Calcule o rotacional e o divergente para os seguintes campos vetoriais:
\vec{F}(x, y, z)=\left(2 y e^{2 x}+e^{z}\right) \hat{i}+\left(3 z e^{3 y}+e^{2 x}\right) \hat{j}+\left(x e^{z}+e^{3 y}\right) \widehat{k}

Passo 1

O enunciado pede para calcular o rotacional (curl) e o divergente (div) do campo \displaystyle \vec F(x,y,z)=\bigl(2y\,e^{2x}+e^{z}\bigr)\,\hat\imath+\bigl(3z\,e^{3y}+e^{2x}\bigr)\,\hat\jmath+\bigl(x\,e^{z}+e^{3y}\bigr)\,\hat k.
O conteúdo aqui é cálculo vetorial em coordenadas cartesianas: vamos usar as definições
\mathrm{div}\,\vec F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z},
\nabla\times\vec F=\Bigl(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\,\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\,\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\Bigr),
onde P=2y\,e^{2x}+e^z, Q=3z\,e^{3y}+e^{2x} e R=x\,e^z+e^{3y}.

Passo 2

Calculemos o divergente term a term:
\frac{\partial P}{\partial x} =\frac{\partial}{\partial x}\bigl(2y\,e^{2x}+e^z\bigr) =2y\cdot2e^{2x}+0 =4y\,e^{2x},
\frac{\partial Q}{\partial y} =\frac{\partial}{\partial y}\bigl(3z\,e^{3y}+e^{2x}\bigr) =3z\cdot3e^{3y}+0 =9z\,e^{3y},
\frac{\partial R}{\partial z} =\frac{\partial}{\partial z}\bigl(x\,e^z+e^{3y}\bigr) =x\,e^z+0 =x\,e^z.
Somando,
\mathrm{div}\,\vec F =4y\,e^{2x}+9z\,e^{3y}+x\,e^z.

Passo 3

Agora o rotacional:
  1. Componente i: \displaystyle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z} =3e^{3y}-3e^{3y}=0.
  2. Componente j: \displaystyle \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x} =e^{z}-e^{z}=0.
  3. Componente k: \displaystyle \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} =2e^{2x}-2e^{2x}=0.
Logo,
\nabla\times\vec F=(0,0,0).

Resposta

rotacional: \displaystyle \nabla\times\vec F=(0,0,0)
divergente: \displaystyle \mathrm{div}\,\vec F=4y\,e^{2x}+9z\,e^{3y}+x\,e^{z}

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