São dadas as seguintes funções escalares e campos vetoriais.
Determine o gradiente das funções escalares. No caso dos campos vetoriais, proceda o cálculo do divergente e rotacional
\vec{K}(r, \theta, \varphi)=r \cos \theta \hat{r}+r \operatorname{sen} \theta \hat{\theta}+r \operatorname{sen} \theta \cos \varphi \hat{\varphi}

Passo 1

O enunciado fornece o campo vetorial em coordenadas esféricas
\vec{K}(r,\theta,\varphi)=K_r\,\hat{r}+K_\theta\,\hat{\theta}+K_\varphi\,\hat{\varphi}
com
K_r=r\cos\theta,\quad K_\theta=r\sin\theta,\quad K_\varphi=r\sin\theta\cos\varphi,
e pede o divergente \nabla\cdot\vec{K} e o rotacional \nabla\times\vec{K}. Vamos usar as fórmulas em coordenadas esféricas.

Passo 2

Para o divergente em coordenadas esféricas vale
\nabla\cdot\vec{K} =\frac{1}{r^2}\frac{\partial}{\partial r}\bigl(r^2K_r\bigr) +\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\bigl(\sin\theta\,K_\theta\bigr) +\frac{1}{r\sin\theta}\frac{\partial K_\varphi}{\partial\varphi}.
Substituindo os componentes:
\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\cdot r\cos\theta) =\frac{1}{r^2}\frac{\partial}{\partial r}(r^3\cos\theta) =3\cos\theta,
\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta\cdot r\sin\theta) =\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(r\sin^2\theta) =2\cos\theta,
\frac{1}{r\sin\theta}\frac{\partial}{\partial\varphi}(r\sin\theta\cos\varphi) =\frac{1}{r\sin\theta}(-r\sin\theta\sin\varphi) =-\sin\varphi.
Somando,
\nabla\cdot\vec{K}=3\cos\theta+2\cos\theta-\sin\varphi=5\cos\theta-\sin\varphi.

Passo 3

Para o rotacional em coordenadas esféricas usamos
(\nabla\times\vec{K})_r =\frac{1}{r\sin\theta}\Bigl[\frac{\partial}{\partial\theta}(\sin\theta\,K_\varphi) -\frac{\partial K_\theta}{\partial\varphi}\Bigr],
(\nabla\times\vec{K})_\theta =\frac{1}{r}\Bigl[\frac{1}{\sin\theta}\frac{\partial K_r}{\partial\varphi} -\frac{\partial}{\partial r}(rK_\varphi)\Bigr],
(\nabla\times\vec{K})_\varphi =\frac{1}{r}\Bigl[\frac{\partial}{\partial r}(rK_\theta) -\frac{\partial K_r}{\partial\theta}\Bigr].
Calculando cada componente:
(\nabla\times\vec{K})_r =\frac{1}{r\sin\theta}\bigl[2r\sin\theta\cos\theta\cos\varphi-0\bigr] =2\cos\theta\cos\varphi,
(\nabla\times\vec{K})_\theta =\frac{1}{r}\bigl[0-2r\sin\theta\cos\varphi\bigr] =-2\sin\theta\cos\varphi,
(\nabla\times\vec{K})_\varphi =\frac{1}{r}\bigl[2r\sin\theta-(-r\sin\theta)\bigr] =3\sin\theta.
Logo
\nabla\times\vec{K} =2\cos\theta\cos\varphi\,\hat{r} -2\sin\theta\cos\varphi\,\hat{\theta} +3\sin\theta\,\hat{\varphi}.

Resposta

\nabla\cdot\vec{K}=5\cos\theta-\sin\varphi, \qquad \nabla\times\vec{K} =2\cos\theta\cos\varphi\,\hat{r}-2\sin\theta\cos\varphi\,\hat{\theta}+3\sin\theta\,\hat{\varphi}.

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