In cylindrical coordinates, the gradient of $f$ is given by $\nabla f\=\left[\begin{array}{c}{f}_r\\ f_{\mathrm{\θ}}\/r\\ f_z\end{array}\right]\equiv \left[\begin{array}{c}u\\ v\\ w\end{array}\right]$, as per Table 9.3.2.

By Table 9.3.3, the curl in cylindrical coordinates is

$\nabla \times \left(\nabla f\right)\=\left|\begin{array}{ccc}\frac{{\mathit{e}}_{\mathit{r}}}{r}& {\mathit{e}}_{\mathbf{\theta }}& \frac{\mathbf{k}}{r}\\ {\partial }_r& {\partial }_{\mathrm{\theta }}& {\partial }_z\\ u& r v& w\end{array}\right|\=\left[\begin{array}{c}\frac{w_{\mathrm{\theta }}-\(r v{\)}_z}{r}\\ u_z-w_r\\ \frac{\(r v{\)}_r-u_{\mathrm{\theta }}}{r}\end{array}\right]$ = $\left[\begin{array}{c}\frac{{\partial }_{\mathrm{\θ}}\(f_z\)-{\partial }_z\(r\(f_{\mathrm{\θ}}\/r\)\) }{r}\\ {\partial }_z\(f_r\)-{\partial }_r\(f_z\)\\ \frac{{\partial }_r\(r\(f_{\mathrm{\θ}}\/r\)\)-{\partial }_{\mathrm{\θ}}\(f_r\)}{r}\end{array}\right]$ = $\left[\begin{array}{c}\frac{f_{z \mathrm{\θ}}-f_{\mathrm{\θ} z}}{r}\\ f_{r z}-f_{z r}\\ \frac{f_{\mathrm{\θ} r}-f_{r \mathrm{\θ}}}{r}\end{array}\right]$ = $\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

where the final simplification to the zero vector hinges on the equality of the mixed partial derivatives. That is why an assumption on the behavior of the function $f$ must be made.

The scalar $f$ does not carry (as an attribute) its coordinate system. Hence, it is necessary to use the Context Panel, not typeset math notation, to obtain the gradient. However, the gradient vector does carry its coordinate system, so the curl operator could be applied either by typeset math notation or by the Context Panel. It is clearly simpler to chain the calculations and continue using the Context Panel in this example.