Para cada item a seguir, calcule rot overrightarrow{\mathbf{F}}.
\vec{F}(x, y, z)=x^{2} y \vec{i}+x \ln (z) \vec{j}+z y \vec{k}

Passo 1

O enunciado pede o rotor (curl) do campo vetorial em \mathbb{R}^3
\vec{F}(x,y,z) = x^{2}y\,\vec{i} \;+\; x\ln(z)\,\vec{j} \;+\; zy\,\vec{k}.
O conteúdo é cálculo vetorial: usaremos a definição
\mathrm{rot}\,\vec{F} \;=\;\nabla\times\vec{F} \;=\;(\partial_{y}F_{z}-\partial_{z}F_{y})\,\vec{i} \;+\;(\partial_{z}F_{x}-\partial_{x}F_{z})\,\vec{j} \;+\;(\partial_{x}F_{y}-\partial_{y}F_{x})\,\vec{k}.

Passo 2

Componente em \vec{i}:
\partial_{y}F_{z}=\partial_{y}(zy)=z, \qquad \partial_{z}F_{y}=\partial_{z}\bigl(x\ln(z)\bigr)=\frac{x}{z},
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(\mathrm{rot}\,\vec{F})_{x} =\partial_{y}F_{z}-\partial_{z}F_{y} =z-\frac{x}{z}.

Passo 3

Componente em \vec{j}:
\partial_{z}F_{x}=\partial_{z}(x^{2}y)=0, \qquad \partial_{x}F_{z}=\partial_{x}(zy)=0,
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(\mathrm{rot}\,\vec{F})_{y} =\partial_{z}F_{x}-\partial_{x}F_{z} =0-0=0.

Passo 4

Componente em \vec{k}:
\partial_{x}F_{y}=\partial_{x}\bigl(x\ln(z)\bigr) =\ln(z), \qquad \partial_{y}F_{x}=\partial_{y}(x^{2}y)=x^{2},
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(\mathrm{rot}\,\vec{F})_{z} =\partial_{x}F_{y}-\partial_{y}F_{x} =\ln(z)-x^{2}.

Resposta

\mathrm{rot}\,\vec{F} =\Bigl(z-\tfrac{x}{z}\Bigr)\,\vec{i} \;+\;0\,\vec{j} \;+\;\bigl(\ln(z)-x^{2}\bigr)\,\vec{k}.

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