Suppose we have a curvilinear coordinate system in $\mathbb{R}^3$, $(q_1, q_2, q_3)$, with $x_i = f_i(q_1, q_2, q_3)$. We define the local basis for this coordinate system by $\hat{e_i} = \frac{\partial\vec r}{\partial q_i}$, and the scale factors in this coordinate system as $h_i = \|\hat{e_i}\|...

I was asked to calculate the curl of the vector field $\theta^2\bf e_\theta$ in spherical coordinates. If we use the typical formula for curl in curvilinear coordinates: $$ \nabla\times \bf F = \left|\begin{matrix}h_1\hat{\bf e_1} & h_2\hat{\bf e_2} & h_3\hat{\bf e_3} \\ \displaystyle\frac{\parti...