Curl of $$$\left\langle y z, x z, x y\right\rangle$$$
Related calculators: Partial Derivative Calculator, Cross Product Calculator, Matrix Determinant Calculator
Your Input
Calculate $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle$$$.
Solution
By definition, $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \nabla\times \left\langle y z, x z, x y\right\rangle$$$, or, equivalently, $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\y z & x z & x y\end{array}\right|$$$, where $$$\times$$$ is the cross product operator.
Thus, $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle \frac{\partial}{\partial y} \left(x y\right) - \frac{\partial}{\partial z} \left(x z\right), \frac{\partial}{\partial z} \left(y z\right) - \frac{\partial}{\partial x} \left(x y\right), \frac{\partial}{\partial x} \left(x z\right) - \frac{\partial}{\partial y} \left(y z\right)\right\rangle.$$$
Find the partial derivatives:
$$$\frac{\partial}{\partial y} \left(x y\right) = x$$$ (for steps, see derivative calculator).
$$$\frac{\partial}{\partial z} \left(x z\right) = x$$$ (for steps, see derivative calculator).
$$$\frac{\partial}{\partial z} \left(y z\right) = y$$$ (for steps, see derivative calculator).
$$$\frac{\partial}{\partial x} \left(x y\right) = y$$$ (for steps, see derivative calculator).
$$$\frac{\partial}{\partial x} \left(x z\right) = z$$$ (for steps, see derivative calculator).
$$$\frac{\partial}{\partial y} \left(y z\right) = z$$$ (for steps, see derivative calculator).
Now, just plug in the found partial derivatives to get the curl: $$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle 0, 0, 0\right\rangle$$$.
Answer
$$$\operatorname{curl} \left\langle y z, x z, x y\right\rangle = \left\langle 0, 0, 0\right\rangle$$$A