Ache o rotacional e o divergente de cada um dos seguintes campos vetoriais:
F(x, y, z)=x z^2 i+y^2 j+x^2 z k.

Passo 1

O enunciado pede para calcular o divergente e o rotacional do campo vetorial
F(x,y,z)=x,z^2,i+y^2,j+x^2,z,k
O conteúdo envolve as definições
nablacdotF e nablatimesF,
que usaremos diretamente.

Passo 2

Para o divergente usamos
nablacdotF = partial Fxpartial x + partial Fypartial y + partial Fzpartial z
Aqui Fx = x z^2, Fy = y^2 e Fz = x^2 z. Logo,
partial Fxpartial x=z^2,
partial Fypartial y=2y,
partial Fzpartial z=x^2
Somando,
nablacdotF = z^2 + 2y + x^2

Passo 3

Para o rotacional usamos
nablatimesF = i & j & k partial/partial x & partial/partial y & partial/partial z Fx & Fy & Fz
Calculamos cada componente:
(nablatimesvec F)x =partial Fzpartial y-partial Fypartial z =partial (x^2 z)partial y-partial (y^2)partial z =0-0=0,
(nablatimesvec F)y =partial Fxpartial z-partial Fzpartial x =partial (x z^2)partial z-partial (x^2 z)partial x =2xz-2xz=0,
(nablatimesvec F)z =partial Fypartial x-partial Fxpartial y =partial (y^2)partial x-partial (x z^2)partial y =0-0=0
Portanto
nablatimesF = 0

Resposta

Divergente: nablacdotF=x^2+z^2+2y
Rotacional: nablatimesF=0

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