\[\mathbf{F} = F_r \mathbf{e_r} + F_{\theta} \mathbf{e_\theta} +F_z \mathbf{e_z}\]
and let \[curl \mathbf{F} = \mathbf{\nabla} \times \mathbf{F} = F_A \mathbf{e_r} + A_{\theta} \mathbf{e_\theta} +A_z \mathbf{e_z}\]
By definition,
\[A_r = \frac{\oint_C \mathbf{F} \cdot d \mathbf{r}}{\int_S dS}\]
where \[C\]
is the boundary curve or the small surface element \[S\]
on the curved surface of the cylinder shown.
\[\begin{equation} \begin{aligned} \oint_C \mathbf{F} \cdot d \mathbf{r} &= \oint_{ABCD} \mathbf{F} d \mathbf{r} \\ &= \int_{AB} \mathbf{F} d \mathbf{r} + \int_{BC} \mathbf{F} d \mathbf{r} + \int_{CD} \mathbf{F} d \mathbf{r} + \int_{DA} \mathbf{F} d \mathbf{r} \\ &= (F_{\theta} r d \theta )_{AB} +(F_z dz)_{BC} - (F_{\theta} d \theta )_{CD} -(F_z dz)_{DA} \\ &= - \frac{\partial F_{\theta}}{\partial z}r dz d \theta + \frac{\partial F_z}{\partial \theta}d \theta dz \end{aligned} \end{equation}\]
The area of the surface element is
\[r d \theta dz\]
.Hence
\[A_r =\frac{ - \frac{\partial F_{\theta}}{\partial z}r dz d \theta + \frac{\partial F_z}{\partial \theta}d \theta dz}{r d \theta dz}= \frac{1}{r} \frac{\partial F_z}{\partial \theta} - \frac{\partial F_{\theta}}{\partial z}\]
\[\begin{equation} \begin{aligned} \oint_C \mathbf{F} \cdot d \mathbf{r} &= \oint_{EFGH} \mathbf{F} d \mathbf{r} \\ &= \int_{EF} \mathbf{F} d \mathbf{r} + \int_{FG} \mathbf{F} d \mathbf{r} + \int_{GH} \mathbf{F} d \mathbf{r} + \int_{HE} \mathbf{F} d \mathbf{r} \\ &= (F_z dz)_{EF} +(F_r dr)_{FG} - (F_z dz)_{GH} -(F_r dr)_{HE} \\ &= - \frac{\partial F_z}{\partial r}dr dz + \frac{\partial F_r}{\partial z}d r dz \end{aligned} \end{equation}\]
The area of the surface element is
\[dr dz\]
.Hence
\[A_\theta =\frac{- \frac{\partial F_z}{\partial r}dr dz + \frac{\partial F_r}{\partial z}d r dz}{dr dz}= \frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r}\]
\[\begin{equation} \begin{aligned} \oint_C \mathbf{F} \cdot d \mathbf{r} &= \oint_{JKLM} \mathbf{F} d \mathbf{r} \\ &= \int_{JK} \mathbf{F} d \mathbf{r} + \int_{KL} \mathbf{F} d \mathbf{r} + \int_{LM} \mathbf{F} d \mathbf{r} + \int_{MJ} \mathbf{F} d \mathbf{r} \\ &= (F_r dr)_{JK} +(F_\theta d \theta)_{KL} - (F_r dr)_{LM} -(F_\theta d \theta)_{MJ} \\ &= - \frac{\partial F_r}{\partial \theta}dr d \theta + \frac{\partial (r F_\theta )}{\partial r}d r d \theta \end{aligned} \end{equation}\]
The area of the surface element is
\[r d \theta dr\]
.Hence
\[A_z =\frac{- \frac{\partial F_r}{\partial \theta}dr d \theta + \frac{\partial r F_\theta}{\partial r}d r d \theta}{r d \theta dr}= \frac{1}{r}(\frac{\partial (r F_\theta )}{\partial r} - \frac{\partial F_r}{\partial \theta})\]
Hence
\[\mathbf{\nabla} \times \mathbf{F} = (\frac{1}{r} \frac{\partial F_z}{\partial \theta} - \frac{\partial F_{\theta}}{\partial z}) \mathbf{e_r} + (\frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r}) \mathbf{e_\theta} + \frac{1}{r}(\frac{\partial (r F_\theta )}{\partial r} - \frac{\partial F_r}{\partial \theta}) \mathbf{e_z}\]