Determine a rotacional dos campos vetoriais abaixo:
u=(rho sen phi) rho+(rho^2 z) phi+(z cos phi) z

Passo 1

O enunciado nos dá o campo vetorial em coordenadas cilíndricas
u = (rhosinphi),hatrho + (rho^2z),hatphi + (zcosphi),hat z
e pede o rotacional nablatimesu. Para isso vamos usar a fórmula do rotacional em cilindricas.

Passo 2

Seja vec A = Arho,hatrho + Aphi,hatphi + Az,hat z. Em coordenadas cilíndricas o rotacional é dado por
(nablatimesvec A)rho = 1rho,partial Azpartial phi - partial(rho,Aphi)partial z
(nablatimesvec A)phi = partial Arhopartial z - partial Azpartial rho
(nablatimesvec A)z = 1rho,partial(rho,Aphi)partial rho - 1rho,partial Arhopartial phi

Passo 3

No nosso caso identificamos
Arho = rhosinphi, Aphi = rho^2z, Az = zcosphi

Passo 4

Cálculo do componente radial:
(nablatimesvec u)rho = 1rhopartial Azpartial phi - partial(rho,Aphi)partial z
partial Azpartial phi = partial(zcosphi)partial phi = -,zsinphi, partial(rho,Aphi)partial z = partial(rho^3z)partial z = rho^3
(nablatimesvec u)rho = 1rhol(-zsinphi - rho^3r) = -zsinphirho - rho^2

Passo 5

Cálculo do componente azimutal:
(nablatimesvec u)phi = partial Arhopartial z - partial Azpartial rho
partial Arhopartial z = 0, partial Azpartial rho = 0
(nablatimesvec u)phi = 0 - 0 = 0

Passo 6

Cálculo do componente axial:
(nablatimesvec u)z = 1rhopartial(rho,Aphi)partial rho - 1rhopartial Arhopartial phi
partial(rho,Aphi)partial rho = partial(rho^3z)partial rho = 3rho^2z, partial Arhopartial phi = partial(rhosinphi)partial phi = rhocosphi
(nablatimesvec u)z = 1rhol(3rho^2z - rhocosphir) = 3rho z - cosphi

Passo 7

Reunindo as três componentes, obtemos
nablatimesvec u = l(-zsinphirho - rho^2r),hatrho + 0,hatphi + (3rho z - cosphi),hat z

Resposta

nablatimesvec u = l(-zsinphirho - rho^2r),hatrho + 0,hatphi + (3rho z - cosphi),hat z

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