Example 1. Find the rotor of the vector $\mathbf{v}=(x+z) \mathbf{i}+(y+z) \mathbf{j}+\left(x^{2}+z\right) \mathbf{k}$.
Solution. Using formula (2), we have

$$
\operatorname{rota}=\left|\begin{array}{ccc}
1 & j & k \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
x+z & y+z & x^{2}+z
\end{array}\right|
$$

expanding the determinant by the elements of the first row and understanding the operation of, say, $\frac{\partial}{\partial y}$ on $x^{2}+z$, as an operation of partial differentiation,
we find

$$
\text { rot } \mathbf{a}=-\mathbf{i}-(2 x-1) \mathbf{j}
$$