Rotasi dari $$$\left\langle y z, x z, x y\right\rangle$$$

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

Menurut definisi, $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \nabla\times \left\langle y z, x z, x y\right\rangle$$$, atau, secara ekuivalen, $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\y z & x z & x y\end{array}\right|$$$, di mana $$$\times$$$ adalah operator hasil kali silang.

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

Tentukan turunan parsial:

$$$\frac{\partial}{\partial y} \left(x y\right) = x$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).

$$$\frac{\partial}{\partial z} \left(x z\right) = x$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).

$$$\frac{\partial}{\partial z} \left(y z\right) = y$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).

$$$\frac{\partial}{\partial x} \left(x y\right) = y$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).

$$$\frac{\partial}{\partial x} \left(x z\right) = z$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).

$$$\frac{\partial}{\partial y} \left(y z\right) = z$$$ (untuk langkah-langkahnya, lihat kalkulator turunan).

Sekarang, cukup masukkan turunan parsial yang telah ditemukan untuk mendapatkan rotasi (curl): $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle 0, 0, 0\right\rangle$$$

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