Rotacional de $$$\left\langle y z, x z, x y\right\rangle$$$

Calcule $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle$$$.

Por definição, $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \nabla\times \left\langle y z, x z, x y\right\rangle$$$, ou, de forma equivalente, $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\y z & x z & x y\end{array}\right|$$$, onde $$$\times$$$ é o operador do produto vetorial.

Logo, $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle \frac{\partial}{\partial y} \left(x y\right) - \frac{\partial}{\partial z} \left(x z\right), \frac{\partial}{\partial z} \left(y z\right) - \frac{\partial}{\partial x} \left(x y\right), \frac{\partial}{\partial x} \left(x z\right) - \frac{\partial}{\partial y} \left(y z\right)\right\rangle.$$$

Encontre as derivadas parciais:

$$$\frac{\partial}{\partial y} \left(x y\right) = x$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial z} \left(x z\right) = x$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial z} \left(y z\right) = y$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial x} \left(x y\right) = y$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial x} \left(x z\right) = z$$$ (para ver os passos, veja a calculadora de derivadas.)

$$$\frac{\partial}{\partial y} \left(y z\right) = z$$$ (para ver os passos, veja a calculadora de derivadas.)

Agora, basta substituir as derivadas parciais encontradas para obter o rotacional: $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle 0, 0, 0\right\rangle$$$

$$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle 0, 0, 0\right\rangle$$$A