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Forum Studenti •Formula di Stokes
Pagina 1 di 2

Formula di Stokes

Inviato: lunedì 9 giugno 2014, 12:02
da Gabe
Ho dei problemi con questi due esercizi:

Indico con [tex]S[/tex] una superficie, con [tex]P[/tex] un punto appartenente o meno alla superficie, con [tex]v_p[/tex] la direzione normale ad [tex]S[/tex] nel punto [tex]P[/tex], calcolare il flusso del vettore rotore di [tex]F[/tex].

1) [tex]S= x^2+y^2+z=1, z\geq0[/tex], [tex]P=(1, 0, 0)[/tex], [tex]v_p=(2, 0, 1)[/tex], [tex]F=(xy, z, arctan(x+y))[/tex].

2) [tex]S= 4x^2+e^zy^2+z^2=1, y\geq0[/tex], [tex]P=(0, 0, 1)[/tex], [tex]v_p=(0, 0, 1)[/tex], [tex]F=(sin (y), xe^z, xz^2)[/tex].

Mi blocco subito per colpa della parametrizzazione, non riesco ad individuare la figura e nemmeno a parametrizzarla

Re: Formula di Stokes

Inviato: lunedì 9 giugno 2014, 13:41
da ghisi

Re: Formula di Stokes

Inviato: lunedì 9 giugno 2014, 22:46
da GIMUSI

Re: Formula di Stokes

Inviato: martedì 10 giugno 2014, 14:01
da Gabe
Ora va meglio :D grazie mille

Re: Formula di Stokes

Inviato: martedì 10 giugno 2014, 17:47
da Angelica27
Gimusi, perdonami, ma nel primo esercizio, quello in cui abbiamo una specie di semisfera, non capisco perché, quando ti calcoli il rotore, ti venga 0 come prima componente. C di y - B di z non fa 0 :/

Re: Formula di Stokes

Inviato: martedì 10 giugno 2014, 21:45
da GIMUSI

Re: Formula di Stokes

Inviato: domenica 29 giugno 2014, 15:59
da Filippo.ingrasciotta
Testo ese:

S:([tex]x^2+ 2 y^2 + z^2 =9 , y\ge 0[/tex])

F ([tex]x e^x +y , x z^2 ,x[/tex])

P:(3,0,0) vp(1,0,0)

Mediante la formula di stokes mi ritrovo a calcolare il flusso sul bordo di S.

Bordo di S ([tex]x^2 + z^2 =9[/tex]) parametrizzato diventa ([tex]3cos\theta , 0 , 3 sen\theta[/tex])

il versore tangente sarà ([tex]-3 sen\theta ,0, 3 cos\theta[/tex]) {Molto probabilmente mi manca un fratto 3 poichè il modulo del versore vale 3.. ma a regola tanto dovrei rimoltiplicare per 3 quando faccio il passaggio da ds a [tex]d\theta[/tex] quindi dovrebbe essere ininfluente sul risultato finale}

Facendo F*vt = [tex]-9e^{3sen\theta} cos\theta sen\theta + 9 cos\theta ^2[/tex]

e svolgento [tex]\int_{0}^{2\pi}-9e^{3sen\theta} cos\theta sen\theta + 9 cos\theta ^2 \, d\theta = 9\pi[/tex]

il risultato del testo è [tex]-9\pi[/tex] quindi penso che abbia sbagliato l'orientazione, però ricontrollando vedo che per [tex]\theta =0[/tex] ottengo il vettore (0,0,3) che è tangente al bordo con orientazione positiva....

Re: Formula di Stokes

Inviato: domenica 29 giugno 2014, 23:46
da GIMUSI

Re: Formula di Stokes

Inviato: lunedì 30 giugno 2014, 9:11
da Filippo.ingrasciotta

Re: Formula di Stokes

Inviato: lunedì 30 giugno 2014, 9:40
da GIMUSI

Re: Formula di Stokes

Inviato: lunedì 30 giugno 2014, 10:49
da Filippo.ingrasciotta
scrivo il testo di 4 ese che non mi sono tornati..

1) S= ([tex]x + y^2 + z= 1, x\ge0 , z\ge0[/tex]) P(1,0,0) vp(1,0,1) F([tex]xz,x,senx senz[/tex])
2) S=([tex]x + y^2 -z = 4 , x\ge0 , z\le0[/tex]) P(0,2,0) vp(1,4,-1) F([tex]x-y, y^2, xyz[/tex])
3) S=([tex]x^2 + y^2 =4 , x\ge0, 0\le z\le1[/tex]) P(2,0,0) vp(1,0,0) F([tex]x+y, z, x-y[/tex])
4) S=([tex]x^2 + y^4 +z^4 =1 , y\ge0[/tex]) P(1,0,0) vp(1,0,0) F([tex]1,2x,2y[/tex])

Re: Formula di Stokes

Inviato: lunedì 30 giugno 2014, 15:44
da GIMUSI

Re: Formula di Stokes

Inviato: lunedì 30 giugno 2014, 16:17
da Filippo.ingrasciotta
[tex]S= (x + y^2 + z= 1, x\ge0 , z\ge0)[/tex] P(1,0,0) vp(1,0,1) F(xz,x,senx senz)

parametrizzo sul piano y=0 ottenendo

[tex]\partial S = (x+z=1) =(t,0,1-t) t\in[0,1][/tex]
il vt mi viene (1,0,-1)

[tex]\int_{\partial S}^{} F*vt \,ds= - \int xz - senx senz \,ds[/tex] = [tex]\int_{0}^{1} -t*(1-t) + sent*sen(1-t) dt= -1/6 + 1/2 (sen1-cos1)[/tex]

ho provato a parametrizzare anche in un altro modo, sul piano z=0

[tex]\partial S = (x+y^2=1) =(t,\sqrt{1-t},0) t\in[0,1][/tex]
vt ([tex]1, \frac{-1}{2\sqrt{1-t}},0[/tex])

facendo l'integrale mi viene

[tex]\int_{0}^{1} \frac{-t}{2\sqrt{1-t}} \, dt = -2/3[/tex]

Mi pare strano che mi vengano due valori diversi e inoltre il risultato giusto è [tex]4/3[/tex]

Re: Formula di Stokes

Inviato: lunedì 30 giugno 2014, 16:30
da GIMUSI

Re: Formula di Stokes

Inviato: lunedì 30 giugno 2014, 16:41
da Filippo.ingrasciotta