SUNARTHSEIS POLLWN METABLHTWN. 5h Seirˆ Ask sewn. Allag metablht n sto diplì olokl rwma

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1 PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II SUNARTHSEIS POLLWN METABLHTWN 5h Seirˆ Ask sewn Allag metablht n sto diplì olokl rwma Jèma. Qrhsimopoi ntac to metasqhmatismì: e (x+y) dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic grammèc: y x, x + y, y. Jèma. Qrhsimopoi ntac to metasqhmatismì: u x + y v x + 4y (x + y)(x + 4y) dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic grammèc: y x +, y x +, y 4 x kai y 4 x +.

2 Jèma. Qrhsimopoi ntac to metasqhmatismì: u x y v x + y (x y)(x + y) dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic grammèc: y x + 4, y x + 7, y x, y x +. Jèma 4. Qrhsimopoi ntac to metasqhmatismì: u xy v y x (x + y) dxdy ìpou τ { (x, y) R : xy 4, x y 4x }, (x > ). Jèma 5. Qrhsimopoi ntac to metasqhmatismì: u x y v y x ( x + y ) dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic kampôlec: xy, xy, y x, y x.

3 Jèma 6. Qrhsimopoi ntac to metasqhmatismì: u y + x v y x y + x x + xy dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic kampôlec: kai brðsketai sthn pr th gwnða twn axìnwn. y x +, y x + 8, y, y x Jèma 7. Qrhsimopoi ntac to metasqhmatismì: (x y) sin (x + y) dxdy ìpou eðnai o q roc pou frˆssetai apì tic korufèc: tou parallhlogrˆmmou. (π, ), (π, π), (π, π), (, π) ShmeÐwsh. UpenjumÐzetai ìti h exðswsh thc eujeðac pou dièrqetai apì ta shmeða (x, y ) kai (x, y ) dðnetai apì ton tôpo: x x y y. x x y y Jèma 8. Qrhsimopoi ntac to metasqhmatismì: cos ( ) x y dxdy x + y ìpou eðnai o tìpoc pou perikleðetai apì tic kampôlec: x, y, x + y.

4 Jèma 9. Qrhsimopoi ntac to metasqhmatismì: ìpou to trapèzio me korufèc ta shmeða v y x (x + y) dxdy A(, ), B(4, ), Γ(, 4) kai (, ). Jèma. Qrhsimopoi ntac to metasqhmatismì: x + y sin(x y) dxdy ìpou to parallhlìgrammo me korufèc ta shmeða ( π ) ( A(, ), B, π, Γ + π, π ) kai (, ). Anaplhrwt c Kajhght c : Dr. Pappˆc G. Alèxandroc 4

5 . Lumènec Ask seic 5

6 Jèma. Qrhsimopoi ntac to metasqhmatismì: e (x+y) dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic grammèc: y x, x + y, y. LUSH f(x, y) dxdy g(u, v) J(u, v) dudv B ma o. O metasqhmatismìc eðnai dosmènoc: MporoÔme na epilôsoume wc proc x, y : x u + v, y u v B ma o. H Iakwbian J(u, v) (x, y) (u, v) x u + v y u v x u x v y u y v. 6

7 H Iakwbian J(u, v) 4 4. J(u, v). B ma o. & g(u, v) e u. f(x, y) e (x+y) B ma 4 o. O tìpoc faðnetai sto sq ma. y y x (τ ) B x + y O y A x Sqediˆzoume t ra sto epðpedo uv thn eikìna tou sunìrou tou tìpou me bˆsh to dosmèno metasqhmatismì, apaleðfontac se kˆje perðptwsh ta x, y. 7

8 SÔnoro y x : y x u v u + v v. & y x x y v. SÔnoro x + y : x + y u + v + u v u. & x + y u. SÔnoro y : y u v v u. & y u x v x v u. 'Etsi prokôptei o tìpoc D sto epðpedo uv pou eðnai h eikìna tou dia mèsou tou dosmènou metasqhmatismoô. 8

9 v u u v u O v u B ma 5 o. 'Eqoume, sômfwna me th sqèsh f(x, y) dxdy e (x+y) dxdy g(u, v) J(u, v) dudv e u dudv O tìpoc D eðnai kanonikìc wc proc u (eujeðec u, u ) en kleðnei pˆnw apì thn v u kai kˆtw apì th v. e u dudv ( u (e u u ) e u dv du e u u du e u du 4 [ ] 4 eu ( e 4 ). 4 ) dv du Parat rhsh: Gia na broôme thn eikìna miac gramm c tou tìpou, apaleðfoume ta x, y apì thn exðswsh tic gramm c kai tic exis seic metasqhmatismoô. 9

10 Jèma. Qrhsimopoi ntac to metasqhmatismì: u x + y v x + 4y (x + y)(x + 4y) dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic grammèc: y x +, y x +, y 4 x kai y 4 x +. LUSH Ja broôme tic sqèseic twn x, y me ta u, v. Opìte kai H Iakwbian J(u, v) (x, y) (u, v) x + 4y v x + y v. u v x + y x y y u + v ( x v 4 u + v ) u 5 v 5. ( u 5 v 5 ) ( u + v ) ( u 5 v 5 ) ( u + v ) Apì ta sônora twn x, y tou ja brw ta sônora thc eikìnac pou prokôptei me to metasqhmatismì. Apì Apì Apì y x + ( u + v) y x + ( u + v) y 4 x ( u + v) 4 ( u 5 v ) + u 5 ( u 5 v ) + u 6 5 ( u 5 v ) v 5

11 Apì Opìte h perioq eðnai h y 4 x + ( u + v) 4 ( u 5 v ) + v 4 5 v u u 6 v 4 O v 6 u To olokl rwma (x + y)(x + 4y)dxdy uv J(u, v) dudv uv dudv ( 4 ) uv dv du ( 4 u [ v u ] 4 6 u du 4 [ ] u ) v dv du du

12 Jèma 4. Qrhsimopoi ntac to metasqhmatismì: u xy ìpou v y x (x + y) dxdy τ { (x, y) R : xy 4, x y 4x }, (x > ). LUSH f(x, y) dxdy g(u, v) J(u, v) dudv B ma o. O metasqhmatismìc eðnai dosmènoc: u xy v y x MporoÔme na epilôsoume wc proc x, y : x u v, y uv. B ma o. H Iakwbian J(u, v) (x, y) (u, v) (u,v) (x,y) u x y v y x u x y u y x v x y x v y x.

13 H Iakwbian J(u, v) (x, y) (u, v) (u,v) (x,y) y y x x x y x + y x y x v. J(u, v) v v. B ma o. x u v, y uv & f(x, y) x + y g(u, v) u v + uv. B ma 4 o. O tìpoc faðnetai sto sq ma.

14 Sqediˆzoume t ra sto epðpedo uv thn eikìna tou sunìrou tou tìpou me bˆsh to dosmèno metasqhmatismì. u xy v y x SÔnoro xy : xy u. SÔnoro xy 4 : xy 4 u 4. SÔnoro y x y x : y x v. SÔnoro y 4x y x 4 : y x 4 v 4. v u u 4 v 4 v O 4 u 4

15 B ma 5 o. 'Eqoume, sômfwna me th sqèsh f(x, y) dxdy (x + y) dxdy g(u, v) J(u, v) dudv ( u v + ) uv v dudv. O tìpoc D eðnai kanonikìc wc proc u (eujeðec u, u 4) en kleðnei pˆnw apì thn v 4 kai kˆtw apì th v. ( u v + ) uv v dudv u 4 u 4 [ 4 ( u v + ) ] uv v dv du [ ( ) ] 4 u + u v dv du u u [ u [ u ] 4 [ v [ 4 [ v v ( ) ] v + v dv du + v + v ] 4 ] 4 du du ( + + ) du u du 4 ]

16 Jèma 5. Qrhsimopoi ntac to metasqhmatismì: u x y v y x ( x + y ) dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic kampôlec: xy, xy, y x, y x. LUSH f(x, y) dxdy g(u, v) J(u, v) dudv B ma o. O metasqhmatismìc eðnai dosmènoc: u x y v y x LÔnontac wc proc x, y, brðskoume: x u v, y u v. B ma o. H Iakwbian J(u, v) (x, y) (u, v) (u,v) (x,y) u x y v y x u x y u y x v x y x v y x. 6

17 H Iakwbian J(u, v) (x, y) (u, v) (u,v) (x,y) y y x x x y x + y x y x v. J(u, v) v v. B ma o. f(x, y) x + y & x u v, y u v g(u, v) u v + u v. B ma 4 o. O tìpoc faðnetai sto sq ma. y y x y x (τ ) xy y x (τ τ τ ) O (τ ) xy y x x 7

18 Sqediˆzoume t ra sto epðpedo uv thn eikìna tou sunìrou tou tìpou me bˆsh to dosmèno metasqhmatismì. u x y v y x SÔnoro xy : SÔnoro xy : SÔnoro y x : SÔnoro y x : xy u. xy u. y x y x v. y x y x v. v v v O u 8

19 B ma 5 o. oc Trìpoc: 'Eqoume, sômfwna me th sqèsh f(x, y) dxdy ( x + y ) dxdy g(u, v) J(u, v) dudv ( u v + u v ) v dudv O tìpoc D eðnai kanonikìc wc proc u (eujeðec u, u ) en kleðnei pˆnw apì thn v kai kˆtw apì th v. ( u v + u v ) v dudv ( u v + u) dudv [ ( u dv] v + u) du [ u v + u v ] du ( u + u + u u ) du ( u + u) du [ ] u 4 + u [ ]

20 oc Trìpoc: 'Eqoume, sômfwna me th sqèsh f(x, y) dxdy ( x + y ) dxdy g(u, v) J(u, v) dudv ( u v + u v ) v dudv O tìpoc D eðnai kanonikìc wc proc v (eujeðec v, v ) en kleðnei pˆnw apì thn u kai kˆtw apì th u ( u v + u v ) v dudv ( u v + u) dudv [ ( u du] v + u) dv [ ] u v + u dv [ v + 8 v ( ) dv v + 7 [ v + 7 ] v ( ) ] dv

21 Jèma 6. Qrhsimopoi ntac to metasqhmatismì: u y + x v y x y + x x + xy dxdy ìpou eðnai o tìpoc pou perikleðetai apì tic kampôlec: kai brðsketai sthn pr th gwnða twn axìnwn. y x +, y x + 8, y, y x LUSH B ma o. f(x, y) dxdy u y + x v y x g(u, v) J(u, v) dudv Epeid h epðlush twn parapˆnw sqèsewn wc proc x, y den dðnei aplèc ekfrˆseic, en h upì olokl rwsh sunˆrthsh gðnetai arketˆ perðplokh wc proc tic metablhtèc u, v, gi> autì upojètoume ìti èqoume brei tic sunart seic x x(u, v), y y(u, v). B ma o. H Iakwbian J(u, v) (x, y) (u, v) (u,v) (x,y) u y + x v y x u x x u y v x y x v y x.

22 H Iakwbian J(u, v) (x, y) (u, v) (u,v) (x,y) x y x x + y x x +y x x x + y. J(u, v) x x + y x x + y. B ma o. Exetˆzoume m pwc h sunˆrthsh f(x(u, v), y(u, v)) J(u, v) èqei apl èkfrash wc proc u, v. f(x(u, v), y(u, v)) J(u, v) y + x x + xy x x + y x x + xy + y x g(u, v) J(u, v). + v B ma 4 o. O tìpoc faðnetai sto sq ma. y y x + 8 y x + y x B A (τ ) (τ ) (τ ) O x (τ τ τ τ )

23 Sqediˆzoume t ra sto epðpedo uv thn eikìna tou sunìrou tou tìpou me bˆsh to dosmèno metasqhmatismì. u y + x v y x SÔnoro y x + y + x : y + x u. SÔnoro y x + 8 y + x 8 : y + x 8 u 8. SÔnoro y x y x : y x v. SÔnoro y y x : y v. v u v u 8 O v 8 u

24 B ma 5 o. 'Eqoume, sômfwna me th sqèsh f(x, y) dxdy y + x x + xy dxdy g(u, v) J(u, v) dudv + v dudv O tìpoc D eðnai kanonikìc wc proc u (eujeðec u, u 8) en kleðnei pˆnw apì thn v kai kˆtw apì th v. + v dudv ln ( ) + v dv du [ln( + v)] du (ln ln ) du ln du 8 ln [u] 8 du ln (8 ) 5 ln. 4

25 Jèma. Qrhsimopoi ntac to metasqhmatismì: x + y sin(x y) dxdy ìpou to parallhlìgrammo me korufèc ta shmeða ( π ) ( A(, ), B, π, Γ + π, π ) kai (, ). LUSH f(x, y) dxdy g(u, v) J(u, v) dudv B ma o. O metasqhmatismìc eðnai dosmènoc: MporoÔme na epilôsoume wc proc x, y : x u + v, y u v B ma o. H Iakwbian J(u, v) (x, y) (u, v) x u + v y u v x u x v y u y v. 5

26 H Iakwbian J(u, v) 4 4. J(u, v). B ma o. & f(x, y) x + y sin(x y) g(u, v) u sin(v). B ma 4 o. Sqediˆzoume sto epðpedo uv thn eikìna tou sunìrou tou tìpou me bˆsh to dosmèno metasqhmatismì, apaleðfontac se kˆje perðptwsh ta x, y. A(, ), (, ) A : x y x y y x. SÔnoro y x : y x u v u + v v. & y x x y v. 6

27 (, ), Γ( + π, π ) Γ : x + π y π x y x y + y x + 4. SÔnoro y x + 4 x + y 4 : x + y 4 u + v + u v 4 u 4. & x + y 4 u 4. A(, ), B ( π, ) π AB : x π y π x y y x. SÔnoro y x : y x u v ( u + v ) u v u v u. & y x x + y u B ( π, ) π, Γ( + π, π ) Γ : x π + π π y + π π + π x π y + π y x π. SÔnoro y x π x y π : x y π u + v u + v π v π. & x y π v π. 'Etsi prokôptei o tìpoc D sto epðpedo uv pou eðnai h eikìna tou dia mèsou tou dosmènou metasqhmatismoô. 7

28 v u 4 v π u O v 4 u B ma 5 o. 'Eqoume, sômfwna me th sqèsh f(x, y) dxdy g(u, v) J(u, v) dudv x + y sin(x y) dxdy u sin v dudv O tìpoc D eðnai kanonikìc wc proc u (eujeðec u, u 4) en kleðnei pˆnw apì thn v π kai kˆtw apì th v. u sin v dudv ( π ( u π u sin v dv ) du ) sin v dv du u [ cos v] π du u ( cos π + cos ) du u du u du [ u ] 4 [ ] 4 u ] [

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