Determine (a) o rotacional e (b) o divergente do campo vetorial.
\mathbf{F}(x, y, z)=\left\langle e^{x}, e^{y}, e^{n+y}\right\rangle

Passo 1

O enunciado dá o campo vetorial
\mathbf{F}(x,y,z)=\langle e^{x},\,e^{y},\,e^{n+y}\rangle
e pede (a) o rotacional \nabla\times\mathbf{F} e (b) o divergente \nabla\cdot\mathbf{F}.
O conteúdo é cálculo vetorial: usamos derivadas parciais e as fórmulas de \nabla\times e \nabla\cdot.

Passo 2

Para um campo \mathbf{F}=\langle P,Q,R\rangle, valem as definições
\nabla\times\mathbf{F} =\bigl(\tfrac{\partial R}{\partial y}-\tfrac{\partial Q}{\partial z},\, \tfrac{\partial P}{\partial z}-\tfrac{\partial R}{\partial x},\, \tfrac{\partial Q}{\partial x}-\tfrac{\partial P}{\partial y}\bigr),
\nabla\cdot\mathbf{F} =\tfrac{\partial P}{\partial x}+\tfrac{\partial Q}{\partial y}+\tfrac{\partial R}{\partial z}.
Aqui P=e^x, Q=e^y, R=e^{n+y} (com n constante).

Passo 3

Calculemos as derivadas parciais para o rotacional:
\displaystyle \tfrac{\partial R}{\partial y} =\tfrac{\partial}{\partial y}e^{n+y} =e^{n+y},
\displaystyle \tfrac{\partial Q}{\partial z} =\tfrac{\partial}{\partial z}e^{y} =0,
\displaystyle \tfrac{\partial P}{\partial z} =0,
\displaystyle \tfrac{\partial R}{\partial x} =0,
\displaystyle \tfrac{\partial Q}{\partial x} =0,
\displaystyle \tfrac{\partial P}{\partial y} =0.
Logo,
\nabla\times\mathbf{F} =\bigl(e^{n+y}-0,\;0-0,\;0-0\bigr) =\langle e^{n+y},\,0,\,0\rangle.

Passo 4

Agora o divergente:
\displaystyle \tfrac{\partial P}{\partial x} =\tfrac{\partial}{\partial x}e^{x} =e^{x},
\displaystyle \tfrac{\partial Q}{\partial y} =\tfrac{\partial}{\partial y}e^{y} =e^{y},
\displaystyle \tfrac{\partial R}{\partial z} =\tfrac{\partial}{\partial z}e^{n+y} =0.
Portanto,
\nabla\cdot\mathbf{F} =e^{x}+e^{y}+0 =e^{x}+e^{y}.

Resposta

(a) \displaystyle \nabla\times\mathbf{F}=\langle e^{n+y},\,0,\,0\rangle.
(b) \displaystyle \nabla\cdot\mathbf{F}=e^{x}+e^{y}.

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