Calcule a divergente e o rotacional dos seguintes campos vetoriais:
F(x y z)=z i+(x+y) j+z^{2} k

Passo 1

O enunciado pede para calcular a divergente e o rotacional do campo
\mathbf{F}(x,y,z)=z\,\mathbf{i}+(x+y)\,\mathbf{j}+z^{2}\,\mathbf{k}.
O conteúdo é cálculo vetorial: usaremos as definições de
divergente \nabla\cdot\mathbf{F} e rotacional \nabla\times\mathbf{F}.

Passo 2

A divergente e o rotacional são dados por
\nabla\cdot\mathbf{F} =\frac{\partial F_x}{\partial x} +\frac{\partial F_y}{\partial y} +\frac{\partial F_z}{\partial z},
\nabla\times\mathbf{F} =\Bigl(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\Bigr)\mathbf{i} +\Bigl(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\Bigr)\mathbf{j} +\Bigl(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\Bigr)\mathbf{k}.

Passo 3

Calculando a divergente:
\displaystyle\frac{\partial F_x}{\partial x} =\frac{\partial z}{\partial x}=0,
\displaystyle\frac{\partial F_y}{\partial y} =\frac{\partial(x+y)}{\partial y}=1,
\displaystyle\frac{\partial F_z}{\partial z} =\frac{\partial z^2}{\partial z}=2z.
Logo
\nabla\cdot\mathbf{F}=0+1+2z=1+2z.

Passo 4

Calculando o rotacional:
componente \mathbf{i}:
\displaystyle\frac{\partial F_z}{\partial y} =\frac{\partial z^2}{\partial y}=0,
\displaystyle\frac{\partial F_y}{\partial z} =\frac{\partial(x+y)}{\partial z}=0,
logo 0-0=0.
componente \mathbf{j}:
\displaystyle\frac{\partial F_x}{\partial z} =\frac{\partial z}{\partial z}=1,
\displaystyle\frac{\partial F_z}{\partial x} =\frac{\partial z^2}{\partial x}=0,
logo 1-0=1.
componente \mathbf{k}:
\displaystyle\frac{\partial F_y}{\partial x} =\frac{\partial(x+y)}{\partial x}=1,
\displaystyle\frac{\partial F_x}{\partial y} =\frac{\partial z}{\partial y}=0,
logo 1-0=1.
Portanto
\nabla\times\mathbf{F} =0\,\mathbf{i}+1\,\mathbf{j}+1\,\mathbf{k} =\mathbf{j}+\mathbf{k}.

Resposta

A divergente é \nabla\cdot\mathbf{F}=1+2z.
O rotacional é \nabla\times\mathbf{F}=\mathbf{j}+\mathbf{k}.

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