V .

Pardeigma 1.2.4. Sto eppedo uprqoun do grammik anexrthta diansmata (brete

do!), all opoiadpote tra diansmata enai grammik exarthmna. 'Ara h distash tou

epipdou enai do (dimR2 = 2). Sto qro pou zome uprqoun 3 grammik anexrthta di-

ansmata (brete tra!), all opoiadpote tssera diansmata enai grammik exarthmna.

'Ara h distash tou eukledeiou qrou enai tra (dimR3 = 3).

Jerhma 1.2.3. An v1, . . . ,vn enai grammik anexrthta diansmata en ndistatoudianusmatiko qrou V , tte kje stoiqeo tou V grfetai me monadik trpo w grammik

sunduasm twn v1, . . . ,vn.

Apdeixh. Ta diansmata x,v1, . . . ,vn enai grammik exarthmna diti enai perisstera

ap n, dhlad perisstera ap th distash tou qrou. 'Ara, uprqoun arijmo a, a1, . . . , an,

qi loi mhdn, ttoioi ste

ax+ a1v1 + . . .+ anvn = 0,


me a 6= 0, alli ta v1, . . . ,vn ja tan grammik exarthmna. 'Opw akrib sto jerhma1.2.2, to x grfetai w grammik sunduasm twn upolopwn,

x = c1v1 + . . .+ cnvn, ck = ak/a.

Prpei tra na dexoume ti an to x mpore na grafe kai w

x = b1v1 + . . .+ bnvn,

tte ck = bk gia kje k = 1, 2, . . . , n. Prgmati, afairnta kat mlh ti parapnw

isthte, brskoume

(c1 b1)v1 + . . .+ (cn bn)vn = 0,kai epeid ta v1, . . . ,vn enai grammik anexrthta, prokptei ti ck = bk, gia kje k =

1, 2, . . . , n.

Orism 1.5. To snolo {v1, . . . ,vn} lgetai bsh tou qrou V an ta diansmatav1, . . . ,vn enai grammik anexrthta kai ep plon pargoun ton qro V .

Anakefalainoume: An {v1, . . . ,vn} enai mi bsh tou qrou V kai x tuqao dinusmatou V , tte to x grfetai me monadik trpo w grammik sunduasm twn v1, . . . ,vn,

dhlad

x = a1v1 + . . .+ anvn.

Oi arijmo a1, . . . , an lgontai suntetagmne tou x w pro thn dojesa bsh kai poll

for grfoume to x w nda arijmn, x = (a1, . . . , an). 'Etsi, dexame ti opoiadpoten grammik anexrthta diansmata, mporon na epilegon w bsh tou V . Epomnw le

oi bsei en dianusmatiko qrou qoun to dio pljo stoiqewn, so me th distash tou

qrou.

Pardeigma 1.2.5. Ta {i, j,k} apotelon bsh tou R3. 'Omoia, ta n diansmatae1 = (1, 0, . . . , 0), e2 = (0, 1, . . . , 0), . . . , en = (0, 0, . . . , 1), apotelon bsh tou R

n, ra

dimRn = n.

Pardeigma 1.2.6. To snolo twn poluwnmwn nbajmo me pragmatiko sunte-lest, enai grammik qro me prxei thn prsjesh poluwnmwn kai pollaplasiasm

me arijm. Dexete ti oi sunartsei {1, t, t2, t3, . . . , tn} apotelon bsh tou qrou.

Pardeigma 1.2.7. To snolo twn pragmatikn arijmn enai DQ distash 1. O

upqwro tou R3pou apoteletai ap diansmata me mhdenik zsuntetagmnh, dhlad

stoiqea th morf (x, y, 0), x, y R, qei distash 2. Kje eujea gramm tou R3 poupern ap to shmeo (0, 0, 0) enai dianusmatik upqwro tou R3 distash 1.


Pardeigma 1.2.8. To eppedo me exswsh 2x 3y + 7z = 0, pern ap thn arqtwn axnwn kai periqei ta diansmata u1 = i + 2/3j,u2 = 7/2i k. Kje grammiksunduasm twn u1 kai u2, ankei epsh sto eppedo. To eppedo loipn enai didistato

upqwro tou R3, kai ta (grammik anexrthta) diansmata u1 kai u2, apotelon mia bsh

tou.

Jerhma 1.2.4. An dimV = n, tte opoiodpote snolo m dianusmtwn tou V me

m > n, enai grammik exarthmno.

H apdeixh brsketai se opoiodpote biblo Grammik 'Algebra. To jerhma aut

exasfalzei ti an qoume m diansmata {v1, . . . ,vm} pou pargoun to qro kai n mnoap aut enai grammik anexrthta, tte mporome na petxoume mn diansmata otwste ta apomnonta n (grammik anexrthta) diansmata na enai ex' sou apotelesmatik

gia na pargoun ton qro.

1.2.4 Efarmog sth lsh Grammikn Susthmtwn

'Estw na m n pnaka A,

A =

a.11 a1n.

.

.

.

.

.

am1 amn

H jstlh sumbolzetai me Aj kai sunjw lgetai dinusma jstlh (jewromenhw mia mda arijmn, enai na stoiqeo tou Rm),

Aj =

a1j.

.

.

amj

'Estw na 2 2 pnaka A. An ta do diansmata stlh tou enai grammikanexrthta, tte detA 6= 0. (Isodnama, an ta do diansmata stlh tou enaigrammik exarthmna, tte detA = 0). Genikesete sti trei diastsei. Sth

Grammik 'Algebra apodeiknetai to jerhma:

Jerhma 1.2.5. Ta diansmata stlh en nn pnaka A enai grammik anexrthta,an kai mno an detA 6= 0.

A jewrsoume tra to omogen ssthma m exissewn

a11x1 + a12x2 + ... + a1nxn = 0

am1x1 + a12x2 + ... + amnxn = 0


gia tou n agnstou x1, . . . , xn.

Mia lsh enai h tetrimmnh, x1 = 0, . . . , xn = 0. Anazhtome mh tetrimmne lsei.

To ssthma mpore na grafe w

nj=1

aijxj = 0, i = 1, 2, ..., m,

Ax = 0 w

x1

a11.

.

.

am1

+ x2

a12.

.

.

am2

+ ...+ xn

a1n.

.

.

amn

=

0.

.

.

0

kai tlo, qrhsimopointa ta diansmata stlh Aj, w

x1A1 + x2A

2 + ... + xnAn = 0 (1.2.1)

Aut h teleutaa morf me thn opoa gryame to ssthma, den enai par mia sqsh gram-

mik exrthsh n dianusmtwn (twn Aj), tou Rm. An loipn n > m = dimRm, ap to

jerhma 1.2.4 prokptei ti ta Aj enai grammik exarthmna, dhlad uprqoun arijmo

x1, . . . , xn, qi loi mhdn, ttoioi ste na isqei h (1.2.1). Kat sunpeia:

Jerhma 1.2.6. 'Ena omogen ssthma m exissewn me n agnstou, n > m, qei mh

tetrimmnh lsh.

A ljoume tra sthn pio endiafrousa perptwsh, en mh omogeno sustmato m

exissewn me n agnstou, x1, . . . , xn

a11x1 + a12x2 + ...+ a1nxn = b1

am1x1 + a12x2 + ...+ amnxn = bm

pou grfetai kai w Ax = b, qrhsimopointa ta diansmata stlh Aj, w

x1A1 + x2A

2 + ... + xnAn = b (1.2.2)

A upojsoume ti m = n kai ti ta diansmata stlh Aj, j = 1, 2, . . . , n enai grammik

anexrthta. Sunep apotelon bsh tou Rn, opte opoiodpote dinusma b, grfetai

me monadik trpo w grammik sunduasm tou. Dhlad uprqoun monadiko arijmo

x1, . . . , xn, ttoioi ste na isqei h (1.2.2). Anadiatupnoume to sumprasma ma w ex.

Jerhma 1.2.7. An m = n kai ta diansmata Aj, j = 1, 2, . . . , n enai grammik

anexrthta, tte to ssthma (1.2.2) qei monadik lsh.

Sunduzonta to jerhma 1.2.7 kai to jerhma 1.2.5 prokptei ti to ssthma (1.2.2)

qei monadik lsh an kai mno an detA 6= 0.


1.2.5 Gewmetrik ermhnea twn grammikn susthmtwn

Upenjumzoume ti h exswsh tou epipdou pou pern ap to shmeo p kai enai kjeto sto

dinusma u grfetai w

(x p) .u = 0, Ax+By + Cz = D. (1.2.3)

Arqzoume me na klassik pardeigma.

Pardeigma 1.2.9. 'Estw W to snolo twn x R3 pou enai kjeta sto dinusmau = (1,1, 0). Dexte tiW enai upqwro tou R3 kai brete th distash tou. Perigrytegewmetrik to snolo W.

An x = (x1, x2, x3) R3 enai kjeto sto dinusma u, dhlad x.u = 0, ja qoumex1 x2 + 0x3 = 0 x1 = x2. 'Ara ta stoiqea tou W qoun prth sunistsa sh meth deterh, sunep kje stoiqeo w W enai th morf w = (w1, w1, w3) . Profan0 W kai w = (w1, w1, w3) W. An w,v W, tte w + v = (w1, w1, w3) +(v1, v1, v3) = (w1 + v1, w1 + v1, w3 + v3) , dhlad to w + v qei prth sunistsa sh me

th deterh, ra w + v W. Sumperanoume ti W enai upqwro. Ma bsh tou Wapoteletai ap ta diansmata (1, 1, 0) , (0, 0, 1) , ra h distash tou upoqrou W enai 2.

Ap thn genik exswsh (1.2.3) sumperanoume ti W enai na eppedo pou pern ap thn

arq (0, 0, 0) kai enai kjeto sto u = (1,1, 0) .

Genikeonta, to snolo twn x = (x, y, z) R3 pou enai kjeta sto dinusma a =(a1, a2, a3) , enai upqwro tou R

3distash do. 'Oloi oi upqwroi tou R

3distash do,

enai eppeda pou pernon ap thn arq twn axnwn kai perigrfontai ap exissei th

morf

a1x+ a2y + a3z = 0.

(Prosoq! to eppedo x+ y + z = 1, den enai upqwro).

Pardeigma 1.2.10. Brete thn tom twn epipdwn x y 2z = 0 kai 2x+ y 2z = 0.Ta eppeda tmnontai se mi eujea parllhlh pro kpoio dinusma v pou enai kjeto

kai sto u1 = (1,1,2) kai sto u2 = (2, 1,2). Epomnw w v mporome na proume to

v = u1 u2 =

i j k

1 1 22 1 2

= 4i 2j+ 3k.Brskoume na koin shmeo twn epipdwn, lnonta ti exissei twn epipdwn. 'Ena ttoio

shmeo enai (profan) to (0, 0, 0) . Me bsh ton gnwst tpo (1.1.1) h exswsh eujea

enai

x

4=

y

2 =z

3.


Genikeonta, jewrome do grammik anexrthta diansmata a = (a1, a2, a3) kai b =

(b1, b2, b3) . Tte ta eppeda a1x+ a2y+ a3z = 0 kai b1x+ b2y+ b3z = 0 tmnontai en gnei

se ma eujea pou pern ap thn arq 0. Prgmati, lnonta (me ton kanna tou Cramer)

w pro x kai y to ssthma

a1x+ a2y = a3z,b1x+ b2y = b3z,

brskoume

x a2 a3b2 b3=

y a3 a1b3 b1=

z a1 a2b1 b2,

dhlad qoume thn exswsh eujea pou pern ap thn arq (0, 0, 0) = 0 kai qei th

diejunsh tou a b. Epomnw h tom do didistatwn upoqrwn enai en gnei namonadistato upqwro.

Paratrhsh. An to eppedo den pern ap thn arq twn axnwn, ja qei exswsh

th morf (1.2.3) me D 6= 0. Sthn perptwsh aut dn enai upqwro, afo den periqeito 0 = (0, 0, 0).

Pte h tom do didistatwn upoqrwn tou R3 den enai monadistato upqwro?

Jewrome tra eppeda pou pernon ap thn arq 0. Brete sunjke ste h tomtou na enai to monosnolo {0} .

Poi enai h tom twn epipdwn xy2z = 0, 3x+3y+6z = 0 kai 2x+y2z = 0?

'Estw do grammik anexrthta diansmata u kai v. Smfwna me to jerhma 1.2.1,ta u kai v pargoun na upqwro W . Brete th distash tou W . Perigryte

gewmetrik to snolo W.

Jewrome tra eppeda ttoia ste toulqiston na ap aut den pern ap thn arq0, opte perigrfontai apo exissei th morf

a1x+ a2y + a3z = d1,

b1x+ b2y + b3z = d2,

c1x+ c2y + c3z = d3,

di qi loi mhdn. Poi sunjke prpei na ikanopoion ta ai, bi, ci otw ste:

(a) ta eppeda tmnontai se na mno shmeo (b) h tom tou enai ma eujea (g) den

tmnontai (h tom tou enai to ken snolo).


Genkeush ston Rn

'Estw a = (a1, a2, ..., an) Rn. Tte to snolo twn x Rn me x.a = 0 enai na upqwrotou R

ndistash n 1. Lme ti x.a = 0,

a1x1 + a2x2 + ... + anxn = 0,

enai h exswsh en uperepipdou P1 tou Rn. An P2 enai to upereppedo b1x1+...+bnxn = 0,

tte h tom twn P1 kai P2 enai en gnei na upqwro distash n 2. Genikeonta, oim exissei (m n)

a11x1 + a12x2 + ... + a1nxn = 0

am1x1 + a12x2 + ... + amnxn = 0

paristnounm upereppeda pou h tom tou enai upqwro distash nm. Apodeikne-tai ti upqwro aut qei th mgisth distash n m an m stle tou pnaka A enaigrammik anexrthte, isodnama an kpoia mm upoorzousa tou pnaka A enai mmhdenik (blpe rjro per txh (rank) pnaka sthn Wikipedia).

Jewrome to snolo W twn x R4 me x1+2x2 = 0 kai x3 x4 = 0. Dexete ti Wenai upqwro kai brete th distash tou.

Perigryte to upereppedo x1 = 0 tou R4.

Asksei

1. Ta paraktw uposnola tou R3enai upqwroi

(a) To snolo twn (x, y, z) gia ta opoa x+ y + z = 0.

(b) To snolo twn (x, y, z) gia ta opoa x = y.

2. An u,v enai grammik anexrthta stoiqea tou R3, o upqwro pou pargetai ap

aut enai na eppedo pou pern ap thn arq twn axnwn.

3. Dexete ti ta snola {a = (1, 1, 1),b = (1, 0, 1), c = (0, 0, 1)}, {a = (0, 1, 1),b =(1, 0, 2), c = (2, 0, 1)}kai {a = (1, 1, 0),b = (1, 2,1), c = (0, 0, 1)} apotelonbsei tou R

3kai brete ti suntetagmne twn dianusmtwn x = (1, 0, 0) kai y =

(1, 1, 1) w pro ti bsei aut.

4. Duo stoiqea u,v en d.q. V , enai grammik exarthmna an kai mno an uprqei

arijm ttoio ste u = v v = u.

5. 'Opw edame to snolo twn m n pinkwn enai DQ. Brete mia bsh sto qro twn2 2 pinkwn. To dio kai sto snolo twn trigwnikn n n pinkwn. To dio kai

1.3. GRAMMIKES APEIKONISEIS 19

sto snolo twn 3 3 summetrikn pinkwn. Poia enai h distash tou qrou twnm n pinkwn?

6. An V enai upqwro tou R2 poia mpore na enai h distash tou V ? An V enai

gnsio upqwro tou R2(dhlad den tautzetai me ton R

2), tte ete V = {0} ete

V enai mia eujea pou pern ap thn arq twn axnwn.

7. An V enai upqwro tou R3 poia mpore na enai h distash tou V ? An V enai

gnsio upqwro tou R3, tte ete V = {0} ete V enai mia eujea pou pern ap

thn arq twn axnwn, ete V enai na eppedo pou pern ap thn arq twn axnwn.

8. Poia enai h distash tou upoqrou tou Rnpou sunstatai ap la ta diansmata

(x1, ..., xn) gia ta opoa x1 + x2 + ...+ xn = 0?

9. Se omogen ssthma upojtoume ti m = n kai ti ta diansmata stlh tou pnaka

A enai grammik anexrthta. Dexte ti h mnh lsh enai h tetrimmnh.

10. Kataskeuste do 3 3 grammik m omogen sustmata tsi ste to na na qeimonadik lsh kai to llo na mn qei lsh.

11. Kataskeuste do 3 3 grammik omogen sustmata tsi ste to na na qeimonadik lsh kai to llo na qei peire lsei.

1.3 Grammik apeikonsei

Orism 1.6. Mia apeiknish

F : V W

ap na dianusmatik qro V se na llo dianusmatik qroW , lgetai grammik apeikn-

ish, an gia kje R kai x,y V isqoun

F(x + y) = F(x) + F(y),

F(x) = F(x).

Sunjw paralepetai h parnjesh sto risma th apeiknish F kai grfoume apl Fx.

Mia grammik apeiknish pnw s' na dianusmatik qro V me eikna msa ston V,

F : V V,

lgetai grammik telest ston V .


To updeigma twn grammikn apeikonsewn enai h apeiknish (x, y) 7 (ax+ by, cx+ dy)ston R

2. Prkeitai dhlad gia thn apeiknish F : R2 R2, me tpo

Fx = F

[x1

x2

]=

[ax1 + bx2

cx1 + dx2

]. (1.3.1)

Parathrome amsw ti

F(x+ y) =

[a (x1 + y1) + b (x2 + y2)

c (x1 + y1) + d (x2 + y2)

]=

[ax1 + bx2

cx1 + dx2

]+

[ay1 + by2

cy1 + dy2

]= Fx + Fy,

kai

F(x) =

[ax1 + bx2

cx1 + dx2

]=

[ax1 + bx2

cx1 + dx2

]= Fx.

Shmantik paratrhsh. H apeiknish F parstatai ap ton 2 2 pnaka A, pouqei w stoiqea tou tou arijmo a, b, c, d:

Ax =

[a b

c d

][x1

x2

]=

[ax1 + bx2

cx1 + dx2

].

Pardeigma 1.3.1. 'Estw na 3 3 pnaka A. An x = (x1, x2, x3)T na dinusma touR3, tte Ax enai epsh na dinusma tou R3. Ekola prokptei ti o pnaka A enai na

grammik telest ston R3, dhlad A (x + y) = Ax+Ay kai A (x) = Ax, R.

Pardeigma 1.3.2. 'Estw V o dianusmatik qro twn pragmatikn sunartsewn

orismnwn sto (a, b), pou qoun paraggou kje txh (sumbolzetai me C(a, b)). O

telest th paraggish, D = d/dt, orzetai w

Df = f ,

dhlad gia kje stoiqeo f tou V ,

(Df)(t) = f (t), t (a, b).

Tte o D enai na grammik telest ston V . Prgmati, D (f + g) = Df +Dg, diti

(D (f + g))(t) = (f(t) + g(t)) = f (t) + g(t) = (Df)(t) + (Dg)(t), t (a, b),

kai moia apodeiknetai ti D (f) = Df, R.

Pardeigma 1.3.3. 'Estw V na DQ distash n kai {v1, . . . ,vn} mia bsh tou, dhladkje stoiqeo x tou V grfetai w

x = x1v1 + . . .+ xnvn.

1.3. GRAMMIKES APEIKONISEIS 21

Orzoume thn apeiknish

L : V Rn,me tpo

L(x) = (x1, . . . , xn)

dhlad h L, apeikonzei kje stoiqeo tou V , sth nda twn suntetagmnwn tou w proth dojesa bsh. H apeiknish L enai grammik.

Pardeigma 1.3.4. 'Estw V = C[a, b], o qro twn suneqn sunartsewn sto disthma

[a, b] kai W = R. H apeiknish I : V W , pou se kje suneq sunrthsh f antistoiqzeiton arijm

baf , dhlad

I (f) =

ba

f (t) dt,

enai grammik.

Pardeigma 1.3.5. A jewrsoume to snolo L(V, V ) lwn twn grammikn telestnston V . Orzoume prsjesh telestn kai pollaplasiasm me bajmwt pw sti prag-

matik sunartsei: An F kai T enai do telest kai c na pragmatik arijm, tte

x V(F+T)(x) = F(x) +T(x),

(cF)(x) = cF(x).

Me ti prxei aut, to snolo L(V, V ) gnetai grammik qro. Mporome na genike-soume, kajistnta to snolo L(V,W ) lwn twn grammikn apeikonsewn ap ton V stonW , grammik qro.

Asksei

1. 1) Poie ap ti paraktw apeikonsei enai grammik?

(a) F : R3 R2me tpo F(x, y, z) = (x, y).(b) F : R4 R4 me tpo F(u) = u.(g) F : R3 R3 me tpo F(u) = u+ (1, 1, 1).(d) F : R2 R2 me tpo F(x, y) = (2x, y x).(e) F : R3 R me tpo F(x, y, z) = xy.

2. Upenjumzoume ti kje sunrthsh f : R R grfetai w jroisma mia rtiasunrthsh A kai mia peritt , dhlad f = A + . 'Estw V o qro lwn twn

pragmatikn sunartsewn. Orzoume thn apeiknish P : V V me tpo

(Pf) (x) =f (x) + f (x)

2.

Dexete ti h P enai grammik.


3. Upenjumzoume ti na pnaka S lgetai summetrik an S = ST kai na pnaka

A lgetai antisummetrik an A = AT . 'Estw V = Mnn, o qro twn n npinkwn kai P : V V h apeiknish me tpo

PC =C + CT

2, C V.

Dexte ti: H apeiknish P enai grammik. Brete th distash tou qrou twn n nantisummetrikn pinkwn.

4. Brete ton pnaka pou anaparist kje apeiknish:

a) F : R3 R2 me tpo F(x, y, z) = (x, y) (probol).b) F : R2 R2 me tpo F(x, y) = (3x, 3y).g) F : R3 R3 me tpo F(u) = u.d) F : R3 R2 me F (e1) = (1,3)T , F (e2) = (4, 2)T , F (e3) = (3, 1)T

Keflaio 2

SUNARTHSEIS POLLWN

METABLHTWN

To antikemeno tou Apeirostiko Logismo enai oi pragmatik sunartsei pragmatik

metablht, dhlad sunartsei tou tpou

f : R R.

Sto Logism Polln Metablhtn ( Dianusmatik Logism) ja exetsoume sunartsei

me pedo orismo kpoio uposnolo tou Rnkai tim msa sto R

m, dhlad dianusmatik

peda:

F : Rn Rm (2.0.1)Eidik periptsei enai:

(A) Dianusmatik sunartsei mia metablht (kample) pou edame sto prohgo-

meno keflaio

r : R Rm.(B) Oi pragmatik sunartsei polln metablhtn (bajmwt peda)

: Rn R

Gia pardeigma, gia na perigryoume th jermokrasa mia jalssia perioq U R3,qreiazmaste mia sunrthsh

T : U Rdhlad T (x, y, z) enai h jermokrasa sto shmeo (x, y, z) tou U .

Sunartsei th morf (2.0.1) me m,n > 1, dhlad dianusmatik peda emfanzontai

suqn sti fusik epistme kai w pardeigma anafroume thn taqthta v = (v1, v2, v3)

en reusto pou exarttai kai ap th jsh kai ap to qrno, enai dhlad mia sunrthsh

v : R4 R3

23

24 KEFALAIO 2. SUNARTHSEIS POLLWN METABLHTWN

tsi ste v (t, x, y, z) = (v1 (t, x, y, z) , v2 (t, x, y, z) , v3 (t, x, y, z)) enai h taqthta tou

reusto sto shmeo (x, y, z) th stigm t.

2.1 Pragmatik sunartsei polln metablhtn

H katanhsh th sumperifor mia sunrthsh mia metablhth, f : R R, gnetai meth bojeia tou grafmato th, {(x, y) R2 : y = f (x)} . Sti do diastsei mporomena genikesoume kai na apoktsoume mia asjhsh th sumperifor en bajmwto pedou

: R2 R, all sti perisstere diastsei h gewmetrik aut prosggish den enaipnta dunat.

2.1.1 Grfhma kai snolo stjmh

Gia na apoktsoume mia optik parstash a jewrsoume mia sunrthsh do metablhtn

orismnh se na qwro U tou epipdou, U R2

: U R

Grfonta z = (x, y), to grfhma th enai la ta shmea (x, y, z) tou R3 me (x, y) U ,sumbolik,

={(x, y, z) R3 : z = (x, y), (x, y) U} .

PSfrag replaements

y

z

x

U

z = f(x; y)

Sqma 2.1: En gnei, to grfhma mia bajmwt sunrthsh do metablhtn enai mia

epifneia.

Mia llh qrsimh nnoia enai aut tou sunlou stjmh (level set) en bajmwto

pedou. An U enai mia perioq tou Rn, : U R kai c mia stajer, tte to snolostjmh Sc me tim c apoteletai ap ta shmea r = (x1, ..., xn) U Rn gia ta opoa(r) = c, sumbolik,

Sc = {r U : (r) = c} .Eidik gia mia sunrthsh do metablhtn to snolo stjmh enai en gnei mia kamplh

pou lgetai isostajmik kamplh. 'Etsi, an (x, y) enai mia sunrthsh do metablhtn,

2.1. PRAGMATIKES SUNARTHSEIS POLLWN METABLHTWN 25

oi isostajmik kample th orzontai w

Ck ={(x, y) R2 : (x, y) = k} .

Eidik gia mia sunrthsh trin metablhtn to snolo stjmh enai en gnei mia

epifneia pou lgetai isostajmik epifneia. 'Etsi, an (x, y, z) enai mia sunrthsh trin

metablhtn, oi isostajmik epifneie th orzontai w

Sk = {(x, y, z) R3 : (x, y, z) = k}.

Meletste ta grafmata twn paradeigmtwn twn Marsden and Tromba kai peirama-tistete sth Mathematica me sunartsei do metablhtn, pq

Plot3D[Sin[x y], {x, 0, 4}, {y, 0, 4}]

Pardeigma 2.1.1. Gia thn (x, y) = x2 + y2 oi isostajmik kample orzontai w

Ck ={(x, y) R2 : x2 + y2 = k}

kai gia k > 0 enai omkentroi kkloi me kntro thn arq twn axnwn kai aktnak. To

grfhma th enai paraboloeid ek peristrof.

Pardeigma 2.1.2. Gia thn (x, y) = x2 + 2y2 oi isostajmik kample orzontai w

Ck ={(x, y) R2 : x2 + 2y2 = k}

kai gia k > 0 enai elleyei me hmixonek kai

k/2. To grfhma th enai paraboloei-

d kai oi tom tou me eppeda z = const enai elleyei.

Pardeigma 2.1.3. Gia thn (x, y) = x2 y2 oi isostajmik kample orzontai w

Ck ={(x, y) R2 : x2 y2 = k}

kai enai uperbol. To grfhma th enai sgma.

Pardeigma 2.1.4. An jewrsoume thn atmosfairik pesh sthn epifneia th jlas-

sa w sunrthsh th jsh, p (x, y) , tte oi isostajmik kample {(x, y) R2 : p (x, y) = c}enai oi isobare pou apeikonzontai s' na qrth barometrikn susthmtwn. Anloga or'-

izontai oi isjerme, oi isobaje klp.

Pardeigma 2.1.5. Gia thn (x, y, z) = x2+y2+z2, oi isostajmik epifneie orzontai

w

Sk = {(x, y, z) R3 : x2 + y2 + z2 = k} (2.1.1)kai gia k > 0 enai omkentre sfare me kntro thn arq twn axnwn kai aktna

k.


Pardeigma 2.1.6. Gia thn (x, y, z) = x2+ y2 z2, h mhdenik isostajmik epifneia,

{(x, y, z) R3 : x2 + y2 z2 = 0}

enai kno. H isostajmik epifneia me tim 1 enai to dqwno uperboloeid kai me tim+1 enai to monqwno uperboloeid.

H nnoia th sunqeia gia bajmwt peda enai anlogh aut pou gnwrzoume ap

ton Apeirostik Logism. Upenjumzoume ti ma sunrthsh f enai suneq an geitonik

shmea tou pedou orismo th qoun geitonik eikne (msw th f). Sthn perptwsh tou

Logismo polln metablhtn, do shmea a kai r sto Rn enai geitonik, an h norm th

diafor tou enai mikr. Posotik, h parapnw ida ekfrzetai me ton akloujo orism:

Orism 2.1. Mia sunrthsh : U R enai suneq sto a U Rn an gia kje > 0, uprqei > 0 ttoio ste gia r U

r a < |(r) (a)| < .

Sthn perptwsh twn do diastsewn, to snolo twn shmewn r me r a < , enaikuklik dsko (qwr to snoro tou), kntrou a = (a, b) kai aktna . An sumbolsoume

to dsko aut me D (a) ja qoume

D (a) ={r R2 : r a < } = {(x, y) R2 :(x a)2 + (y b)2 < } .

Sti trei diastsei, to snolo twn shmewn r me r a < , enai mia mpla (qwrto snoro th), kntrou a = (a, b, c) kai aktna . An sumbolsoume thn mpla aut me

B (a) ja qoume

B (a) ={r R3 : r a < } = {(x, y, z) R3 :(x a)2 + (y b)2 + (z c)2 < } .

Qrhsimopoiome ton ro mpla, diti sfara kntrou a = (a, b, c) kai aktna enai h

epifneia

S (a) ={r R3 : r a = } = {(x, y, z) R3 :(x a)2 + (y b)2 + (z c)2 = } .

2.1.2 Merik pargwgo

Mia prosggish th nnoia th paraggou mia omal sunrthsh p.q trin metabl-

htn, (x, y, z), enai h akloujh. An jewrsoume pro stigmn ti metablht y kai z

w stajer, tte mporome na paragwgsoume thn w pro x, pw akrib kai sthn


perptwsh sunrthsh mia metablht. H pargwgo w pro x lgetai merik pargwgo

w pro x kai sumbolzetai me /x, dhlad

x=

d

dx

(y,z)=const

= limh0

(x+ h, y, z) (x, y, z)h

.

'Omoia orzontai kai oi merik pargwgoi th w pro y kai w pro z. Gia pardeigma,

gia thn (x, y) = x2 + xy5, /x = 2x + y5 kai /y = 5xy4. 'Omoia genikeetai h

nnoia th merik paraggou gia bajmwt peda : Rn R.H merik pargwgo w pro x, ekfrzei to rujm metabol th tan metabanoume

ap to shmeo (x, y, z), se na geitonik shmeo pou brsketai sthn eujea pou pern ap

to (x, y, z) kai enai parllhlh me ton xona x.

Orism 2.2. Ma sunrthsh : U R lgetai paragwgsimh sto U Rn an le oimerik pargwgoi th uprqoun kai enai suneqe sto U . Sthn perptwsh aut lme

ti to bajmwt pedo enai txh C1 (U).

Oi detere merik pargwgoi orzontai w

2

x2=

x

(

x

),

2

y2=

y

(

y

),

2

xy=

x

(

y

),

2

yx=

y

(

x

).

P.q. sto parapnw pardeigma (gia thn (x, y) = x2 + xy5) prokptei 2/x2 =

2, 2/y2 = 20xy3, 2/xy = 5y4, 2/yx = 5y4. Oi meikt pargwgoi 2/xy

kai 2/yx enai se gia le ti kal sunartsei. Akribstera (bl p.q. Finney et

al), isqei ti:

Jerhma 2.1.1. An 2/x, 2/y2 kai 2/xy enai suneqe se mia perioq tou

(x, y), tte oi meikt pargwgoi 2/xy kai 2/yx enai se sto shmeo (x, y).

Sto ex, ja upojtoume ti la ta bajmwt peda enai paragwgsima (txh C1). An

qreiazmaste detere trte merik paraggou, ja upojtoume ti ta bajmwt peda

enai txh C2 C3, akma kai C.

2.1.3 Bajmda gradient

Upenjumzoume ti se mia distash, df/dx ekfrzei to rujm metabol th f . Se trei

diastsei, mporome na jewrsoume trei rujmo metabol mia sunrthsh : R3 R,pou antistoiqon sti trei merik paraggou. Se kje shmeo r = (x, y, z) R3, htrida twn merikn paraggwn (/x, /y, /z) orzei na dinusma tou R3, ra

enai na dianusmatik pedo ap to R3sto R

3. To dianusmatik aut pedo lgetai klsh,

bajmda, gradient tou , kai sumbolzetai me grad(r) (r),

= x

i +

yj+

zk.


Gia pardeigma, gia thn (x, y, z) = xey + z, qoume = (ey,xey, 1).Se mia distash, to diaforik df = f (x)dx ekfrzei thn apeirost metabol th

sunrthsh (proseggistik, tan h anexrthth metablht metablletai katx, h metabol

th sunrthsh enai f f (x)x). Se trei diastsei, to diaforik th grfetai

d =

xdx+

ydy +

zdz. (2.1.2)

To diaforik d ekfrzei thn apeirost metabol th , me thn ex nnoia: tan oi

anexrthte metablht metabllontai kat x, y, z , h metabol th sunrthsh

dnetai proseggistik ap th sqsh

x

x+

yy +

zz. (2.1.3)

H austhr shmasa th (2.1.3) ja gnei katanoht se epmenh pargrafo (bl Jerhma

Taylor). Shmeinoume ti to diaforik (2.1.2) mpore na grafe kai w

d = (r).dr. (2.1.4)

Brete to gradient th (x, y, z) = r =x2 + y2 + z2. Parathreste ti enai kjeto

sthn epifneia sfara aktna r.

H idithta aut tou grad enai genik: enai orjognio sti isostajmik epifneieS tou . Austhr apdeixh ja dome sthn epmenh pargrafo, all mporome na peisjome

ap to akloujo epiqerhma. Pnw se mia isostajmik epifneia S,

S = {(x, y, z) R3 : (x, y, z) = const},

h sunrthsh enai stajer, dhlad d = 0, pou ta diaforik dx, dy, dz sthn (2.1.2)

upologzontai pnw sthn epifneia S. Me lla lgia, to eswterik ginmeno (r).drenai mhdn, ra enai kjeto sthn epifneia S.

Sthn hlektrostatik, h ntash tou hlektriko pedou E sundetai me to dunamik msw th E = . Elgxete th sqsh aut gia thn perptwsh (r) = kQ/r.

Sunpeie tou orismo enai oi paraktw idithte.

An = const, tte = 0

(+ ) = +

() = +

(/) = ( ) /2, eke pou 6= 0.


2.1.4 Pargwgo kat mko kamplh kai

kateujunmenh pargwgo

Sthn pargrafo aut ja orsoume to rujm metabol en bajmwto pedou kat

mko mia kamplh r (t). Jewrome th snjesh tou bajmwto pedou : R3 Rme thn kamplh r : R R3 (ousiwd prkeitai gia ton periorism th sthn kamplhC = {r R3 : r = r (t)}). Tte h snjesh tou, r, enai mia pragmatik sunrthshpragmatik metablht me tpo

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

pou r (t) = (x(t), y(t), z(t)). Sto ex ja qrhsimopoiome to dio grmma, , gia th sn-

jesh. O kanna th alusda genikeetai me profan trpo, dhlad

d

dt=

x

dx

dt+

y

dy

dt+

z

dz

dt, (2.1.5)

, suntomografik

d

dt= .r, (2.1.6)

pou ennoetai ti = (r (t)) kai r = (x(t), y(t), z(t)). O tpo (2.1.6) parqei thnpargwgo th kat mko th kamplh r (t) kai paristnei to rujm metabol th

kat mko th kamplh r (t).

Sthn eidik perptwsh pou h r (t) enai h eujea pou pern ap to shmeo a kai qei

th diejunsh tou diansmato v, dhlad r (t) = a + tv, tte qoume thn pargwgo kat

thn katejunsh v, kateujunmenh pargwgo, Dv, pou ja orsoume genik se epmenh

pargrafo, bl. (2.1.8). To detero mlo th (2.1.6) enai v. (a) kai sunep qoumeton tpo

Dv (a) = v. (a) . (2.1.7)Ap thn (2.1.6) prokptei loipn ti gia tuqosa kamplh r (t), to eswterik ginmeno

.r enai h pargwgo th kat thn katejunsh r.

Idithte tou Emaste tra se jsh na dome th gewmetrik shmasa th klsh (gradient).

Jerhma 2.1.2. An (r) 6= 0, tte h klsh (r) deqnei pro ekenh thn katejunshkat thn opoa h auxnei grhgortera.

Apdeixh. 'Estw n na monadiao dinusma. Tte o rujm metabol th kat mko

tou n enai n. (r) = | (r)| cos , pou enai h gwna metax n kai (r). 'Ara orujm gnetai mgisto tan n enai omrropo tou (r) , ( = 0).


PSfrag replaements

S

C

r(a)

(x; y +y)

_r(0)

a = r(0)

x

y

z

Sqma 2.2: To (a) enai kjeto se kje kamplh th epifneia stjmh pou pernap to a.

H llh idithta th klsh en bajmwto , enai ti enai kjeto sti epifneiestjmh th sunrthsh. Akribstera, isqei to paraktw jerhma.

Jerhma 2.1.3. 'Estw mia paragwgsimh sunrthsh trin metablhtn kai Sk h

epifneia stjmh me tim k, Sk = {r R3 : (r) = k}. 'Estw na shmeo a th Sk kai miaopoiadpote kamplh ep th epifneia Sk, r : (, ) Sk, pou th stigm t = 0 pernap to a. Tte (a) .r (0) = 0.

Apdeixh. 'Estw ti r(t) periqetai sthn Sk kai r(0) = a. Tte r (0) enai efaptmeno

dinusma sthn Sk sto shmeo a. H sunrthsh enai stajer ep th Sk, ra (r(t)) = k.

Paragwgzonta brskoume

0 =d

dt

t=0

= (a) .r (0) ,

dhlad (a) r (0).

To apotlesma isqei gia kje efaptmeno dinusma sthn Sk sto shmeo a, dedomnou

ti h kamplh r enai tuqosa.

Epanextash th kateujunmenh paraggou

'Opw edame oi merik pargwgo th w pro x, y, z, ekfrzoun tou rujmo metabol

th kat ti dieujnsei x, y, z. Me th sqsh (2.1.7), epitqame na perigryoume to rujm

metabol th kat mi opoiadpote diejunsh v. 'Opw ja dome euj amsw, sto

apotlesma aut mporome na ftsoume jewrhtik qwr na persoume ap thn nnoia th

merik paraggou.

Endiafermaste na orsoume thn pargwgo se na shmeo a tou U , en bajmwto

pedou pou orzetai se mia perioq U Rn. H pargwgo ekfrzei to rujm metabol


th tan metabanoume ap to a se na geitonik tou shmeo. O rujm aut exarttai

ap th diejunsh pou kinomaste xekinnta ap to a. Jewrnta loipn na geitonik

shmeo tou a pou brsketai sthn eujea pou pern ap to shmeo a kai qei th diejunsh tou

diansmato v, aut ja grfetai a+ tv, me t R. 'Ena elogo orism th paraggouth sto shmeo a, enai to rio (an uprqei) kaj t 0 tou phlkou

limt0

(a+ tv) (a)t

. (2.1.8)

To rio aut lgetai kateujunmenh pargwgo th sto shmeo a sthn katejunsh tou

diansmato v kai sumbolzetai me Dv (a). Gia thn kateujunmenh pargwgo Dn mia

paragwgsimh sunrthsh kat th diejunsh tou monadiaou n, qrhsimopoietai kai to

smbolo /n.

H sqsh (2.1.7) apotele tpo upologismo th Dv (a). Shmeinoume ti o genik

orism (2.1.8) emperiqei thn nnoia th merik paraggou diti gia na bajmwto pedo

, p.q. trin metablhtn, prokptei

Di (a) = (a)

x, Dj (a) =

(a)

y, Dk (a) =

(a)

z.

Me lla lgia, h merik pargwgo p.q. w pro x, den enai par h kateujunmenh

pargwgo kat thn diejunsh tou diansmato i = (1, 0, 0). Prgmati, gia na bajmwto

pedo , p.q. do metablhtn, ja qoume sto shmeo a = (a, b)

Di (a) = limt0

(a+ ti) (a)t

= limt0

(a + t, b) (a, b)t

= (a)

x.

Teleinoume me mi teqnik fsew paratrhsh. Upenjumzoume ti h pargwgo

mia pragmatik sunrthsh mia metablht, orzetai sto eswterik kpoiou diastmato

[a, b]. An epijumome na epektenoume thn pargwgo sto na kro tou diastmato, to pol

na orzetai h pleurik pargwgo. Sthn perptwsh tou bajmwto pedou , gia na orzetai

h pargwgo sto shmeo a tou U , prpei a+ tv na paramnei sto U (gia pol mikr t).

Me lla lgia, prpei to a na enai eswterik shmeo tou U , me thn nnoia ti mia mikr

mpla me kntro to a prpei na ketai ex' oloklrou msa sto U . Ja lme loipn ti to

a U enai eswterik shmeo tou U an uprqei > 0, ttoio ste to snolo twn shmewnr me r a < , na periqetai sto U .

Asksei

1. Sqediste merik isostajmik kai merik isouye kample th (x, y) = x2+2y2.

2. Sqediste merik isostajmik kample th sunrthsh me tpo

(x, y) = 2x2 + y2.


Brete to gradient sta shmea (1, 1) , (1, 1) . Sqediste to up klmaka (kat'ektmhsh) se tra shmea tou sqmato. Upologste thn pargwgo th kat mko

th kamplh

r(t) = x(t) i + y(t) j me x(t) =12cos t kai y(t) = sin t, t [0, 2] .

Exhgeste giat to apotlesma tan anamenmeno.

3. Brete ta z/u, z/ an z = x2 + 2xy, x = u cos , y = u sin . Brete ta z/u,

z/ an z = ex + xy2, x = u+ , y = eu+. Brete thn dw/dt an w = x2 + y2 + z2,

x (t) = et cos t, y (t) = et sin t, z (t) = et.

4. An z = f (u) , u = (x+ y) /xy, dexte ti

x2z

x= y2

z

y.

5. An w = x2 + y z + sin t kai x+ y = t, brete ta(w

y

)x,t

,

(w

y

)z,t

,

(w

z

)x,y

,

(w

t

)x,z

,

(w

z

)x,t

.

6. Dexte ti, an (x, y) = 0, tte

y

x= /x

/y

arke /y 6= 0. An x2 + y2 + z2 = 4, brete ti z/x, z/y.

7. Dexte ti Dn = Dn. An n, tte Dn = 0.

8. Kat pso ja metablhje h (x, y, z) = lnx2 + y2 + z2 an to shmeo (3, 4, 12)

metakinhje kat s = 0.1 kat mko tou 3i+ 6j 2k?

9. Gnwrzoume ti an S = {(x, y, z) R3 : (x, y, z) = c} enai mia isostajmik epifneiath , tte to enai kjeto se kje kamplh ep th S. Gryte thn exswsh touefaptomnou epipdou shn S sto shmeo (x0, y0, z0) . Brete to efaptmeno eppedo

kai thn kjeth sthn epifneia x2 xy + z3 = 1 sto shmeo (1, 1, 1) .

10. To uperboloeid x2 + y2 z2 = 1 kai to eppedo x + y + z = 5 tmnontai se miakamplh C. Brete thn efaptomnh th C sto shmeo (1, 2, 2) .

11. 'Ena swmatdio kinetai kat mko th kamplh r (t) = t2 i t j + ln 2tk. Dexte

ti th stigm t = 1 qtupei kjeta thn epifneia z = ln (y + 2x2 y2).

12. Perigryte ti paraktw epifneie.

2.2. DIANUSMATIKA PEDIA 33

a) Elleiptik klindro y2 + 4x2 = 4, 2 z 2b) Elleiyoeid

x2

a2+y2

b2+z2

c2= 1

g) Elleiptik paraboloeid z = a2x2 + b2y2

d) Elleiptik kno z2 = a2x2 + b2y2

e) Monqwno uperboloeid

x2

a2+y2

b2 z

2

c2= 1

st) Dqwno uperboloeid

x2

a2+y2

b2 z

2

c2= 1

z) Uperbolik paraboloeid z = a2x2 b2y2

2.2 Dianusmatik peda

'Opw edame sthn eisagwg, na dianusmatik pedo (d.p.) ston Rnenai mia apeiknish

F : U Rn (pou U Rn) pou se kje shmeo r tou U antistoiqe na dinusma F(r).Sti trei diastsei, grfoume suntomografik, F = (Fx, Fy, Fy) , F = (F1, F2, F3) ,

F = F1i+F2j+F3k kai ennoome fusik ti oi sunistse Fi enai bajmwt sunartsei,

F(r) = F (x, y, z) = F1 (x, y, z) i+ F2 (x, y, z) j+ F3 (x, y, z)k.

To aplostero pardeigma d.p. enai mia grammik apeiknish, p.q. to d.p. F : R2 R2me tpo (bl. (1.3.1) kai epmenh paratrhsh)

F (x, y) = (ax+ by, cx+ dy) .

Mia optik parstash gia dianusmatik peda se do diastsei mporome na proumeap th Mathematica. Dokimste ta dianusmatik peda

x i+ y j, yi+ x j, yx2 + y2

i+x

x2 + y2j, F(r) =

r

r3.

Sto ex, ja upojtoume ti la ta dianusmatik peda enai paragwgsima txhtoulqiston C1 (U), dhlad ja upojtoume ti an F enai na dianusmatik pedo,

le oi merik pargwgoi twn sunistwsn Fi tou F, uprqoun kai enai suneqe

sto U . Sta parapnw paradegmata prosdiorste to mgisto uposnolo tou R2 sto

opoo ta dianusmatik peda enai txh C1. Poia ap aut mporon na grafon w

grad, kpoia bajmwt sunrthsh : R2 R?


Mia kamplh pou se kje shmeo th r, to dinusma F(r) enai efaptmeno sthn kamplh,

lgetai oloklhrwtik kamplh tou F. Akribstera:

Orism 2.3. Ma kamplh r(t) lgetai oloklhrwtik kamplh en dianusmatiko

pedou F, an isqei

r(t) = F(r(t)).

Oi oloklhrwtik kample lgontai akma pediak gramm (field lines), dunamik

gramm.

Gia pardeigma, oi oloklhrwtik kample tou dianusmatiko pedou F(x, y) = r =

(x, y) enai eujee pou pernon ap thn arq twn axnwn. 'Omoia, oi oloklhrwtik kam-

ple tou pedou Coulomb

E(r) = Qr

r3= Q

(x

(x2 + y2 + z2)3/2i+

y

(x2 + y2 + z2)3/2j+

z

(x2 + y2 + z2)3/2k

), Q R,

enai eujee pou pernon ap thn arq twn axnwn.

Gia na brome thn oloklhrwtik kamplh tou F pou pern ap to shmeo r0 th stigm

t = 0, prpei na lsoume to prblhma arqikn timn

r(t) = F(r(t)), r(0) = r0. (2.2.1)

Me lla lgia, to prblhma angetai sthn eplush tou sustmato twn diaforikn ex-

issewn

x = F1 (x, y, z) , y = F2 (x, y, z) , z = F3 (x, y, z) ,

me arqik sunjke x (0) = x0, y (0) = y0, z (0) = z0.

Oi oloklhrwtik kample en dianusmatiko pedou qoun do basik idithte.

Ap kje shmeo r0 (msa sto pedo orismo en dianusmatiko pedou F), pern mia

oloklhrwtik kamplh.

Oi oloklhrwtik kample tou F den tmnontai.

Oi parapnw isqurismo gnontai profane an jumhjome to jerhma parxh lsh

kai monadikthta th lsh sustmato diaforikn exissewn: To prblhma arqikn

timn (2.2.1) qei monadik lsh.

Brete ti oloklhrwtik kample twn (grammikn) dianusmatikn pedwn F metpou

F(x, y) = (y,x), F(x, y) = (x,y).


2.2.1 O telest andelta. Apklish kai strobilism

Orism 2.4. O dianusmatik telest

= i x

+ j

y+ k

z,

lgetai telest andelta.

'Eqoume dei ti tan o telest andelta, , dr pnw se na bajmwt pedo , pargeito dianusmatik pedo grad = (/x, /y, /z) . Mporome na jewrsoume th

drsh tou telest pnw se dianusmatik peda. To apotlesma exarttai ap to an otelest dra eswterik exwterik pnw sto d. p. 'Etsi gia na dianusmatik pedo

F = (F1, F2, F3) orzoume

Apklish (divergence):

divF = .F = F1x

+F2y

+F3z

Strobilism (curl):

curlF = F =

i j kx

y

z

F1 F2 F3

Shmeinoume ti h apklish enai bajmwto pedo en o strobilism enai dianusmatik

pedo.

Pardeigma 2.2.1. Gia to d.p. F(x, y, z) = r, div r = 3 kai curl r = 0. Gia to d.p.

F(x, y, z) = (y, x, 0) , > 0, enai divF = 0 kai curlF = 2k.

'Ena dianusmatik pedo F lgetai astrbilo an curlF = 0. To paraktw jerhma

deqnei ti to gradient kje bajmwt sunrthsh enai astrbilo d. p.

Jerhma 2.2.1. Gia kje bajmwt pedo (pou enai toulqiston txh C2) isqei

curl (grad) = = 0.

Apdeixh. Ap ton orism tou strobilismo ja qoume

= i

y

z

y

z

j

x

z

x

z

+ k

x

y

x

y

= i

(2

yz

2

zy

)+ j

(2

zx

2

xz

)+ k

(2

xy

2

yx

),

kai to apotlesma prokptei ap thn isthta twn meiktn paraggwn.


Se antjesh me ton strobilism tou gradient en bajmwto pedou pou enai pnta

mhdn, h apklish tou gradient enai na no bajmwt pedo . pou sumbolzetai me2, kai diabzetai Laplacian . Epomnw h Laplacian tou grfetai

2 = 2

x2+2

y2+2

z2.

O telest

2 = 2

x2+

2

y2+

2

z2,

lgetai telest Laplace. Tlo h diaforik exswsh

2 = 0, 2

x2+2

y2+2

z2= 0,

lgetai exswsh Laplace. 'Opw ja dome, h exswsh Laplace enai mi ap ti spoudaitere

diaforik exissei me merik paraggou. Profan se do diastsei h exswsh

Laplace grfetai2

x2+2

y2= 0.

To dunamik tou hlektriko pedou pou pargetai ap shmeiak forto Q, dnetaiap ton tpo

(x, y, z) =Q

r, r = |r| .

Dexte ti h ntash tou hlektriko pedou dnetai ap th sqsh E = . Breteto divE kai to curlE. Apodexte ti to dunamik ikanopoie thn exswsh Laplace.

O telest Laplace enai grammik, dhlad an kai enai bajmwt peda (dofor paragwgsima) kai c arijm, tte

2 (+ ) = 2+2, kai 2 (c) = c2.

Toto shmanei ti h exswsh Laplace enai grammik DE, dhlad an u kai v enai

lsei th exswsh Laplace, tte kai u+ v enai lsh.

'Estw : R3 R bajmwt pedo txh C3. Poi ap ti paraktw parastseienai lanjasmne? Exhgeste.

div (curl (grad)) , curl (div (grad)) , div (grad (curl )) ,

grad (curl (div )) , curl (grad (div )) , grad (div (curl )) .

'Ena dianusmatik pedo A lgetai asumpesto an divA = 0. To paraktw jerhma

deqnei ti o strobilism opoioudpote d.p. enai asumpesto d. p.


Jerhma 2.2.2. Gia kje dianusmatik pedo F (pou enai toulqiston txh C2) isqei

div (curlF) = . ( F) = 0.

Apdeixh. H apdeixh enai parmoia me autn tou jewrmato 2.2.1.

Oi paraktw tautthte mporon na deiqjon me ektlesh twn prxewn.

div (F+G) = divF+ divG, curl (F+G) = curlF+ curlG

. (F) = .F+ .F (F) = F+ F ( F) = (.F)2F

Asksei

1. 'Estw bajmwt sunrthsh kai a stajer dinusma. Brete ta (a.) kai (a).'Estw F kai G dianusmatik peda. Giat (a.)F den enai so me F(a.)? Dexte ti(G).F = G.( F). Dexte ti

(F.)F = ( F) F+12 |F|2 .

2. Knete prqeiro sktso twn d.p.

yi xj, xi + yj, yri x

rj,

y

r3i+

x

r3j, me r = |r| .

Poi ap aut enai astrbila?

3. Upenjumzoume ti h oloklhrwtik kamplh r (t) = (x (t) , y (t) , z (t)) en d.p. F =

(F1, F2, F3) ikanopoie ti diaforik exissei

dx

dt= F1 (x, y, z) ,

dy

dt= F2 (x, y, z) ,

dz

dt= F3 (x, y, z) .

Dexte ti

dx

F1=

dy

F2=

dz

F3,

pou ennoetai ti an pq F1 = 0 tte dx = 0. Brete ti oloklhrwtik kample twn

d.p.

F (x, y, z) = (2 cosh x, 2y sinh x, sinh x) ,G (x, y, z) = yi xj.


4. H knhsh idaniko reusto perigrfetai ap ti exissei Euler

u

t+ (u.)u = p

,

pou u h taqthta tou reusto, h puknthta tou, p h pesh kai to exwterik

dunamik (sunjw barthta). Orzoume na no d.p. , w = u. Qrhsi-mopointa thn tautthta (u.)u = ( u) u+ 1

2 (u.u) , dexte ti h exswsh

twn reustn grfetai

u

t+ u+1

2u2 = p

,

kai parnonta ton strobilism kai sta duo mlh prokptei

t+ ( u) = 0.

5. H Laplacian en bajmwto pedou f se polik suntetagmne x = r cos , y = r sin

dnetai ap

2f = 2f

2r+

1

r

f

r+

1

r22f

2.

(Updeixh:

fr

= fx

xr

+ fy

y

= fx

cos + fysin , klp).

6. Apodexte ti oi paraktw sunartsei enai armonik (dhl. ikanopoion thn exswsh

Laplace)

(x, y, z) = x2 + y2 2z2, (x, y) = lnx2 + y2, (x, y) = tan1 (y/x) .

2.2.2 Efarmog: Metdosh jermthta

A upojsoume ti jermanoume mia perioq en smato. Tte jermthta rei ap shmea

uyhl jermokrasa se shmea qamhl jermokrasa. H jermokrasa enai mia sunrthsh

th jsh, T (x, y, z). H ro jermthta qei diaforetik katejunsh se kje shmeo, ra

parstatai ap na dianusmatik pedo h. Akribstera, to mtro tou h enai to phlko th

jermik enrgeia an monda qrnou I = dQ/dt pou pern kjeta ap mia stoiqeidh

epifneia, S, dia th epifneia aut

h =dI

dS.

An loipn n enai to monadiao kjeto sto dS, tte h ro jermthta (an monda qrnou

kai an monda embado) enai h.n, dhlad enai h sunistsa tou h sthn kjeth sthn

epifneia. Poia enai h sqsh metax ro jermthta h kai jermokrasa T ? Ap to

perama gnwrzoume ti an metax twn pleurn mia plka pqou l kai embado S,

2.3. TO JEWRHMA TAYLOR 39

epikrate diafor jermokrasa T = T2 T1, tte h jermthta an monda qrnou poupern ap thn plka dnetai ap th sqsh

I = kST

l,

pou k enai o suntelest jermik agwgimthta tou uliko. Se poio polploka smata,

p.q. se mia jalssia perioq, prosanatolzoume topik ti pleur mia stoiqeidou

plka embado S parllhla pro ti isjerme epifneie sthn perioq th plka,

opte ja qoume I/S = kT/l, me l to pqo th plka. To I/S, enai

to mtro th ro jermthta h, pou h katejunsh tou enai ap thn isjermh T + T

pro thn isjermh T , dhlad enai kjeth pro ti isjerme. Ap thn llh, T/l, o

rujm metabol th jermokrasa, enai o mgisto rujm, diti h metabol jsh l

enai kjeth sti isjerme. Kat sunpeia, o rujm T/l, enai to mtro tou T .Sumperanoume ti ta diansmata h kai T enai suggrammik, ra

h = kT.

To prshmo meon, tjetai diti h jermthta rei pro thn katejunsh pou h jermokrasa

elattnetai. H exswsh pou katalxame apotele, pw ja dome, eidik perptwsh tou

nmou tou Fick.

2.3 To Jerhma Taylor

To Jerhma Taylor se prosggish prth txh mia sunrthsh mia metablht grfetai

w

f (x) = f (a) + f (a) (x a) +R1 (x) , me limxa

R1 (x)

x a = 0.

Aut shmanei ti to sflma enai mikr se sgkrish me thn dh mikr posthta x a. Sedeterh txh prosggish to Jerhma Taylor grfetai

f (x) = f (a) + f (a) (x a) + f (a)2!

(x a)2 +R2 (x) , me limxa

R2 (x)

(x a)2 = 0.

'Alloi trpoi graf tou Jewrmato Taylor enai

f (x+x) = f (x) + f (x)x+f (x)

2!x2 +

f (x)

3!x3 + ...

f (x+x) = f (x) + f (x)x+O (x2) .Se do diastsei, to Jerhma Taylor diatupnetai w ex.


Jerhma 2.3.1. 'Estw : U R na bajmwt pedo me suneqe merik parag goukje txh sto pedo orismo tou U R2. An a = (a, b) kai r = (x, y) enai duo geitonikshmea sto U tte

(x, y) = (a, b) +

x(x a) +

y(y b) +R1 (x, y) ,

pou oi merik pargwgoi upologzontai sto (a, b) kai

limra

R1 (r)

|r a| = 0.

Se deterh txh prosggish ja qoume

(x, y) = (a, b) +

x(x a) +

y(y b) + 1

2

2

x2(x a)2 + 1

2

2

y2(y b)2 +

2

xy(x a) (y b) +R2 (x, y)


limra

R2 (r)

|r a|2 = 0.

Se trth txh prosggish ja qoume

(x, y) = (a, b) +

[(x a)

x+ (y b)

y

] (x, y) +

1

2!

[(x a)

x+ (y b)

y

]2 (x, y) +

1

3!

[(x a)

x+ (y b)

y

]3 (x, y) +R3 (x, y)


limra

R3 (r)

|r a|3 = 0.

An to uploipo Rn (r) tenei sto mhdn kaj n , tte h seir Taylor sugklneikai mporome na gryoume

(x, y) = (a, b) +

[(x a)

x+ (y b)

y

] (x, y) +

1

2!

[(x a)

x+ (y b)

y

]2 (x, y) + ...

'Allo trpo graf enai o akloujo:

(x+x, y +y) = (x, y) +

xx+

yy +

1

2!

[x

x+y

y

]2 (x, y) + ...

2.3. TO JEWRHMA TAYLOR 41

akma

(x+x, y +y) = (x, y) +

xx+

yy +O (2) ,

pou oi merik pargwgoi upologzontai sto (x, y).

To apotlesma genikeetai se trei diastsei kat profan trpo:

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

(x, y, z) +

xx+

yy +

zz +

1

2!

[x

x+y

y+z

z

]2 (x, y, z) + ...

Sunjw grfoume

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

xx+

yy +

zz +O (2) .

Paraleponta tou rou anterh txh ja qoume gia thn metabol tou bajmwto

pedou

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

x+

yy +

zz,

pou oi merik pargwgoi upologzontai sto (x, y, z). 'Etsi h shmasa th (2.1.3) gnetai

katanoht me austhr trpo.

Sumbolik morf th seir Taylor

Me th bojeia tou telest paraggish

D =d

dx, D2 =

d2

dx2, ... Dn =

dn

dxn,

h seir Taylor mia sunrthsh mia metablht f, grfetai

f (x+ h) = f (x) +hD

1!f (x) +

h2D2

2!f (x) +

h3D3

3!f (x) + ...

=

(I +

hD

1!+h2D2

2!+h3D3

3!+ ...

)f (x) ,

kai sumbolik

f (x+ h) = ehDf (x) .

Dexte ti sti trei diastsei to parapnw apotlesma genikeetai w

(x+ h1, y + h2, z + h3) = eh. (x, y, z) ,

pou h = (h1, h2, h3).


2.4 Akrtata bajmwtn pedwn

'Estw : U R na bajmwt pedo orismno se na qwro U Rn. To a U lgetaishmeo topiko megstou gia thn , an uprqei mpla, B (a) , kntrou a kai aktna

ttoia ste

x B (a) (x) (a) .Anloga orzetai to shmeo topiko elaqstou.

Sto Logism sunartsewn mia metablht, h pargwgo mhdenzetai sta shmea top-

iko akrottou. To anlogo jerhma sthn perptwsh perissotrwn diastsewn diatupne-

tai w ex.

Jerhma 2.4.1. An a enai shmeo topiko akrottou gia thn , tte le oi merik

pargwgoi th sto a mhdenzontai,

xi

x=a

= 0, i = 1, 2, ..., n.

An /xi|x=a = 0, i = 1, 2, ..., n, tte lme ti to a enai krsimo shmeo gia thn .Pardeigma 2.4.1. (a) Gia thn (x, y) = 2 x2 y2 to (0, 0) enai shmeo topiko (kaioliko) megstou diti (x, y) 2 = (0, 0) (x, y) R2.

(b) H sunrthsh (x, y) = x2 + y2 qei (olik) elqisto sto (0, 0).

(g) Gia thn (x, y) = xy to (0, 0) den enai shmeo ote megstou ote elaqstou diti

sto 1o kai 3o tetarthmrio enai (x, y) > (0, 0) = 0 en sto 2o kai 4o tetarthmrio

enai (x, y) < (0, 0) = 0. To (0, 0) qarakthrzetai w sagmatoeid shmeo.

(d) Gia thn (x, y) = x2y2 to (0, 0) enai shmeo topiko elaqstou.

Sto Logism mia metablht na krsimo shmeo qarakthrzetai ap to prshmo th

deterh paraggou. To anlogo jerhma sthn perptwsh twn do diastsewn diatupne-

tai w ex.

Jerhma 2.4.2. 'Estw ti h : U R enai do for paragwgisimh kai a = (a, b) U R2 enai shmeo topiko akrottou gia thn , dhlad

x

(a,b)

= 0 =

y

(a,b)

.

Jewrome ton pnaka

H () =

[2x2

2xy

2xy

2y2

](a,b)

An detH() > 0 kai 2/x2|(a,b) > 0, tte to (a, b) enai shmeo topiko elaqstou.An detH() > 0 kai 2/x2|(a,b) < 0, tte to (a, b) enai shmeo topiko megstou.An detH() < 0, tte to (a, b) enai sagmatoeid shmeo.

An detH() = 0, tte den mporome na apofanjome.

2.5. TO JEWRHMA TWN PEPLEGMENWN SUNARTHSEWN 43

Pardeigma 2.4.2. Gia thn sunrthsh ln (x2 + y2 + 1), monadik krsimo shmeo enai

to (0, 0). Efarmog tou Jewrmato deqnei ti to (0, 0) enai topik elqisto.

Pardeigma 2.4.3. (pro apofugn tou Jewrmato). Suqn oi merik pargwgoi d-

nontai ap polploke ekfrsei me sunpeia akma kai o entopism twn krsimwn shmewn

na enai dusqerstato. Prospajome tte na antimetwpsoume to prblhma me pio xupnh

prosggish. 'Etsi, gia thn sunrthsh

(x, y) = exp

(1

x2 + 2 + cos2 y 2 cos y)

mporome pol apl na meletsoume thn g (x, y) = x2 + 2 + cos2 y 2 cos y pou grfetaiw g (x, y) = x2 + 1 + (1 cos y)2. H g parnei elqisth tim tan x2 + (1 cos y)2 = 0,dhlad sta shmea (0, 2k). 'Ara sta shmea aut h qei mgisto.

2.5 To jerhma twn peplegmnwn sunartsewn

Se mia distash, an mia paragwgsimh sunrthsh f ikanopoie thn sunjkh f (a) 6= 0 sekpoio shmeo a, tte uprqei h antstrofh f1, dhlad topik kont sto a, mporome na

lsoume thn exswsh y = f(x) w pro x,

x = f1 (y) .

I II III IV

PSfrag replaements

x

x

x

x

yy

yyyy

Sqma 2.3: ( I) Uprqei lsh y = g(x). ( II) Uprqei lsh x = h(y). ( III) kai ( IV)

Mno topik uprqei lsh y = g(x) x = h(y).

Se do diastsei, to C snolo stjmh mia sunrthsh : R2 R enai en gneimia kamplh pou paristnetai ap thn exswsh

(x, y) = C. (2.5.1)

(Upojtoume sto ex ti le oi sunartsei qoun suneqe merik paraggou). T-

jetai to erthma an mporome na lsoume thn (2.5.1) w pro y ( w pro x), dhlad


na brome mia sunrthsh g : y = g(x) ( antstoiqa x = h(y)) pou to grfhma th na

tautzetai me to snolo stjmh th . Sto sqma fanontai merik paradegmata.

'Allo pardeigma: Sthn x2+y2 = 1, mporome na lsoume w pro y (gia y > 0) kai na

qoume topik, dhlad sto disthma (1, 1), y = 1 x2 . 'Omoia, mporome na lsoumew pro x kai na qoume gia y (1, 1), x =

1 y2.

Qwr blbh th genikthta, mporome na jewrome to mhdenik snolo stjmh en

bajmwto pedou, dedomnou ti h (x, y) = C mpore pnta na grafe w (x, y)C = 0.Jerhma 2.5.1. 'Estw : U R kai na shmeo (a, b) U R2 ttoio ste

(1) (a, b) = 0,

(2)

y

(a,b)

6= 0.

Tte uprqoun perioq twn a kai b ttoie ste na uprqei lsh w pro y th exswsh

(x, y) = 0, dhlad uprqei monadik sunrthsh g ttoia ste to grfhma th y = g(x)

tautzetai me to mhdenik snolo stjmh th . Akribstera,

g (a) = b kai (x, g (x)) = 0

sthn perioq tou a. Ep plon, h pargwgo dy/dx = g (x) mpore na breje me mmesh

parag gish th (x, y) = 0. Prgmati, paragwgzonta w pro x thn (x, y) = 0,

parnoume

x+

y

dy

dx= 0 dy

dx= /x

/y.

To jerhma genikeetai kai se perisstere diastsei. P.q. an h exswsh

(x, y, z) = C (2.5.2)

paristnei epifneia S, enai dunatn na brome mia sunrthsh do metablhtn tsi ste

z = (x, y)

na paristnei aut thn epifneia? (ousiwd na lsoume thn (2.5.2) w pro z).

Apnthsh: An se kpoio shmeo (a, b, c) S isqei/z 6= 0,

tte topik uprqei monadik sunrthsh ttoia ste h (2.5.2) na mpore na luje w

pro z,

z = (x, y) .

Ep plon, oi pargwgoi /x kai /y mporon na brejon me mmesh paraggish th

(x, y, z) = C,z

x= /x

/z,

z

y= /y

/z

2.5. TO JEWRHMA TWN PEPLEGMENWN SUNARTHSEWN 45

Pardeigma 2.5.1. Jewrome th monadiaa sfara x2+y2+z2 = 1, w mia isostajmik

epifneia th (x, y, z) = x2 + y2 + z2. Sto shmeo (0, 0, 1) enai z 6= 0, ra mporomena lsoume w pro z kai na qoume kont sto breio plo (0, 0, 1), z = 1 (x, y) 1 x2 y2 me x2 + y2 < 1. 'Omoia, kont sto ntio plo (0, 0,1), z = 2 (x, y)

1 x2 y2me x2 + y2 < 1. Sto shmeo (1, 0, 0) enai z = 0, ra den mporome na

lsoume w pro z kont sto shmeo aut.

Asksei

1. Brete to polunumo Taylor 2ou bajmo grw ap to (0, 0) th (x, y) = cos (x+ y2) .

2. Brete ta akrtata th (x, y) = x2 + y2 x y + 1 sto dsko D : x2 + y2 1.Krsima shmea: /x = 0 = /y, ra 2x1 = 0 = 2y1. Epomnw na krsimoshmeo enai to (1/2, 1/2) . H Hessian enai

H (x, y) =

[2/x2 2/xy

2/xy 2/y2

]=

[2 0

0 2

],

ra 2/x2|(1/2,1/2) > 0 kai detH (1/2, 1/2) = 4 > 0, epomnw to (1/2, 1/2) enaishmeo topiko elaqstou. Exetzoume tra ti gnetai sto snoro D : x2 + y2 = 1

pou parametropoietai ap thn r (t) = (cos t, sin t) , t [0, 2] . Tte

(r (t)) = cos2 t+ sin2 t cos t sin t+ 1 = 2 cos t sin t g (t) .

Parathrome ti arke na brome ta akrtata th g. 'Eqoume g (t) = 0 mno tan

sin t = cos t t = /4, 5/4. 'Ara upoyfia krsima shmea enai ta r (/4) , r (5/4)kai ta kra r (0) = r (2) . Brskoume ti tim th sta shmea aut kai sumpera-

noume (elgxete!) ti to (1/2, 1/2) enai shmeo elaqstou kai to(2/2,2/2)

enai shmeo megstou.

3. Exetste an h kamplh (x, y) = 0 me (x, y) = ecos(xy) + x2 + y e perigrfetaikont sto (0, 0) ap kpoia sunrthsh y = f (x) .

'Eqoume (0, 0) = 0 kai /x (0, 0) = 0, /y (0, 0) = 1. 'Ara kat to jerhma

peplegmnwn sunartsewn uprqei perioq (, ) tou 0 kai monadik sunrthshy = f (x) :

f (0) = 0, (x, f (x)) = 0 x (, ) .Epiprosjtw h klsh th y = f (x) sto 0 enai 0 :

dy

dx= f (x) = /x

/y=y sin (xy) ecos(xy) + 2xx sin (xy) ecos(xy) + 1 ,

ra f (0) = 0.


4. Apodexte ti h exswsh (x, y, z) = 0 me

(x, y, z) = ln (x+ y + z 2) ex+y 2x+ y + z

orzei mia epifneia sthn perioq tou (1, 1) pou pern ap to (1, 1, 1) kai perigrfetai

ap mia sunrthsh z = f (x, y) . Brete to efaptmeno eppedo sto (1, 1, 1).

-1.5 -1 -0.5 0 0.5 1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

5. To sqma paristnei ti isostajmik kample th sunrthsh

(x, y) =y2

2 x

2

2 x

3

3, x [1.8, 1.2] , y [0.8, 0.8] .

Dexte ti to (1, 0) enai shmeo topiko elaqstou th . Dikaiologeste th morftwn isostajmikn kampuln. Brete to gradient sta shmea (1

2, 0),(12, 0). Sto

prto shmeo jewrome ti kateujnsei twn monadiawn dianusmtwn n1 = (1, 0) kai

n2 =(

12, 1

2

). Kat mko poia ap ti do kateujnsei metablletai taqter-

a h ? Sqediste to up klmaka (kat' ektmhsh) se tra shmea tou sq-mato. Apodexete ti h paramnei stajer kat mko twn troqin tou dunamiko

sustmatox = y, y = x+ x2.

6. 'Estw (x, y, z) = x3y+x/y z kai P to shmeo (2, 1,1) . Upologste to gradient sto shmeo P. Upologste thn (kateujunmenh) pargwgo th sto shmeo Pkat th diejunsh tou i+ j k. Proseggste grammik thn tim tou sto shmeoQ : (2.1, 0.9,0.9) (parathreste ti to shmeo Q apqei l = 0.1 ap to P katth diejunsh u = i + j k). Jewrome thn isostajmik epifneia S th pouperiqei to shmeo P . Aut tmnei to eppedo z = 1 se ma kamplh C. Brete namonadiao efaptmeno th C sto shmeo P .

Keflaio 3

OLOKLHRWSH SUNARTHSEWN

POLLWN METABLHTWN

Ja meletsoume oloklhrmata pnw se kample (epikamplia oloklhrmata), pnw se

qwra tou R2(dipl oloklhrmata), pnw se perioq tou R

3(tripl oloklhrmata)

kai pnw se epifneie (epifaneiak oloklhrmata). Se kje perptwsh, jewrome ma

diamrish tou sunlou oloklrwsh, sqhmatzoume to katllhlo jroisma Riemann kai

parnoume to rio tan h leptthta th diamrish tenei sto mhdn.

3.1 Epikamplia oloklhrmata

A upojsoume ti na ulik shmeo kinetai kat mko mia kamplh

r(t) = x(t)i + y(t)j+ z(t)k, t I = [t1, t2]

kai dqetai mia dnamh F. Sta epmena ja jewrome kample apl kai lee, dhlad

kample pou den autotmnontai kai dr/dt 6= 0, t I. H dnamh exarttai en gnei ap thjsh, enai loipn na dianusmatik pedo

F : R3 R3.

Endiafermaste gia to rgo th dnamh, kat mko tou drmou A B. Upojtoumeti to swmatdio pern ap ta shmea A kai B ti stigm t1 kai t2 antstoiqa. Qwrzoume

kat ta gnwst to drmo se n stoiqeidh tmmata ri pou proseggistik jewrontai

eujgramma kai sqhmatzoume to jroisma Riemann

Wn =

ni=1

Fi.ri.

To rio tou ajrosmato Riemann tan n (kai tan h leptthta th diamrish|ri| tenei sto mhdn), lgetai epikamplio oloklrwma th F kat mko th kamplh

47

48 KEFALAIO 3. OLOKLHRWSH SUNARTHSEWN POLLWN METABLHTWN

C kai sumbolzetai C

F.dr,

BA

F.dr.

O trpo upologismo en epikampulou oloklhrmato parqetai ap to paraktw

jerhma pou parajtoume qwr apdeixh.

Jerhma 3.1.1. An to dianusmatik pedo F enai suneq kai h kamplh r = r(t) qei

suneq pargwgo, tte to epikamplio oloklrwma uprqei kai mlistaC

F.dr =

t2t1

F (r(t)) .r(t)dt.

Parathrsei. (a) To diaforik r(t)dt paristnei thn apeirost metatpish dr

kai ma jumzei th sqsh taqthtaqrno = metatpish(b) Enai profan ti an C enai h dia kamplh, all me ton antjeto prosanatolism,

tte C

F.dr = C

F.dr

(g) An h kamplh C enai kleist, tte to epikamplio oloklrwma tou F kat mko

th kamplh C sumbolzetai C

F.dr.

Pardeigma 3.1.1. An na swmatdio kinetai kat mko tou hmikuklou r(t) = (x(t), y(t))

me x(t) = 1+ cos t kai y(t) = 1+ sin t, t [0, ], kai dqetai dnamh F(x, y) = y2i x2j,tte to rgo ja enaiC

F.dr =

pi0

F (r(t)) .r(t)dt =

pi0

[(1 + sin t)2 i (1 + cos t)2 j] .( sin t i + cos t j)dt =

=

pi0

[ sin t (1 + sin t)2 cos t (1 + cos t)2] dt = 2 10/3.An to swmatdio kinetai kat mko tou eujgrammou tmmato A B, (pou parstataiparametrik ap ti x(t) = 2 t kai y(t) = 1, t [0, 2]), tte to rgo enai

C

F.dr =

20

F (r(t)) .r(t)dt =

20

[i (2 t)2 j] .(i)dt = 2

0

(1)dt = 2.

Ap to pardeigma aut prokptei ti to epikamplio oloklrwma en dianusmatiko

pedou ap na shmeo A se na llo shmeo B, exarttai en gnei ap ton drmo pou

akoloujome, dhlad ap thn kamplh C pou sundei ta shmea A kai B. En totoi, pw

ja dome sto jerhma 3, uprqoun dianusmatik peda pou to epikamplio oloklrwma

tou ap na shmeo A se na llo shmeo B, enai pnta anexrthto tou drmou. Ta

peda aut onomzontai sunthrhtik peda. Ekola mporome na apodexoume thn akloujh

isodunama.

3.1. EPIKAMPULIA OLOKLHRWMATA 49

Jerhma 3.1.2. To epikamplio oloklrwma en dianusmatiko pedou F enai anexrth-

to tou drmou an kai mno an to epikamplio oloklrwma tou F kat mko opoiasdpote

kleist kamplh C enai mhdn.

Apdeixh. 'Estw ti gia to dianusmatik pedo F isqei ti to epikamplio oloklrwma

tou enai anexrthto tou drmou. 'Estw akma mia opoiadpote kleist kamplh C kai do

shmea th, A kai B.

PSfrag replaements

A

B

C

1

C

2

C

C

a

C

b

Sqma 3.1: Do kample Ca kai Cb me koin kra.

Jewrome thn C w sunistmenh ap do kample Ca kai Cb me koin kra A kai B

kai grfoume sumbolik

C = Ca Cbopte

C

F.dr =

Ca

F.dr+

Cb

F.dr.

Epeid to oloklrwma

BAF.dr enai anexrthto tou drmou ja qoume

Ca

F.dr =

C

b

F.dr,

kai epeid

C

b

F.dr = CbF.dr, sumperanoume ti

CF.dr = 0. 'Omoia apodeiknetai kai

to antstrofo.

An to dianusmatik pedo F enai to gradient mia bajmwt sunrthsh , dhlad

F = , tte to epikamplio oloklrwma tou enai anexrthto tou drmou. Akribsteraisqei:

Jerhma 3.1.3. Gia kje bajmwt sunrthsh (pou enai toulqiston C1) kai gia

opoiadpote kamplh C pou sundei do opoiadpote shmea A kai B, isqei ti BA

.dr = (B) (A) .


Apdeixh. 'Estw mia tuqosa kamplh C pou sundei do opoiadpote shmea A kai B,

me parametrik anaparstash r = r(t), t [t1, t2]. Profan r(t1) = A, r(t2) = B.Smfwna me to jerhma 3.1.1, ja qoume

BA

.dr = t2t1

(r(t)) .r(t)dt = t2t1

d (r(t))

dtdt = (r(t2)) (r(t1))

= (B) (A) .

To apotlesma aut deqnei ti to epikamplio oloklrwma en gradient exarttai

mno ap thn arqik kai telik jsh, enai dhlad anexrthto th diadrom.

'Ameso prisma tou parapnw jewrmato enai to ex:

Gia opoiadpote kleist kamplh CC

.dr = 0.

Efarmog

'Estw na ulik shmeo pou kinetai kat mko th kamplh r(t) up thn epdrash mia

dnamh F. To swmatdio pern ap ta shmea A kai B ti stigm t1 kai t2 antstoiqa.

Sumbolzoume me v to mtro th taqthta, dhlad v (t) = |v(t)| = |r(t)| . To rgo thdnamh, kat mko tou drmou A B, dnetai kat ta gnwst ap

WAB =

BA

F.dr =

t2t1

F (r(t)) .r(t)dt =

t2t1

mv.vdt =

t2t1

md

dt

(1

2v2)dt

=1

2mv2 (t2) 1

2mv2 (t1) .

H sqsh aut anafretai sunjw w jerhma metabol th kinhtik enrgeia K

(JMKE), kai grfetai sumbolik

WAB = K (B)K (A) .

An uprqei sunrthsh dunamiko , dhlad F = , tte, B

A

F.dr = B

A

.dr,

opte ap to jerhma 3 prokptei ti to (JMKE) grfetai

(B) + (A) = K (B)K (A) K (B) + (B) = K (A) + (A) .

H teleutaa sqsh ekfrzei th diatrhsh th mhqanik enrgeia kai gia to lgo aut

ta peda pou parrqontai ap sunrthsh dunamiko, dhlad F = , onomzontai sun-thrhtik (diathrhtik, conservative).

3.1. EPIKAMPULIA OLOKLHRWMATA 51

Pardeigma 3.1.2. 'Estw mia kamplh C parametropoihmnh ap thn r = r(t), t [t1, t2]. 'Ena llo trpo graf tou epikampliou oloklhrmato tou d.p. F =

(F1, F2, F3) kat mko th kamplh C enaiC

F.dr =

C

F1dx+ F2dy + F3dz.

'Etsi, gia to epikamplio oloklrwma

I =

C

x2dx+ xydy + dz

kat mko th C : r(t) = (t, t2, 1) t [0, 1], ja qoume

I =

10

(x2 (t) x+ x (t) y (t) y + z

)dt =

10

(t2 + 2t4

)dt =

11

15.

Asksei

Upenjumzoume ti gia ton upologism en epikampliou oloklhrmato

CF.dr apaitetai

h parametropohsh tou drmou oloklrwsh,

C : r (t) = x (t) i+ y (t) j+ z (t) k, t [a, b] ,C

F (x, y, z) .dr =

ba

F (x (t) , y (t) , z (t)) .r (t) dt

1. Upologste ta paraktw oloklhrmataC4xyzdx + zdy + x2y2dz pou C enai to tmma th parabol y = x2, z = 1 ap

to (0, 0, 1) mqri to (1, 1, 1) .

CF.dr pou F (x, y) = y2i+x2j kai C h permetro tou trignou me koruf (0, 0) ,

(1, 0) , (1, 1) . 'Omoia toCG.dr pou G (x, y) = x2i + y2j.

2. Brete ta mkh twn paraktw kampuln

a) Parabol y = x2 ap to (0, 0) sto (2, 4) .

b) Uperbol x2 y2 = 1 ap to (0,1) mqri to (1,2) .g) 'Elleiyh x2/a2 + y2/b2 = 1 me a > b (to sqetik oloklrwma den upologzetai

sunartsei stoiqeiwdn sunartsewn, mpore mw na proseggiste se dunmei tou

b/a).

3. H dnamh pou aske h gh se diasthmploio dnetai w gnwstn ap thn

F (x, y, z) = GMm rr3,


pou M kai m oi mze g kai diasthmoploou, r enai to dinusma jsh kai r =

r. Dexte ti aut prorqetai ap gradient, dhlad uprqei sunrthsh dunamikoU(x, y, z): F = U . Upologste to rgo th F kat mko th kamplh r (t) =(2Ret, 2R sin t, 2R

pit), t [0, ] , pou R enai h aktna th gh.

4. Upologste to epikamplio oloklrwma

I =

C

cos z dx+ exdy + eydz

kat mko th C : r(t) = (1, t, et) t [0, 2] .

3.2 Dipl oloklhrmata

Jewrome to orjognio A = [a, b][c, d] kai mia suneq sunrthsh ap na anoiqt snoloU tou R2 pou periqei to A me tim sto R, : A R. Upojtoume ti (x, y) 0 stopedo orismo th U .

PSfrag replaements

y

z

x

A

z = f(x; y)

Sqma 3.2: Grfhma th z = f(x, y), (x, y) A

Tte o gko ktw ap thn epifneia z = (x, y), (x, y) A, dnetai ap to diploloklrwma

A

A

,

A

(x, y)dxdy,

pou A

(x, y) dxdy =

ba

( dc

(x, y)dy

)dx.

Pardeigma 3.2.1. (x, y) = x2 + y2, A = [1, 1] [0, 1]. Tte, o gko ktw apto paraboloeid dnetai ap

A

(x, y)dxdy =

11

( 10

(x2 + y2

)dy

)dx =

11

(x2 +

1

3

)dx =

4

3

3.2. DIPLA OLOKLHRWMATA 53

Dexte ti an ektelsoume ti oloklhrsei kat antstrofh txh, prokptei to dio apot-

lesma, dhlad

11

( 10

(x2 + y2

)dy

)dx =

10

( 11

(x2 + y2

)dx

)dy.

Genik an : U R enai suneq (qi aparathta 0), gia na orsoume to diploloklrwma th se na orjognio A = [a, b] [c, d] msa sto U jewrome mia diamrishtou A, dhlad

a = x0 x1 ... xn = b, c = y0 y1 ... yn = d

me

xi = xi xi1 = b an

, yj = yj yj1 = d cn

.

Se kje orjognio Aij = [xi1, xi] [yj1, yj], jewrome na tuqao shmeo (i, j) kaisqhmatzoume to jroisma Riemann

Sn =ni=1

nj=1

(i, j)xiyj .

PSfrag replaements

a

b

d

x

y

(

i

;

j

)

x

i

y

j

Sqma 3.3: Diamrish tou orjogwnou A

Apodeiknetai ti to rio limn Sn uprqei kai enai to dio gia opoiadpote epilog

shmewn (i, j) msa se kje orjognio Aij . To rio lgetai oloklrwma th sto A,

sumbolzetai me A

=

A

=

A

(x, y) dxdy,

kai lme ti h enai oloklhrsimh sto A.

Shmewsh. Jewrsame diamersei se sa upodiastmata (dhlad xi = x kai

yj = y, i, j = 1, ...n), qwr aut na enai aparathto gia thn parxh tou orou. Arke


h lepthta th diamrish na tenei sto mhdn tan n (dhlad maxi=1,...nxi 0 kaimaxj=1,...nyj 0 tan n).

Ap ton orism prokptoun oi paraktw idithte tou diplo oloklhrmato.

(a)

A

(+ ) =

A

+

A

, kai

A

c = c

A

, c R. (Grammikthta)

(b) An (x, y) 0 (x, y) A tteA

0.

Sunpeia:

An (x, y) (x, y) (x, y) A tteA

A

.

(g) An h enai oloklhrsimh sta orjognia A kai B tte enai oloklhrsimh sto

A B kai mlista AB

=

A

+

B

. (Prosjetikthta)

O trpo upologismo en diplo oloklhrmato dnetai ap to Jerhma Fubini:

Jerhma 3.2.1. An h enai suneq sto A = [a, b] [c, d], tte ba

( dc

(x, y) dy

)dx =

dc

( ba

(x, y) dx

)dy.

PSfrag replaements

x

x

x

y

y

1

(y)

2

(y)

1

(x)

2

(x)

a

b

d

Sqma 3.4: Qwra tpou I kai II

Jewrome tra na qwro D tou R2 pou mpore na enai na ek twn tpwn:

(I) x [a, b] , 1 (x) y 2 (x) ,(II) y [c, d] , 1 (y) x 2 (y) .

Tlo, ja lme ti to qwro D enai tpou (III) an mpore na perigrafe ete w tpou

(I) ete w tpou (II).

3.2. DIPLA OLOKLHRWMATA 55

To dipl oloklrwma th f se kje na tpo ap aut ta qwra dnetai ap:

(I)

D

f =

ba

( 2(x)1(x)

f (x, y)dy

)dx,

(II)

D

f =

dc

( 2(y)1(y)

f (x, y) dx

)dy,

(III)

D

f =

ba

( 2(x)1(x)

f (x, y) dy

)dx =

dc

( 2(y)1(y)

f (x, y) dx

)dy.

Pardeigma 3.2.2. Na upologisje to

I =

D

(x3y + cos x

)dxdy

sto trigwnik qwro D me koruf (0, 0), (/2, 0), (/2, /2).

To D orzetai w

x [0, 2] , 1 (x) 0 y x 2 (x) ,

ra

I =

pi/20

x0

(x3y + cos x

)dydx =

pi/20

[x3y2

2+ y cosx

]y=xy=0

dx =

pi/20

(x5

2+ x cosx

)dx

=

[1

12x6 + x sin x+ cosx

]pi/20

=1

2 +

1

7686 1.

Pardeigma 3.2.3. Na upologisje to

I =

D

a2 y2 dxdy

sto tetartokklio D kntrou (0, 0) kai aktna a pou ketai sto prto tetarthmrio.

To D orzetai w

0 x a 0 y a2 x2 w 0 y a 0 x

a2 y2.

Upologste to oloklrwma me tou do trpou. P.q. ja qoume

I =

a0

a2y20

a2 y2dxdy =

a0

[xa2 y2

]x=a2y2x=0

dy =

a0

(a2 y2) dy

=

[a2y y

3

3

]a0

=2a3

3.


3.3 Tripl oloklhrmata

'Estw mia perioq U tou R3 kai : U R suneq sunrthsh. Jewrome na parallh-leppedo V pou periqetai msa sthn perioq U . Tte to tripl oloklrwma ( oloklrwma

gkou) th sto VV

V

(x, y, z) dV

V

(x, y, z) dxdydz,

orzetai w to rio en katllhlou ajrosmato Riemann. H kataskeu enai anlogh

th kataskeu tou diplo oloklhrmato. Dhlad jewrome mia diamerish tou V se n3

stoiqeidh parallhleppeda gkou V = xyz, kai se kajna ap aut jewrome na

tuqao shmeo (i, j, k). 'Etsi sqhmatzoume to jroisma Riemann

Sn =

ni=1

nj=1

nk=1

(i, j , k)xiyjzk.

Apodeiknetai ti to rio limn Sn uprqei kai enai to dio gia opoiadpote epilog

shmewn (i, j, k) msa se kje stoiqeide parallhleppedo. To rio lgetai oloklrwma

th sto V kai lme ti h enai oloklhrsimh sto V .

Isqoun oi gnwst idithte tou diplo oloklhrmato kaj kai to Jerhma Fubini.

Pardeigma 3.3.1. 'Estw V = [0, 1] [0, 1] [0, 1], tteV

(x+ y) z1 + z2

dxdydz =

10

dx

10

dy

10

dz(x+ y) z1 + z2

=

10

dx

10

dy[(x+ y)

1 + z2

]z=1z=0

=(

2 1) 1

0

dx

[xy +

y2

2

]y=1y=0

=2 1.

Gia qwra geniktera ap parallhleppeda, efarmzoume mejdou anloge autn pou

qrhsimopoiontai gia ton upologism dipln oloklhrwmtwn.

Pardeigma 3.3.2. 'Estw V h mpla x2 + y2 + z2 R2, tteV

dV =

RR

dx

R2x2R2x2

dy

R2x2y2R2x2y2

dz = ... = 4R3

Pardeigma 3.3.3. An jewrsoume mia suneq katanom hlektriko fortou se mia

perioq tou qrou, tte puknthta fortou sto shmeo (x, y, z) (fortou an monda gkou)

orzetai w

(x, y, z) =dq

dVSunep, msa se gko V perikleetai forto

q (V ) =

V

vector calculus

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