Answer
$\text{curl } \mathbf{F} = \mathbf{0}$
Work Step by Step
$F(x, y, z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$
$\text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix}$
where $P = x$, $Q = y$, and $R = z$.
$\text{curl } \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
x & y & z
\end{vmatrix}$
$\text{curl } \mathbf{F} = \left( \frac{\partial}{\partial y}(z) - \frac{\partial}{\partial z}(y) \right)\mathbf{i} - \left( \frac{\partial}{\partial x}(z) - \frac{\partial}{\partial z}(x) \right)\mathbf{j} + \left( \frac{\partial}{\partial x}(y) - \frac{\partial}{\partial y}(x) \right)\mathbf{k}$
$= \left( 0 - 0 \right)\mathbf{i} - \left( 0 - 0 \right)\mathbf{j} + \left( 0 - 0 \right)\mathbf{k}$
$= 0\mathbf{i} - 0\mathbf{j} + 0\mathbf{k}$
$= \mathbf{0}$
