Answer
$\text{curl } \mathbf{F} = \frac{1}{\sqrt{x^2 + y^2 + z^2}} \left[ (y - z)\mathbf{i} + (z - x)\mathbf{j} + (x - y)\mathbf{k} \right]$
Work Step by Step
$F(x, y, z) = \sqrt{x^2 + y^2 + z^2}(\mathbf{i} + \mathbf{j} + \mathbf{k})$
$\text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix}$
where $P = \sqrt{x^2 + y^2 + z^2}$, $Q = \sqrt{x^2 + y^2 + z^2}$, and $R = \sqrt{x^2 + y^2 + z^2}$.
$\text{curl } \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
\sqrt{x^2 + y^2 + z^2} & \sqrt{x^2 + y^2 + z^2} & \sqrt{x^2 + y^2 + z^2}
\end{vmatrix}$
$\text{curl } \mathbf{F} = \left( \frac{\partial}{\partial y}(\sqrt{x^2 + y^2 + z^2}) - \frac{\partial}{\partial z}(\sqrt{x^2 + y^2 + z^2}) \right)\mathbf{i} - \left( \frac{\partial}{\partial x}(\sqrt{x^2 + y^2 + z^2}) - \frac{\partial}{\partial z}(\sqrt{x^2 + y^2 + z^2}) \right)\mathbf{j} + \left( \frac{\partial}{\partial x}(\sqrt{x^2 + y^2 + z^2}) - \frac{\partial}{\partial y}(\sqrt{x^2 + y^2 + z^2}) \right)\mathbf{k}$
$= \left( \frac{y}{\sqrt{x^2 + y^2 + z^2}} - \frac{z}{\sqrt{x^2 + y^2 + z^2}} \right)\mathbf{i} - \left( \frac{x}{\sqrt{x^2 + y^2 + z^2}} - \frac{z}{\sqrt{x^2 + y^2 + z^2}} \right)\mathbf{j} + \left( \frac{x}{\sqrt{x^2 + y^2 + z^2}} - \frac{y}{\sqrt{x^2 + y^2 + z^2}} \right)\mathbf{k}$
$= \frac{1}{\sqrt{x^2 + y^2 + z^2}} \left[ (y - z)\mathbf{i} - (x - z)\mathbf{j} + (x - y)\mathbf{k} \right]$
$= \frac{1}{\sqrt{x^2 + y^2 + z^2}} \left[ (y - z)\mathbf{i} + (z - x)\mathbf{j} + (x - y)\mathbf{k} \right]$
