Question

Determine o rotacional para o seguinte campo vetorial a seguir: Cos X sen Y Î - Z^J ( X,Y,Z) = (0,π,1)

Answer 1

Pretendemos calcular o rotacional do campo vetorial $\vec{f}: \mathbb{R}^3 \to \mathbb{R}^3$ dado por:

$\vec{f}(x,y,z) = \cos x \sin y \, \hat{i} - z\, \hat{j},$

no ponto de coordenadas $(x, y, z) = (0, \pi, 1)$.

Escrevendo $\vec{f} = f_x\,\hat{i} +f_y\,\hat{j} + f_z\,\hat{k}$ e utilizando a mnemónica do determinante, temos:

$\vec{\nabla} \times \vec{f} = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ \\\dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\\\ f_x & f_y & f_z}\end{vmatrix} = \left(\dfrac{\partial f_z}{\partial y}-\dfrac{\partial f_y}{\partial z}\right)\hat{i} + \left(\dfrac{\partial f_x}{\partial z}-\dfrac{\partial f_z}{\partial x}\right)\hat{j} + \left(\dfrac{\partial f_y}{\partial x}-\dfrac{\partial f_x}{\partial y}\right)\hat{k}.$

Neste caso, temos:

$f_x(x,y,z) = \cos x \sin y \implies \begin{cases}\dfrac{\partial f_x}{\partial y} = \cos x \cos y \\ \\ \dfrac{\partial f_x}{\partial z} = 0\end{cases}.$

$f_y(x,y,z) = -z \implies \begin{cases}\dfrac{\partial f_y}{\partial x} = 0 \\ \\ \dfrac{\partial f_y}{\partial z} = -1\end{cases}.$

$f_z(x,y,z) = 0 \implies \begin{cases}\dfrac{\partial f_z}{\partial x} = 0 \\ \\ \dfrac{\partial f_z}{\partial y} = 0\end{cases}.$

Substituindo, vem:

$\vec{\nabla} \times \vec{f} = \left(\underbrace{\dfrac{\partial f_z}{\partial y}}_{=0}-\underbrace{\dfrac{\partial f_y}{\partial z}}_{=-1}\right)\hat{i} + \left(\underbrace{\dfrac{\partial f_x}{\partial z}}_{=0}-\underbrace{\dfrac{\partial f_z}{\partial x}}_{=0}\right)\hat{j} + \left(\underbrace{\dfrac{\partial f_y}{\partial x}}_{=0}-\underbrace{\dfrac{\partial f_x}{\partial y}}_{=\cos x \cos y}\right)\hat{k} =\\\\= \hat{i} - \cos x \cos y \, \hat{k}.$

No ponto pretendido, vem por fim:

$\vec{\nabla} \times \vec{f}\Big\vert_{(0,\pi, 1)} =  \hat{i} - \underbrace{\cos 0}_{=1} \underbrace{\cos \pi}_{=-1} \, \hat{k} = \hat{i} + \hat{k}.$

Resposta:

$\boxed{\vec{\nabla} \times \vec{f}\Big\vert_{(0,\pi, 1)} = \hat{i} + \hat{k}}.$