Matematici Speciale

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  MATEMATICI SPECIALE Adrian CARABINEANU  Cuprins I Calcul integral 5 1 Integrale curbilinii 7 1.1 Recti…carea curbelor . . . . . . . . . . . . . . . . . . . . . . . 71.1.1 Drumuri parametrizate . . . . . . . . . . . . . . . . . . 71.1.2 Curbe parametrizate . . . . . . . . . . . . . . . . . . . 101.1.3 Recti…carea curbelor plane . . . . . . . . . . . . . . . . 111.1.4 Recti…carea curbelor în spa¸tiu . . . . . . . . . . . . . . 121.2 Integrale curbilinii de primul tip (de prima spe¸t¼a) . . . . . . . 131.3 Integrale curbilinii de al doilea tip . . . . . . . . . . . . . . . . 16 2 Integrale duble 19 2.1 Integrale Riemann duble în plan . . . . . . . . . . . . . . . . . 192.1.1 Clase de functii integrabile Riemann . . . . . . . . . . 202.1.2 Reducerea integralei duble la o integral¼a iterat¼a . . . . 202.1.3 Exemple . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.4 Schimbarea de variabil¼a în integrala dubl¼a . . . . . . . 232.2 Integrale de suprafa¸t¼a . . . . . . . . . . . . . . . . . . . . . . . 252.2.1 Aria unei suprafe¸te în spa¸tiul tridimensional . . . . . . 252.2.2 Integrale de suprafa¸ta. Reducerea la o integral¼a dubl¼aîn plan . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Integrale triple 29 3.1 Integrala Riemann tripl¼a . . . . . . . . . . . . . . . . . . . . . 293.1.1 Clase de functii integrabile . . . . . . . . . . . . . . . . 293.1.2 Reducerea integralei triple la o integral¼a iterat¼a . . . . 303.1.3 Schimbarea de variabil¼a în integrala tripl¼a . . . . . . . 333  4  CUPRINS  4 Formule integrale 35 4.1 Formula ‡ux-divergen¸t¼a (Gauss-Ostrogradski) . . . . . . . . . 354.2 Formula lui Stokes (rotor-circula¸tie) . . . . . . . . . . . . . . . 384.2.1 Formula lui Green . . . . . . . . . . . . . . . . . . . . 384.2.2 Formula lui Stokes . . . . . . . . . . . . . . . . . . . . 394.3 Exerci¸tii ¸si probleme . . . . . . . . . . . . . . . . . . . . . . . 434.3.1 Operatori diferen¸tiali . . . . . . . . . . . . . . . . . . . 43 5 Aplica¸tii ale formulelor integrale 45 5.1 Condi¸tia ca o integral¼a curbilinie de al doilea tip s¼a …e inde-pendent¼a de drum . . . . . . . . . . . . . . . . . . . . . . . . . 455.1.1 Func¸tia poten¸tial . . . . . . . . . . . . . . . . . . . . . 465.1.2 Calculul integralelor curbilinii de al doilea tip . . . . . 475.2 Derivarea integralelor depinzând de un parametru . . . . . . . 485.2.1 Cazul unidimensional . . . . . . . . . . . . . . . . . . . 485.2.2 Cazul bidimensional . . . . . . . . . . . . . . . . . . . 485.2.3 Cazul tridimensional . . . . . . . . . . . . . . . . . . . 51 II Func¸tii complexe 55 6 Corpul numerelor complexe 57 6.1 Modulul ¸si argumentul unui numar complex. Forma trigono-metric¼a a unui num¼ar complex . . . . . . . . . . . . . . . . . . 586.1.1 Opera¸tii cu numere complexe scrise sub form¼a trigono-metric¼a . . . . . . . . . . . . . . . . . . . . . . . . . . 606.2 Topologia pe mul¸timea numerelor complexe . . . . . . . . . . 616.2.1 ¸Siruri de numere complexe . . . . . . . . . . . . . . . . 636.2.2 Proprietati ale ¸sirurilor convergente . . . . . . . . . . . 646.2.3 Serii de numere complexe . . . . . . . . . . . . . . . . 64 7 Func¸tii complexe de o variabil¼a complex¼a 67 7.1 Exemple de func¸tii complexe de o variabil¼a complex¼a . . . . . 687.1.1 Func¸tia exponential¼a . . . . . . . . . . . . . . . . . . . 687.1.2 Propriet¼a¸ti ale func¸tiei exponen¸tiale . . . . . . . . . . . 697.1.3 Func¸tii trigonometrice ¸si func¸tii hiperbolice de o vari-abil¼a complex¼a . . . . . . . . . . . . . . . . . . . . . . 707.1.4 Func¸tia logaritm . . . . . . . . . . . . . . . . . . . . . 71
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