Calcule o rotacional.
F(x, y, z)=y z i+x z j+x y k

Passo 1

O enunciado pede para calcular o rotacional do campo vetorial
F(x,y,z)=yz,i+xz,j+xy,k.
O conteúdo é o operador rotacional (ou curl) em campos vetoriais tridimensionais, e vamos usar a fórmula
nablatimesF =l(partial Rpartial y-partial Qpartial zr)i +l(partial Ppartial z-partial Rpartial xr)j +l(partial Qpartial x-partial Ppartial yr)k,
onde F=(P,Q,R).

Passo 2

Identificamos as componentes de F:
P(x,y,z)=yz, Q(x,y,z)=xz, R(x,y,z)=xy

Passo 3

Calculamos as derivadas parciais para cada componente do rotacional:
(nablatimesF)1 =partial Rpartial y-partial Qpartial z =partial (xy)partial y-partial (xz)partial z =x - x = 0,
(nablatimesF)2 =partial Ppartial z-partial Rpartial x =partial (yz)partial z-partial (xy)partial x =y - y = 0,
(nablatimesF)3 =partial Qpartial x-partial Ppartial y =partial (xz)partial x-partial (yz)partial y =z - z = 0

Passo 4

Como todas as componentes do rotacional são zero, concluímos que
nablatimesF=0.

Resposta

nablatimesF(x,y,z) = (0,0,0)

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