Teorema de Stokes Calculo

[pic 1]

TEOREMA DE STOKES

Sean S, [pic 2], n como antes indicadas y sea F=Mi+Nj+Pk un campo vectorial en el que M, N, P tienen derivadas de primer orden continuas en S y su frontera [pic 3]. Si T designa el vector tangente unitario de [pic 4], entonces:

[pic 5]

Ejemplos

1)        Verifique el teorema F=yi – xj+yzk, si S es el paraboloide z=x2 +y2 , con el círculo x2 +y2 =1, z=1 y su frontera.

        X=cost, y=sent, z=1

        [pic 6]

        Rot F=zi+0 – 2k

        [pic 7]                

   Solución:

1.  Sea Z=f(x1,x2,…,xn), el rotor es un campo vectorial

F(x,y,z)=M(x,y,z)i+N(x,y,z)j+P(x,y,z)k, entonces:

ROT F=[pic 8][pic 9]

 [pic 10]

      F=yi – xj+yzk

[pic 11][pic 12][pic 13][pic 14]

Rot F=[pic 15][pic 16][pic 17][pic 18][pic 19]

1. El vector n [pic 20][pic 21][pic 22][pic 23][pic 24][pic 25]

[pic 26][pic 27]

  . n= (-2x , -2y, 1)

1. (Rot F ) . n= (z , 0 , -2) . ( -2x , -2y,1) = -2xz +0  -2

1. [pic 28]

[pic 29]

[pic 30][pic 31][pic 32]

[pic 33][pic 34]

[pic 35]

[pic 36]

Sabemos que     x= r cos  ,   y =r sen    🡺   0 ≤ r ≤ 1    .     0  ≤     ≤ 2[pic 37][pic 38][pic 39][pic 40]

[pic 41][pic 42][pic 43][pic 44]

[pic 45][pic 46][pic 47]

[pic 48][pic 49]

[pic 50]

[pic 51]

[pic 52]

[pic 53][pic 54]

[pic 55][pic 56]

[pic 57]

[pic 58][pic 59]

[pic 60][pic 61][pic 62][pic 63]

[pic 64]

[pic 65]

[pic 66][pic 67][pic 68][pic 69][pic 70][pic 71]

[pic 72][pic 73]

[pic 74][pic 75][pic 76][pic 77][pic 78][pic 79]

[pic 80][pic 81]

[pic 82][pic 83]

[pic 84][pic 85][pic 86]

[pic 87][pic 88]

1. Grafico y restricciones  [pic 89][pic 90][pic 91]

[pic 92][pic 93][pic 94][pic 95]

[pic 96][pic 97][pic 98]

[pic 99][pic 100][pic 101][pic 102]

[pic 103][pic 104][pic 105]

[pic 106][pic 107][pic 108]

[pic 109][pic 110]

[pic 111][pic 112][pic 113]

[pic 114][pic 115][pic 116][pic 117]

[pic 118][pic 119]

1. ROT F=[pic 120][pic 121]

             F(x,y,z)=(x-z)i+(y-z)j+x2k

Rot F= (0 –(0-1))i + (0-1-2x)j + (0 -0)k= (1 , -1-2x , 0)

...