Calcule o rotacional e o divergente para os seguintes campos vetoriais:
\vec{F}(x, y, z)=\sqrt{x^{2}+y^{2}+1} \hat{i}+\sqrt{y^{2}+z^{2}+1} \hat{j}+\sqrt{x^{2}+z^{2}+1} \hat{k}

Passo 1

O enunciado dá o campo vetorial em \mathbb{R}^3
\vec{F}(x,y,z) =\sqrt{x^{2}+y^{2}+1}\,\hat{i} +\sqrt{y^{2}+z^{2}+1}\,\hat{j} +\sqrt{x^{2}+z^{2}+1}\,\hat{k}
e pede o divergente \nabla\cdot\vec{F} e o rotacional \nabla\times\vec{F}. Vamos usar as definições
\nabla\cdot\vec{F} =\frac{\partial F_{1}}{\partial x} +\frac{\partial F_{2}}{\partial y} +\frac{\partial F_{3}}{\partial z},
\nabla\times\vec{F} =\bigl(\partial_{y}F_{3}-\partial_{z}F_{2},\; \partial_{z}F_{1}-\partial_{x}F_{3},\; \partial_{x}F_{2}-\partial_{y}F_{1}\bigr).

Passo 2

Identificamos as componentes
F_{1}(x,y,z)=\sqrt{x^{2}+y^{2}+1},\quad F_{2}(x,y,z)=\sqrt{y^{2}+z^{2}+1},\quad F_{3}(x,y,z)=\sqrt{x^{2}+z^{2}+1}.

Passo 3

Calculamos as derivadas parciais para o divergente:
\frac{\partial F_{1}}{\partial x} =\frac{x}{\sqrt{x^{2}+y^{2}+1}},\quad \frac{\partial F_{2}}{\partial y} =\frac{y}{\sqrt{y^{2}+z^{2}+1}},\quad \frac{\partial F_{3}}{\partial z} =\frac{z}{\sqrt{x^{2}+z^{2}+1}}.

Passo 4

Somando esses termos obtemos
\nabla\cdot\vec{F} =\frac{x}{\sqrt{x^{2}+y^{2}+1}} +\frac{y}{\sqrt{y^{2}+z^{2}+1}} +\frac{z}{\sqrt{x^{2}+z^{2}+1}}.

Passo 5

Agora calculamos as derivadas parciais que entram no rotacional:
\partial_{y}F_{3}=0,\quad \partial_{z}F_{2}=\frac{z}{\sqrt{y^{2}+z^{2}+1}},\quad \partial_{z}F_{1}=0,\quad \partial_{x}F_{3}=\frac{x}{\sqrt{x^{2}+z^{2}+1}},\quad \partial_{x}F_{2}=0,\quad \partial_{y}F_{1}=\frac{y}{\sqrt{x^{2}+y^{2}+1}}.

Passo 6

Montando as componentes de \nabla\times\vec{F}:
(\nabla\times\vec{F})_{1} =\partial_{y}F_{3}-\partial_{z}F_{2} =-\frac{z}{\sqrt{y^{2}+z^{2}+1}},
(\nabla\times\vec{F})_{2} =\partial_{z}F_{1}-\partial_{x}F_{3} =-\frac{x}{\sqrt{x^{2}+z^{2}+1}},
(\nabla\times\vec{F})_{3} =\partial_{x}F_{2}-\partial_{y}F_{1} =-\frac{y}{\sqrt{x^{2}+y^{2}+1}}.

Resposta

Divergente:
\nabla\cdot\vec{F} =\frac{x}{\sqrt{x^{2}+y^{2}+1}} +\frac{y}{\sqrt{y^{2}+z^{2}+1}} +\frac{z}{\sqrt{x^{2}+z^{2}+1}}.
Rotacional:
\nabla\times\vec{F} =-\frac{z}{\sqrt{y^{2}+z^{2}+1}}\,\hat{i} -\frac{x}{\sqrt{x^{2}+z^{2}+1}}\,\hat{j} -\frac{y}{\sqrt{x^{2}+y^{2}+1}}\,\hat{k}.

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