Determine o rotacional e o divergente do campo vetorial F.
F(x, y)=x y z i-x^2 y j

Passo 1

O enunciado pede para calcular o divergente e o rotacional do campo vetorial vec F(x,y,z)=x,y,z,vec i - x^2,y,vec j + 0,vec k. Vamos usar o operador nabla nabla=(partial/partial x,partial/partial y,partial/partial z) e as definições padrão de divergente e rotacional.

Passo 2

Para o divergente usamos nablacdotvec F=dfracpartial Fxpartial x+dfracpartial Fypartial y+dfracpartial Fzpartial z. Calculamos cada termo:
partial Fxpartial x=partialpartial x(x y z)=y z,
partial Fypartial y=partialpartial y(-x^2 y)=-x^2,
partial Fzpartial z=partialpartial z(0)=0
Somando, vem
nablacdotvec F=y z - x^2

Passo 3

Para o rotacional usamos (nablatimesvec F)x=partial Fzpartial y-partial Fypartial z=0-0=0,. Calculamos cada componente:
$$
(\nabla\times\vec F)_x=\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}=0-0=0,$$
$$
(\nabla\times\vec F)_y=\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}=x y-0=x y,$$
$$
(\nabla\times\vec F)_z=\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}=-2x y - x z. (nablatimesvec F)y=partial Fxpartial z-partial Fzpartial x=x y-0=x y,é\nabla\times\vec F=0,\vec i + x y,\vec j +(-2x y - x z),\vec k.$$

Resposta

Divergente: nablacdotvec F=y z - x^2. Rotacional: nablatimesvec F=lang<=0,,x y,,-2x y - x zrang<=.

Exercícios relacionados