Calcule o Rotaciona do campo vetorial \vec{f}(x, y, z)=\left(z x^{2}\right) \vec{i}+\left(x y^{2}\right) \vec{j}+\left(x z^{2}\right) \vec{j}.

Passo 1

O enunciado pede o rotacional (ou curl) do campo vetorial \displaystyle \vec f(x,y,z) = (z x^{2})\,\vec i + (x y^{2})\,\vec j + (x z^{2})\,\vec k.
O conteúdo aqui é cálculo vetorial: usamos derivadas parciais e a definição \nabla\times\vec f.

Passo 2

Definimos as componentes
f_x = z x^{2},\quad f_y = x y^{2},\quad f_z = x z^{2}
e recordamos a fórmula do rotacional:
\nabla\times\vec f = \Bigl(\frac{\partial f_z}{\partial y}-\frac{\partial f_y}{\partial z}\Bigr)\vec i + \Bigl(\frac{\partial f_x}{\partial z}-\frac{\partial f_z}{\partial x}\Bigr)\vec j + \Bigl(\frac{\partial f_y}{\partial x}-\frac{\partial f_x}{\partial y}\Bigr)\vec k.

Passo 3

Calculamos as derivadas para as componentes i e j:
\frac{\partial f_z}{\partial y} = \frac{\partial}{\partial y}(x z^{2}) = 0, \quad \frac{\partial f_y}{\partial z} = \frac{\partial}{\partial z}(x y^{2}) = 0 \;\Longrightarrow\; (\nabla\times\vec f)_x = 0-0 = 0.
\frac{\partial f_x}{\partial z} = \frac{\partial}{\partial z}(z x^{2}) = x^{2}, \quad \frac{\partial f_z}{\partial x} = \frac{\partial}{\partial x}(x z^{2}) = z^{2} \;\Longrightarrow\; (\nabla\times\vec f)_y = x^{2}-z^{2}.

Passo 4

Calculamos a componente k:
\frac{\partial f_y}{\partial x} = \frac{\partial}{\partial x}(x y^{2}) = y^{2}, \quad \frac{\partial f_x}{\partial y} = \frac{\partial}{\partial y}(z x^{2}) = 0 \;\Longrightarrow\; (\nabla\times\vec f)_z = y^{2}-0 = y^{2}.

Resposta

\nabla\times\vec f = 0\,\vec i + \bigl(x^{2}-z^{2}\bigr)\,\vec j + y^{2}\,\vec k.

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