Let $h,g:[a,b]\to \mathbb{R}$ be differential functions with $h(s) > 0$ and $\alpha : [a, b] \to \mathbb{R}^3$ a curve parameterized by arc length, $$\alpha(s) = \left(0, h(s), g(s)\right). $$ Consider the parametrized surface $$x(s, \theta) = \left(h(s)\sin \theta, h(s)\cos \theta, g(s)\right).$$ I want to calculate the Gaussian Curvature of this surface using the moving frame method.
My purse is use the equation $d\omega_{12} = - K \omega_1 \wedge\omega_2$, where $\omega_{12}$ is one of the connection forms and $\omega_1, \omega_2$ are the coframe.
First, I took the filds $e_1:=\frac{\partial x}{\partial s}$ and $e_2:= \frac{1}{h}\frac{\partial x}{\partial \theta}$ to be the frame. Now, how can I find out the coframe $\omega_1$ and $\omega_2$ associated?