PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II SUNARTHSEIS POLLWN METABLHTWN EPIKAMPULIA OLOKLHRWMATA

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1 PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II SUNARTHSEIS POLLWN METLHTWN EPIKAMPULIA OLOKLHRWMATA 1. EpikampÔlio Olokl rwma 1ou eðdouc Efarmogèc 2. Dianusmatikˆ pedða 3. EpikampÔlio Olokl rwma 2ou eðdouc Anaplhrwt c Kajhght c : Dr. Pappˆc G. Alèxandroc 1

2 EPIKAMPULIA OLOKLHRWMATA To kefˆlaio autì, ektìc apì to fusikì endiafèron pou parousiˆzei, mporeð na jewrhjeð san mia epèktash thc ènnoiac tou orismènou oloklhr matoc se sqèsh me to pedðo olokl rwshc. Sto orismèno olokl rwma to pedðo olokl rwshc eðnai èna diˆsthma thc eujeðac twn pragmatik n arijm n R. Ed ja asqolhjoôme me oloklhr mata, twn opoðwn to pedðo olokl rwshc eðnai èna tm ma miac kampôlhc (pou mporeð na eðnai epðpedh ìqi). Ta oloklhr mata autˆ onomˆzontai epikampôlia oloklhr mata. 1. EpikampÔlio Olokl rwma 1ou eðdouc Efarmogèc Orismìc 1.1 : EpikampÔlio olokl rwma epikampôlio olokl rwma 1ou eðdouc thc f(x, y) katˆ m koc tou to sumbolðzoume f(x, y) dl. Prìtash 1.1 : An h pragmatik sunˆrthsh f(x, y) eðnai suneq c sto pedðo D pou perièqei to Ðqnoc miac eujugrammðsimhc kampôlhc r(t) x(t)i + y(t)j, t [α, β], tìte upˆrqei to epikampôlio olokl rwma f(x, y) dl. An h kampôlh èqei suneq parˆgwgo, tìte isqôei h isìthta tb f(x, y) dl f [x(t), y(t)] ẋ 2 (t) + ẏ 2 (t) dt, ìpou ẋ(t) dx dt t A, ẏ(t) dy dt. Prìtash 1.2 : An h pragmatik sunˆrthsh f(x, y, z) eðnai suneq c sto pedðo D pou perièqei to Ðqnoc miac eujugrammðsimhc kampôlhc r(t) x(t)i + y(t)j + z(t)k, t [α, β], tìte upˆrqei to epikampôlio olokl rwma f(x, y, z) dl. An h kampôlh èqei suneq parˆgwgo, tìte isqôei h isìthta f(x, y, z) dl tb t A f [x(t), y(t), z(t)] ẋ 2 (t) + ẏ 2 (t) + ż 2 (t) dt, ìpou ẋ(t) dx dt, ẏ(t) dy dt, ż(t) dz dt. 2

3 Idiìthtec tou EpikampulÐou Oloklhr matoc 1ou eðdouc Oi parakˆtw idiìthtec prokôptoun apì ton orismì tou epikampulðou oloklhr matoc kai apì th sqèsh tou me to orismèno olokl rwma. DÐnontai oi pragmatikèc sunart seic f, g, oi opoðec eðnai suneqeðc sto pedðo D kai D to Ðqnoc kampôlhc me suneq parˆgwgo. Tìte isqôoun: (a) (λf + µg) dl λ f dl + µ g dl, λ, µ R stajeroð. (b) f dl ÂΓ f dl + ΓB f dl, ìpou G èna shmeðo tou Ðqnoc. (g) f dl BA f dl. (d) f dl f dl. Pìrisma 1.1 : An h sunˆrthsh f eðnai suneq c sto pedðo D kai D to Ðqnoc miac katˆ ta tm mata ÂA 1, Â 1 A 2,..., Â k 1 B eujugrammðsimhc kampôlhc, tìte isqôei f dl f dl + f dl + + f dl. ÂA 1 Â 1 A 2 A k 1 B Gia ton upologismì tou oloklhr matoc f(x, y, z) dl tb t A f [x(t), y(t), z(t)] ẋ 2 (t) + ẏ 2 (t) + ż 2 (t) dt an eðnai dosmènh h sunˆrthsh f(x, y, z) kai h kampôlh akoloujoôme ta parakˆtw b mata : B ma 1 o BrÐskoume tic parametrikèc exis seic thc kampôlhc. Grˆfoume thn exðswsh thc kampôlhc se parametrik morf. An M(x, y, z) eðnai tuqaðo shmeðo pˆnw sthn kampôlh mporoôme na grˆyoume: x x(t), y y(t), z z(t) dhlad oi suntetagmènec jèshc tou shmeðou eðnai sunart seic thc paramètrou t, ste se kˆje tim thc paramètrou t, na antistoiqeð èna shmeðo pˆnw sthn kampôlh. 3

4 B ma 2 o UpologÐzoume tic timèc t A, t B pou antistoiqoôn sta ˆkra A, B thc kampôlhc. ArkeÐ gia autì mða apì tic sqèseic x x(t), y y(t), z z(t), (t A t t B ). Protimˆme fusikˆ thn aploôsterh ap> autèc. B ma 3 o UpologÐzoume tic parag gouc: ẋ(t) dx dt, dy ẏ(t) dt, dz ż(t) dt. B ma 4 o UpologÐzoume thn sunˆrthsh f [x(t), y(t), z(t)]. B ma 5 o UpologÐzoume to orismèno olokl rwma. tb t A f [x(t), y(t), z(t)] ẋ 2 (t) + ẏ 2 (t) + ż 2 (t) dt. TonÐzoume ìti prèpei t A < t B. ParathroÔme dhlad ìti èna epikampôlio olokl rwma 1ou eðdouc èqei anˆlogec idiìthtec me to orismèno olokl rwma. 4

5 Gia na broôme thn parametrik morf, eðnai qr simec oi akìloujec peript seic: (a) Parametrik parˆstash eujugrˆmmou tm matoc me ˆkra: A(a 1, a 2, a 3 ), B(b 1, b 2, b 3 ). JewroÔme to tuqaðo shmeðo M(x, y, z) anˆmesa sta A, B. Epeid ta dianôsmata AM, BM eðnai profan c suggrammikˆ, upˆrqei t R : AM t (1) Z B. O. A M (x,y,z) Y X 'Etsi gia kˆje shmeðo M antistoiqeð mða tim tou t, apì th sqèsh (1). 'Omwc AM (x, y, z) (a 1, a 2, a 3 ) (x a 1, y a 2, z a 3 ) afoô oi suntetagmènec tou dianôsmatoc AM prokôptoun an apì tic suntetagmènec (x, y, z) tou pèratoc afairèsoume tic antðstoiqec a 1, a 2, a 3 thc arq c. 'Omoia: (b 1 a 1, b 2 a 2, b 3 a 3 ) kai h sqèsh (1) grˆfetai: ap> ìpou prokôptoun: (x a 1, y a 2, z a 3 ) t(b 1 a 1, b 2 a 2, b 3 a 3 ) (x a 1, y a 2, z a 3 ) [t(b 1 a 1 ), t(b 2 a 2 ), t(b 3 a 3 )] x a 1 t(b 1 a 1 ) x a 1 + t(b 1 a 1 ) (2) y a 2 t(b 2 a 2 ) y a 2 + t(b 2 a 2 ) (3) z a 3 t(b 3 a 3 ) z a 3 + t(b 3 a 3 ) (4) Oi (2), (3), (4) apoteloôn thn parametrik parˆstash tou tm matoc. JewroÔme t ra mða ap> autèc, p.q thn (2). Gia x x A a 1, dðnei t A en gia x x B b 1 dðnei t B 1. 'Etsi br kame ta t A, t B. Ja mporoôsame na broôme (ta Ðdia) jewr ntac tic sqèseic (3) (4) antð thc (2). 5

6 (b) Parametrik parˆstash tìxou kôklou, pou brðsketai sto epðpedo Oxy. Upojètoume ìti o kôkloc èqei kèntro to shmeðo C(a, b) kai aktðna R. To tuqaðo shmeðo M(x, y) thc perifèreiac èqei: x a + R cos t, y b + R sin t ìpou t eðnai h gwnða pou sqhmatðzei h aktðna CM me to jetikì hmiˆxona Ox. Jetik forˆ thc gwnðac t eðnai h antiwrologiak. y y M(x, y) R b C(a, b) t O a x x Sthn eidik perðptwsh pou o kôkloc èqei kèntro thn arq twn axìnwn, eðnai: x R cos t, y R sin t. (g) Parametrik parˆstash tìxou èlleiyhc, pou èqei kèntro thn arq twn axìnwn kai hmiˆxonec a, b. y b M(x, y) O t a x To tuqaðo shmeðo M(x, y) èqei: x a cos t, y b sin t ìpou h gwnða t èqei jetik forˆ thn antiwrologiak, en eðnai t sto jetikì hmiˆxona Ox. 6

7 Gewmetrik ermhneða Briskìmaste sto q ro twn tri n diastˆsewn, ìpou èqoume mia gramm C kai dôo shmeða thc A, B. 'Estw ìti èqoume mia suneq sunˆrthsh f(x, y, z) epð thc kampôlhc C. A. C Z O.. B dl (x,y,z) f(x,y,z) Y X Th gramm th jèloume me parametrikèc exis seic x x(t), y y(t), z z(t). An f(x, y, z) 1 tìte ajroðzoume ta stoiqei dh mèrh tou kai to epikampôlio olokl rwma mac dðnei to m koc L tou tìxou : L dl. Fusik ermhneða An h f(x, y, z) dhl nei thn grammik puknìthta tou sôrmatoc, tìte to epikampôlio olokl rwma mac dðnei th mˆza M tou sôrmatoc: M f(x, y, z) dl. Oi suntetagmènec tou kèntrou mˆzac C(x C, y C, z C ) eðnai: x C 1 M y C 1 M z C 1 M xf(x, y, z) dl yf(x, y, z) dl zf(x, y, z) dl. 7

8 Shmantik parat rhsh 1: 'Otan h gramm C eðnai kleist (A B), tìte to epikampôlio olokl rwma sumbolðzetai f(x, y, z) dl. Shmantik parat rhsh 2: C Den prèpei na sugqèoume tic parametrikèc suntetagmènec me tic polikèc suntetagmènec. Stic parametrikèc suntetagmènec èqoume mða parˆmetro, èstw thn t, en stic polikèc dôo paramètrouc, tic t, ϑ, (ìpou ϑ gwnða). Tic parametrikèc suntetagmènec tic qrhsimopoioôme sta epikampôlia oloklhr mata, en tic polikèc suntetagmènec sta diplˆ oloklhr mata. 8

9 Parˆdeigma 1.1: Na upologisjeð to epikampôlio olokl rwma I pˆnw sto eujôgrammo tm ma pou èqei ˆkra to A, B. ( 2xyz + z 2 ) dl, A(1,, ), B(2, 2, 1) LÔsh f(x, y, z) dl tb t A f [x(t), y(t), z(t)] ẋ 2 (t) + ẏ 2 (t) + ż 2 (t) dt, ìpou ẋ(t) dx dt, ẏ(t) dy dt, ż(t) dz dt. B ma 1 o BrÐskoume tic parametrikèc exis seic thc kampôlhc. An M(x, y, z) eðnai tuqaðo shmeðo pˆnw sto, tìte: Z X. A O (1,,). M (x,y,z) B(2,2,-1) Y AM t (x 1, y, z ) t(2 1, 2, 1 ) (x 1, y, z) (t, 2t, t). Apì thn teleutaða brðskoume: {x 1 t, y 2t, z t} x 1 + t, y 2t, z t pou eðnai h zhtoômenh parametrik parˆstash. 9

10 B ma 2 o UpologÐzoume tic timèc t A, t B pou antistoiqoôn sta ˆkra A, B thc kampôlhc. JewroÔme mða apì tic teleutaðec isìthtec, p.q thn y 2t. 'Eqoume: y A 2t A t A, y B 2 2 2t B t B 1. B ma 3 o UpologÐzoume tic parag gouc: ẋ(t) dx dt, dy ẏ(t) dt, dz ż(t) dt. Apì tic parametrikèc exis seic brðskoume: ẋ(t) dx dt 1, dy ẏ(t) dt 2, dz ż(t) dt 1. B ma 4 o UpologÐzoume thn sunˆrthsh f [x(t), y(t), z(t)]. f(x, y, z) 2xyz + z 2 f [x(t), y(t), z(t)] 2x(t)y(t)z(t) + z 2 (t) 2(1 + t) 2t( t) + ( t) 2 4t 2 (1 + t) + t 2 3t 2 4t 3. B ma 5 o UpologÐzoume to orismèno olokl rwma. tb t A f [x(t), y(t), z(t)] ẋ 2 (t) + ẏ 2 (t) + ż 2 (t) dt. ( 2xyz + z 2 ) dl tb f [x(t), y(t), z(t)] ẋ 2 (t) + ẏ 2 (t) + ż 2 (t) dt t A ( 3t 2 4t 3) ( 1) 2 dt ( 3t 2 4t 3) dt 3 [ t t 2 dt 4 6 ] 1 4 [ t ] 1 t 3 dt 6 ( 1 3 3) 6 ( 1 4 4)

11 Parˆdeigma 1.2: Na upologisjeð to epikampôlio olokl rwma I pˆnw sto tetartokôklio tou sq matoc, me aktðna Ðsh me 2. x dl y B(, 2) M 2 t O A(2, ) x LÔsh f(x, y) dl tb t A f [x(t), y(t)] ẋ 2 (t) + ẏ 2 (t) dt, ìpou ẋ(t) dx dt, ẏ(t) dy dt. Gia thn paramètrik parˆstash tou tetartokôkliou parathroôme ìti to tuqaðo shmeðo tou M(x, y) èqei: ìpou t h gwnða pou faðnetai sto sq ma. x(t) 2 cos t, y(t) 2 sin t Profan c to shmeðo A antistoiqeð sto t A en to B gia t B π 2. Autì prokôptei apì to sq ma apì mða apì tic x(t) 2 cos t, y(t) 2 sin t an antikatast soume Apì tic exis seic autèc prokôptei: x A 2, y A, x B, y B 2. ẋ(t) dx dt 2 sin t, dy ẏ(t) dt 2 cos t. 11

12 MporoÔme loipìn t ra na grˆyoume: x dl 4 tb t A x(t) ẋ 2 (t) + ẏ 2 (t) dt π 2 π 2 π 2 2 cos t ( 2 sin t) 2 + (2 cos t) 2 dt 2 cos t 4 ( sin 2 t + cos 2 t ) dt cos t dt 4 [sin t] π 2 4 (sin π ) 2 sin 4(1 ) 4. 12

13 Parˆdeigma 1.3: Jèloume na upologðsoume to olokl rwma f(x, y, z) dl pou eðnai to epikampôlio olokl rwma thc sunˆrthshc f(x, y, z) pˆnw sthn kampôlh. EÐnai dosmènh h sunˆrthsh f(x, y, z) x + yz 2 kai h parametrik morf thc kampôlhc : x(t) t, y(t) 2t, z(t) t me t A 1, t B 2. Me dosmènh thn parametrik parˆstash thc brðskoume: ẋ(t) 1, ẏ(t) 2, ż(t) 1 f[x(t), y(t), z(t)] x(t) + y(t) z 2 (t) t + 2t ( t) 2 t + 2t 3 kai akìmh: ẋ2 + ẏ 2 + ż ( 1) 2 6. 'Etsi èqoume: ( x + yz 2 ) dl 2 ( t + 2t 3 ) 6 dt ( t + 2t 3 ) dt t dt ] 2 t 3 dt [ ] t [ t ( ) ( ) Sto parˆdeigmˆ autì tan dosmènh h parametrik parˆstash thc kampôlhc kai ta t A, t B. Se pollèc ìmwc peript seic h kˆmpôlh dðnetai se ˆllh morf. 13

14 Parˆdeigma 1.4: Na upologisjeð to epikampôlio olokl rwma: I KK 2xy dl (1) y B(2, 5) Q R A(6, 2) P O K x ìpou KK eðnai h perðmetroc tou trig nou tou sq matoc. LÔsh Profan c mporoôme na grˆyoume: 2xydl KK KA 2xydl + 2xydl + 2xydl (2) BK opìte arkeð na upologðsoume qwristˆ kajèna apì ta trða oloklhr mata tou b' mèlouc: (a) Gia to pr to olokl rwma jewroôme mða parametrik parˆstash tou euj. tm matoc KA. An P (x, y) eðnai tuqaðo shmeðo tou, èqoume: KP t KA (x, y ) t(6, 2 ) x 6t, y 2t x(t) 6t, y(t) 2t. Apì thn pr th, me x k brðskoume t k, en me x A 6 brðskoume t k 1. Me parag gish prokôptei: ẋ(t) 6, ẏ(t) 2. MporoÔme t ra na grˆyoume: 14

15 AK 2xydl t 2t dt 1 t 2 dt [ 24 ] 1 4 t (3) (b) Gia to deôtero olokl rwma, an Q(x, y) eðnai tuqaðo shmeðo thc, èqoume: AQ t (x 6, y 2) t(2 6, 5 2) x 6 4t, y 2 + 3t x(t) 6 4t, y(t) 2 + 3t. Apì thn pr th ap> autèc, me x A 6 brðskoume t A, en me x B 2 brðskoume t B 1. Akìmh: opìte èqoume: 2xy dl ẋ(t) 4, ẏ(t) 3 2(6 4t)(2 + 3t) ( 4) dt ( 12t 2 + 1t + 12 ) dt 1 [ 12 3 t3 t 2 dt + 1 ] 1 1 [ t2 ] 1 t dt dt 13. (4) (g) Gia to trðto olokl rwma, an R(x, y) tuqaðo shmeðo thc BK èqoume: BR t BK (x 2, y 5) t( 2, 5) x 2 2t, y 5 5t x(t) 2 2t, y(t) 5 5t. H pr th ap> autèc dðnei: Gia x B 2 to t B en gia x K to t K 1. Akìmh 'Eqoume: BK 2xydl ẋ(t) 2, ẏ(t) 5. 2(2 2t)(5 5t) ( 2) 2 + ( 5) 2 dt 1 [ 2 ] 1 29 t ( 1t 2 2t + 1 ) dt [ 4 ] 1 29 [ t ] (5) 15

16 Me antikatˆstash sth sqèsh (2) prokôptei h tim tou I : I Parat rhsh: Profan c isqôei: f(x, y, z) dl BA f(x, y, z) dl gia to epikampôlio olokl rwma pr tou eðdouc. Prosèqoume ìmwc to ˆnw ˆkro tou orismènou oloklhr matoc wc proc t na eðnai pˆnta megalôtero tou kˆtw. ParathroÔme akìmh ìti sthn ˆskhsh aut h kampôlh olokl rwshc eðnai kleist. Gia to lìgo autì sumbolðzoume: KK 2xy dl C 2xy dl, (C KK). EÐnai profanèc akìmh, ìti h kampôlh olokl rwshc eðnai kleist, mporoôme na jewr soume arq (kai pèrac sumpðptoun) èna opoiod pote shmeðo. 16

17 Parˆdeigma 1.5: DÐnetai to sôrma (ulikì tìxo) me parametrik parˆstash: kai grammik puknìthta x t, y t 2, z t 3, (t A, t B 1) f(x, y, z) Na upologisjeð h mˆza tou kai to kèntro mˆzac tou. xyz 1 + 4y + 9zx LUSH H sunolik mˆza tou sôrmatoc: M f(x, y, z)dl. Oi suntetagmènec tou kèntrou mˆzac C eðnai: x C 1 M y C 1 M z C 1 M xf(x, y, z)dl yf(x, y, z)dl zf(x, y, z)dl H mˆza tou tìxou eðnai: M f(x, y, z) dl M Apì thn parametrik parˆstash tou tìxou brðskoume: MporoÔme t ra na grˆyoume: M tb ẋ(t) 1, ẏ(t) 2t, ż(t) 3t 2. f(x, y, z) dl xyz 1 + 4y + 9xz dl. t A f(x(t), y(t), z(t)) ẋ 2 (t) + ẏ 2 (t) + ż 2 (t) dt t t 2 t t2 + 9t t (2t) 2 + (3t 2 ) 2 dt t t2 + 9t t 2 + 9t 4 dt [ t t 6 7 dt 7 ]

18 To kèntro mˆzac tou tìxou èqei suntetagmènec: x C 1 xf(x, y, z) dl M tb 7 t A x(t)y(t)z(t) ẋ2 x(t) (t) + ẏ 2 (t) + ż 2 (t) dt 1 + 4y(t) + 9x(t)z(t) t t 2 t 3 t 1 + (2t)2 + (3t 2 ) 2 dt 1 + 4t2 + 9 t t 3 t t2 + 9t t2 + 9t 4 dt t 7 dt y C 1 M tb 7 t A yf(x, y, z) dl x(t)y(t)z(t) ẋ2 y(t) (t) + ẏ 2 (t) + ż 2 (t) dt 1 + 4y(t) + 9x(t)z(t) t 2 t t 2 t (2t)2 + (3t 2 ) 2 dt 1 + 4t2 + 9 t t 3 t t2 + 9t t2 + 9t 4 dt t 8 dt z C 1 M tb 7 t A ProkÔptei loipìn h jèsh tou zf(x, y, z) dl x(t)y(t)z(t) ẋ2 z(t) (t) + ẏ 2 (t) + ż 2 (t) dt 1 + 4y(t) + 9x(t)z(t) t 3 t t 2 t (2t)2 + (3t 2 ) 2 dt 1 + 4t2 + 9 t t 3 t t2 + 9t t2 + 9t 4 dt t 9 dt C : ( 7 8, 7 9, 7 ). 1 18

19 2. Dianusmatikˆ pedða Mia dianusmatik sunˆrthsh F (F 1, F 2,..., F n ) Φ (R n, R n ) orismènh sto sônolo X R n lème ìti eðnai èna dianusmatikì pedðo, kurðwc ìtan ekfrˆzei èna dianusmatikì mègejoc thc Fusik c. Gia parˆdeigma: i. to dianusmatikì pedðo twn efaptìmenwn dianusmˆtwn miac kampôlhc, ii. to dianusmatikì pedðo twn efaptìmenwn dianusmˆtwn miac kampôlhc. 'Ena dianusmatikì pedðo orismèno s> èna uposônolo X R 2 (ant. X R 3 ) ja to sumbolðzoume sun jwc wc ex c: F (x, y) P (x, y)i + Q(x, y)j (ant. F (x, y, z) P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k). 'Ena dianusmatikì pedðo F, orismèno sto anoiqtì sônolo D, ja lème ìti eðnai èna pedðo klðsewn, ìtan upˆrqei èna bajmwtì pedðo f, diaforðsimo sto X tètoio, ste na isqôei F (x) gradf(x) F j (x) f(x) x j, j 1,..., n, x D. Sthn perðptwsh twn dianusmatik n pedðwn F (x, y) kai F (x, y, z) grˆfetai kai P (x, y) f x (x, y), Q(x, y) f y (x, y) P (x, y, z) f x (x, y, z), Q(x, y, z) f y (x, y, z), R(x, y, z) f z (x, y, z) antðstoiqa. Oi teleutaðec sqèseic ja grˆfontai pollèc forèc kai me thn isodônamh graf : df(x, y) P (x, y)dx + Q(x, y)dy, df(x, y, z) P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz. 19

20 3. EpikampÔlio Olokl rwma 2ou eðdouc Orismìc 3.1 : DÐnetai to dianusmatikì pedðo F (x, y) P (x, y)i + Q(x, y)j suneqèc sto Ðqnoc thc kampôlhc r(t) x(t)i + y(t)j, t [α, β]. EpikampÔlio olokl rwma tou dianusmatikoô pedðou F(x,y) epikampôlio olokl rwma 2 oυ eðdouc thc F(x,y) kata m koc tou tìxou sumbolðzetai P (x, y)dx + Q(x, y)dy. An r xi + yj eðnai to diˆnusma jèsewc tou tuqaðou shmeðou (x, y) thc kampôlhc, tìte to epikampôlio olokl rwma grˆfetai P (x, y)dx + Q(x, y)dy F (x, y) dr. Prìtash 3.1 : An h dianusmatik sunˆrthsh F eðnai suneq c katˆ m koc tou Ðqnoc miac kampôlhc r(t), t [α, β] tìte upˆrqei to epikampôlio olokl rwma thc F katˆ m koc tou kai isqôei β β F dr F (r(t)) ṙ(t) dt [P (x(t), y(t))ẋ(t) + Q(x(t), y(t))ẏ(t)] dt. α α Prìtash 3.2 : Sthn perðptwsh enìc dianusmatikoô pedðou F (x, y, z) tou R 3 kai tou Ðqnouc F dr β α α kampôlhc r(t) (x(t), y(t), z(t)), t [α, β] isqôei: F (r(t)) ṙ(t)dt β miac [P (x(t), y(t), z(t))ẋ(t) + Q(x(t), y(t), z(t))ẏ(t) + R(x(t), y(t), z(t))ż(t)] dt. Prìtash 3.3 : DÐnetai to dianusmatikì pedðo F (x, y) P (x, y)i + Q(x, y)j orismèno sto Ðqnoc C miac kampôlhc r(t). An h kampôlh eðnai parˆllhlh proc ton ˆxona Ox (ant. Oy), tìte upˆrqei to epìmeno olokl rwma kai isqôei ( ) Q(x, y)dy, ant. P (x, y)dx. C C Sthn perðptwsh aut eðnai ẏ (ant. sunˆrthsh P (x, y) (ant. Q(x, y)). ẋ ), en den eðnai aparaðthth opoiad pote proüpìjesh gia th 2

21 Mia ousi dhc diaforˆ tou epikampôlio olokl rwma 2 oυ eðdouc apì to epikampôlio olokl rwma 1 oυ eðdouc, eðnai ìti to epikampôlio olokl rwma 2 oυ eðdouc (h tim tou) exartˆtai apì th forˆ diagraf c tou Ðqnoc thc. Prìtash 3.4 : An h dianusmatik sunˆrthsh F eðnai suneq c sto Ðqnoc t [α, β], pou apoteleðtai apì ta ÂA 1,..., Â n B, tìte isqôei F dr F dr + +. ÂA 1 Â n B miac kampôlhc r r(t), Prìtash 3.5 : DÐnontai oi dianusmatikèc sunart seic F kai G, oi opoðec ikanopoioôn tic proüpojèseic thc protˆsewc 3.4 sto Ðqnoc miac kampôlhc r r(t), t [α, β],. Tìte isqôoun: i. (λf + µg) dr λ F dr + µ G dr, gia kˆje λ, µ R. ii. BA F dr F dr. 'Opwc eðdame sthn isìthta F dr F dr BA h forˆ diagraf c tou Ðqnouc miac kampôlhc paðzei shmantikì rìlo sthn tim tou oloklhr matoc. 'Otan to Ðqnoc eðnai èna eujôgrammo tm ma tìxo apì to A kai B., tìte h forˆ diagraf c eðnai prokajorismènh 'Otan ìmwc prìkeitai gia to Ðqnoc C miac kampôlhc (anoiqt c kleist c), tìte prèpei na upodeðxoume me kˆpoio trìpo. 'Etsi, gia na dhl soume th forˆ diagraf c ja grˆyoume gia thn orj forˆ (antðjeth twn deikt n tou rologioô) kai C C ìtan h diagraf gðnetai katˆ thn antðjeth forˆ. Prìtash 3.6 : DÐnetai h dianusmatik sunˆrthsh F, h opoða eðnai suneq c sto Ðqnoc C miac kleist c kampôlhc r r(t), t [α, β]. Tìte to olokl rwma C F dr eðnai anexˆrthto apì to shmeðo pou jewroôme wc arq. 21

22 Parˆdeigma 3.1: Na upologisjeð to èrgo thc dônamhc F (x, y, z) (y, z, x) gia thn metakðnhsh apì to shmeðo K(,, ), eujôgramma, sto shmeðo Λ(1, 2, 3). LÔsh To zhtoômeno èrgo eðnai Ðso me to epikampôlio olokl rwma: W K Λ Λ K F dl. (1) Z. M (x,y,z). /\ (1,2,3) K Y X 'Omwc eðnai: F (y, z, x), dl (dx, dy, dz) F dl ydx + zdy + xdz. (2) BrÐskoume t ra mða parametrik parˆstash tou eujôgrammou tm matoc KΛ. An M(x, y, z) tuqaðo shmeðo tou, eðnai: KM tkλ (x, y, z ) t(1, 2, 3 ) x t, y 2t, z 3t. Sthn arq K èqw x K, ˆra t K. Sto pèrac Λ eðnai x Λ 1, ˆra t Λ 1. Akìmh apì thc parametrikèc exis seic brðskoume: dx dt, dy 2dt, dz 3dt. H (1) lìgo thc (2) kai twn parametrik n exis sewn dðnei: W K Λ 1 1 (2tdt + 3t 2dt + t 3dt) 11 t dt

23 Parˆdeigma 3.2: Na upologisjeð to epikampôlio olokl rwma I katˆ m koc thc kampôlhc: B A [ (xy + z)dx + ( x 2 y 2) dy + 7z 2 xdz ] x t, y 2t, z t 2, A(,, ), B(2, 4, 4) kai na dojeð mða fusik ermhneða tou. LÔsh Apì th dosmènh parametrik parˆstash, me x A brðskoume t A, en me x B 2 prokôptei t B 2. Akìmh èqoume: Me bˆsh ta parapˆnw, to olokl rwma grˆfetai: An jewr soume th dônamh: profan c eðnai: I dx dt, dy 2 dt, dz 2t dt. [( t2t + t 2 ) dt + ( t 2 4t 2) 2 dt + 7t 4 t2t dt ] ( 14t 6 3t 2) dt [ 14 7 t7 t 6 dt 3 ] [ 3 3 t3 ] 2 t 2 dt F ( xy + z, x 2 y 2, 7z 2 x ) B A F dl I B A B A [ xy + z, x 2 y 2, 7z 2 x ] (dx, dy, dz) (xy + z) dx + ( x 2 y 2) dy + 7z 2 x dz I B A 'Ara to olokl rwma autì eðnai Ðso me to èrgo thc dônamhc F gia metakðnhsh apì to shmeðo A mèqri to B katˆ m koc thc kampôlhc: x t, y 2t, z t 2. F dl. 23