Nos items abaixo, calcule nabla times F e nabla cdot F.

Passo 1

O problema pede para calcular o rotacional ( nabla times F) e a divergência ( nabla cdot F) do campo vetorial F(x,y,z)=x^3ln z,i+x e^-y,j-(y^2+2z),k. Vamos usar as definições de cálculo vetorial e derivadas parciais para obter os resultados.

Passo 2

A divergência de um campo F=(Fx,Fy,Fz) é dada por
nablacdotF=partial Fxpartial x+partial Fypartial y+partial Fzpartial z
Aqui temos
Fx=x^3ln z,
Fy=x e^-y e
Fz=-(y^2+2z).
Calculando as derivadas:
partial Fxpartial x=3x^2ln z,
partial Fypartial y=-x e^-y,
partial Fzpartial z=-2
Somando, achamos
nablacdotF=3x^2ln z - x e^-y - 2

Passo 3

Para o rotacional usamos a fórmula
$$\nabla\times\mathbf{F}=iggl(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\biggr)\mathbf{i}+\biggl(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\biggr)\mathbf{j}+\biggl(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\biggr)\mathbf{k}.
$$
Precisamos então das derivadas parciais:
nablatimesF=iggl(partial Fzpartial y-partial Fypartial zgr)i+gl(partial Fxpartial z-partial Fzpartial xgr)j+gl(partial Fypartial x-partial Fxpartial ygr)k
partial Fzpartial y=-2y,
partial Fypartial z=0,
partial Fxpartial z=x^31z=x^3z,
partial Fzpartial x=0,
partial Fypartial x=e^-y,

Passo 4

Substituindo essas derivadas na expressão do rotacional, obtemos
(nablatimesF)x=-2y-0=-2y,
(nablatimesF)y=x^3z-0=x^3z,
(nablatimesF)z=e^-y-0=e^-y
Logo,
nablatimesF=-2y,i+x^3z,j+e^-y,k

Resposta

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