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Forum Studenti •Applicazione della formula di Gauss-Green
Pagina 1 di 2

Applicazione della formula di Gauss-Green

Inviato: mercoledì 12 febbraio 2014, 11:17
da Giacomo Falcone
Ciao a tutti! Nello svolgere i vecchi testi d'esame ho riscontrato qualche difficoltà quando si trattava di calcolare l'integrale di una funzione su un dominio racchiuso da una curva data in forma parametrica! Per capirci meglio riporto un esercizio:

Sia D il dominio di [tex]R^2[/tex] racchiuso dall'asse delle x, dalla retta x=2 e dalla curva [tex]gamma[/tex] data da (x(t), y(t))=[tex](t+t^3, t+t^2)[/tex] con t compreso fra 0 e 1. Calcolare l'integrale su D di x dxdy

Vorrei più che altro chiarimenti su come trovare il campo vettoriale E che mi permette di trasformare l'integrale sul dominio D in un integrale lungo la curva [tex]gamma[/tex]; ad esempio nell'esercizio citato dovrei considerare divE=x e f=1 oppure f=x e divE=1 ?? E come arrivo a scrivere l'integrale della forma differenziale lungo la curva [tex]gamma[/tex] ??

Grazie a tutti in anticipo :D
P.s. Non so ancora scrivere bene con il LaTex quindi mi scuso in anticipo se non sono riuscito ad essere chiarissimo! :mrgreen:

Re: Applicazione della formula di Gauss-Green

Inviato: mercoledì 12 febbraio 2014, 17:42
da ghisi

Re: Applicazione della formula di Gauss-Green

Inviato: giovedì 13 febbraio 2014, 10:28
da Massimo Gobbino
E io, già che ci sono, sposto nella sezione giusta ...

Re: Applicazione della formula di Gauss-Green

Inviato: giovedì 13 febbraio 2014, 12:22
da Giacomo Falcone
Grazie mille per la spiegazione!

Re: Applicazione della formula di Gauss-Green

Inviato: giovedì 27 marzo 2014, 17:01
da Filippo.ingrasciotta
Sfrutto la discussione già aperta per una spiegazione semmai più dettagliata sul come procedere, perché ho dei grossi dubbi.

Dato il dominio [tex]\Omega=\{(t^2 ,t^3 +t): t \in [0,1]\}\cup\{y=0\}\cup\{x=1\}[/tex]

Devo calcolare l'area. So che devo usare Gauss-Green ma non ho idea di dove mettere mano perché ho il dominio ma non ho la funzione

Re: Applicazione della formula di Gauss-Green

Inviato: giovedì 27 marzo 2014, 19:44
da ghisi

Re: Applicazione della formula di Gauss-Green

Inviato: venerdì 28 marzo 2014, 8:54
da Filippo.ingrasciotta

Re: Applicazione della formula di Gauss-Green

Inviato: venerdì 28 marzo 2014, 11:51
da ghisi

Re: Applicazione della formula di Gauss-Green

Inviato: venerdì 28 marzo 2014, 15:47
da Filippo.ingrasciotta

Re: Applicazione della formula di Gauss-Green

Inviato: domenica 30 marzo 2014, 19:20
da ghisi
Il bordo di [tex]\Omega[/tex] è formato dalla curva [tex]\gamma =(t^2 ,t^3 +t), \; 0\leq t \leq 1[/tex] (che è orientata negativamente) dalla curva [tex]\gamma_1 = (t,0), \; 0\leq t \leq 1[/tex] (che così parametrizzata è orientata positivamente) e dalla curva [tex]\gamma_2 = (1,t), \; 0\leq t \leq 2[/tex] (che così parametrizzata è orientata positivamente).

L'area di [tex]\Omega[/tex] si può scrivere quindi ad esempio come

[tex]-\int_{\partial^{+}\Omega }y \, dx = -\int_{\gamma_1} y \, dx -\int_{\gamma_2} y \, dx + \int_{\gamma} y \, dx[/tex]

Re: Applicazione della formula di Gauss-Green

Inviato: domenica 30 marzo 2014, 20:24
da Filippo.ingrasciotta

Re: Applicazione della formula di Gauss-Green

Inviato: mercoledì 30 aprile 2014, 11:35
da Filippo.ingrasciotta
Altro esercizio altro problema, sempre col solito gauss-green :?

Stavolta mi viene dato il bordo del dominio e la funzione, e vuole sapere l'integrale....

Cito il testo di un esercizio così che magari può essere utile da esempio.

Funzione [tex]\phi = y[/tex]

Bordo del dominio [tex]{(cost, sent) t \in [0, \pi]} \cup {y=0}[/tex]

A questo punto non so quale formula usare perché sempre e comunque mi manca un dato, ovvero E il campo di vettori

Re: Applicazione della formula di Gauss-Green

Inviato: mercoledì 30 aprile 2014, 11:58
da ghisi

Re: Applicazione della formula di Gauss-Green

Inviato: mercoledì 30 aprile 2014, 15:02
da Filippo.ingrasciotta

Re: Applicazione della formula di Gauss-Green

Inviato: giovedì 1 maggio 2014, 9:37
da ghisi