Determine o rotacional e o divergente dos campos abaixo.
F(x, y, z)=(x y^2 z^2, x^2 y z^2, x^2 y^2 z)

Passo 1

O enunciado pede para calcular o divergente e o rotacional (curl) do campo vetorial
F(x,y,z)=l(P,Q,Rr) com P=x,y^2,z^2,;Q=x^2,y,z^2,;R=x^2,y^2,z
O conteúdo usado aqui é o cálculo de operadores diferenciais em campos de três variáveis:
– Para o divergente usamos nablacdotvec F=partial Ppartial x+partial Qpartial y+partial Rpartial z.
– Para o rotacional usamos
nablatimesvec F =l(partial Rpartial y-partial Qpartial z,; partial Ppartial z-partial Rpartial x,; partial Qpartial x-partial Ppartial yr)

Passo 2

Cálculo do divergente:
partial Ppartial x =partial (x,y^2,z^2)partial x =y^2,z^2, partial Qpartial y =partial (x^2,y,z^2)partial y =x^2,z^2,
partial Rpartial z =partial (x^2,y^2,z)partial z =x^2,y^2
Somando essas derivadas parciais, obtemos
nablacdotvec F =y^2z^2+x^2z^2+x^2y^2

Passo 3

Cálculo do rotacional:
  1. Componente x:
partial Rpartial y =partial (x^2y^2z)partial y =2x^2y,z, partial Qpartial z =partial (x^2y,z^2)partial z =2x^2y,z,
logo partial Rpartial y-partial Qpartial z=0.
2. Componente y:
partial Ppartial z =partial (x,y^2z^2)partial z =2x,y^2,z, partial Rpartial x =partial (x^2y^2z)partial x =2x,y^2,z,
logo partial Ppartial z-partial Rpartial x=0.
3. Componente z:
partial Qpartial x =partial (x^2y,z^2)partial x =2x,y,z^2, partial Ppartial y =partial (x,y^2z^2)partial y =2x,y,z^2,
logo partial Qpartial x-partial Ppartial y=0.
Portanto todas as componentes são zero.

Resposta

nablacdotvec F = x^2y^2+x^2z^2+y^2z^2, nablatimesvec F = (0,0,0)

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