Calcule div \mathbf{F} e (\operatorname{rot} \mathbf{F}).
\mathbf{F}(x, y, z)=x^{2} \mathbf{i}-2 \mathbf{j}+y z \mathbf{k}

Passo 1

O enunciado pede para calcular a divergência e o rotacional (curl) do campo vetorial
F(x,y,z)=x^2,i-2,j+y z,k
O conteúdo aqui é cálculo vetorial em R^3. Vamos usar as definições
– divergência: nablacdotF= partial Fxpartial x+partial Fypartial y+partial Fzpartial z,
– rotacional: nablatimesF=l(partial Fzpartial y-partial Fypartial zr)i-l(partial Fzpartial x-partial Fxpartial zr)j+l(partial Fypartial x-partial Fxpartial yr)k

Passo 2

Calcular a divergência:
Fx = x^2, Fy = -2, Fz = y z
Aplicando
nablacdotF =partialpartial x(x^2) +partialpartial y(-2) +partialpartial z(y z)
Temos
partialpartial x(x^2)=2x, partialpartial y(-2)=0, partialpartial z(y z)=y
Logo
nablacdotF=2x+0+y=2x+y

Passo 3

Calcular o rotacional usando a fórmula por componentes:
(nablatimesF)x =partial Fzpartial y-partial Fypartial z =partialpartial y(y z)-partialpartial z(-2) =z-0=z,
(nablatimesF)y =partial Fxpartial z-partial Fzpartial x =partialpartial z(x^2)-partialpartial x(y z) =0-0=0,
(nablatimesF)z =partial Fypartial x-partial Fxpartial y =partialpartial x(-2)-partialpartial y(x^2) =0-0=0
Portanto
nablatimesF=z,i+0,j+0,k =z,i

Resposta

Divergência: nablacdotF=2x+y
Rotacional: nablatimesF=z,i

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