1. Seja w=4 x^2+4 y^2+z^2, x=rho sen theta cos phi, y=rho sen phi sen theta, z=rho cos phi. Determine partial w / partial rho, partial w / partial phi e partial w / partial theta.

Passo 1

O enunciado pede para calcular as derivadas parciais de
w = 4x^2 + 4y^2 + z^2
as variáveis são dadas por transformações (sem esquecer que vamos usar a regra da cadeia):
x = rho,sintheta,cosphi, y = rho,sinphi,sintheta, z = rho,cosphi
O conteúdo é aplicação da regra da cadeia em coordenadas esféricas (dadas). Vamos precisar de partial w/partial x, partial w/partial y, partial w/partial z e dos jacobianos partial x/partial u, partial y/partial u, partial z/partial u para u = rho,phi,theta.

Passo 2

Primeiro, calculamos as derivadas parciais de w em relação a x,y,z e as parciais de x,y,z em relação a rho,phi,theta:
partial wpartial x = 8x, partial wpartial y = 8y, partial wpartial z = 2z
Derivadas de x,y,z:
partial xpartial rho = sintheta,cosphi, partial xpartial phi = -rho,sintheta,sinphi, partial xpartial theta = rho,costheta,cosphi;
partial ypartial rho = sinphi,sintheta, partial ypartial phi = rho,cosphi,sintheta, partial ypartial theta = rho,sinphi,costheta;
partial zpartial rho = cosphi, partial zpartial phi = -rho,sinphi, partial zpartial theta = 0

Passo 3

Agora aplicamos a regra da cadeia para partial w/partial rho:
partial wpartial rho = partial wpartial xpartial xpartial rho + partial wpartial ypartial ypartial rho + partial wpartial zpartial zpartial rho
Substituindo e simplificando (lembre que x = rho,sintheta,cosphi, etc):
partial wpartial rho = 8x,sintheta,cosphi + 8y,sinphi,sintheta + 2z,cosphi = 8rho(sin^2theta)(cos^2phi+sin^2phi) + 2rhocos^2phi = 8rhosin^2theta + 2rhocos^2phi

Passo 4

Para partial w/partial phi, usamos
partial wpartial phi = partial wpartial xpartial xpartial phi + partial wpartial ypartial ypartial phi + partial wpartial zpartial zpartial phi
Substituindo e simplificando:
partial wpartial phi = 8x(-rho,sintheta,sinphi) + 8y(rho,cosphi,sintheta) + 2z(-rho,sinphi) = -2rho^2cosphi,sinphi = -rho^2sin(2phi)

Passo 5

Finalmente, para partial w/partial theta:
partial wpartial theta = partial wpartial xpartial xpartial theta + partial wpartial ypartial ypartial theta + partial wpartial zpartial zpartial theta
Substituindo e simplificando:
partial wpartial theta = 8x(rho,costheta,cosphi) + 8y(rho,sinphi,costheta) = 8rhocosthetal(xcosphi + ysinphir) = 8rho^2costhetasintheta = 4rho^2sin(2theta)

Resposta

partial wpartial rho=8rhosin^2theta+2rhocos^2phi, partial wpartial phi=-rho^2sin(2phi), partial wpartial theta=4rho^2sin(2theta)

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