Calcule o rotacional do campo vetorial F(r, theta, phi)=r^2 cos phi r+r^3 sin theta theta+3 cos phi sin theta phi, em coordenadas esféricas.

Passo 1

O enunciado pede para calcular nablatimesF onde
F(r,theta,phi) =Fr,r+Ftheta,theta+Fphi,phi =r^2cosphi,r+r^3sintheta,theta+3cosphisintheta,phi
O conteúdo usado é a fórmula do rotacional em coordenadas esféricas:
(nablatimesF)r =1rsinthetal(partialpartialtheta(sintheta,Fphi) -partial Fthetapartialphir)
(nablatimesF)theta =1rl(1sinthetapartial Frpartialphi -partialpartial r(r,Fphi)r)
(nablatimesF)phi =1rl(partialpartial r(r,Ftheta) -partial Frpartialthetar)

Passo 2

Calcule (nablatimesF)r:
partialpartialtheta(sintheta,Fphi) =partialpartialthetal(3cosphisin^2thetar) =6cosphi,sinthetacostheta,
partial Fthetapartialphi =partialpartialphi(r^3sintheta)=0
Logo
(nablatimesF)r =1rsinthetal(6cosphi,sinthetacostheta-0r) =6cosphicosthetar

Passo 3

Calcule (nablatimesF)theta:
1sinthetapartial Frpartialphi =1sinthetapartialpartialphi(r^2cosphi) =-r^2sinphisintheta,
partialpartial r(r,Fphi) =partialpartial rl(3rcosphisinthetar) =3cosphisintheta
Então
(nablatimesF)theta =1rl(-r^2sinphisintheta-3cosphisinthetar) =-rsinphisintheta-3cosphisinthetar

Passo 4

Calcule (nablatimesF)phi:
partialpartial r(r,Ftheta) =partialpartial r(r^4sintheta) =4r^3sintheta,
partial Frpartialtheta =partialpartialtheta(r^2cosphi)=0
Logo
(nablatimesF)phi =1r(4r^3sintheta-0)=4r^2sintheta

Resposta

nablatimesF =6cosphicosthetar,r +l(-rsinphisintheta-3cosphisinthetarr)theta +4r^2sintheta,phi

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