I am a physics student with a minimal background in differential geometry and I am trying to determine an area element on an arbitrary surface. Suppose we have a surface parameterized by a function $z=F(\theta,\phi)$. I wish to determine the area elements of a Gaussian pillbox outside of our surface whose top and bottom surfaces "match" the curvature of the surface. Suppose the location of the bottom of the surface is $\mathbf{r}_1 = r \hat{\mathbf{n}}$ and the location of the top of the surface is $\mathbf{r}_2 = r \hat{\mathbf{n}} + \epsilon \hat{\mathbf{n}}$. How can I quantify the area element on each of these surfaces in terms of the principal curvature $\kappa_1, \kappa_2$ of the surface itself?
Is it true that the radii of curvature satisfy the relations $R_1 = \frac{1}{\kappa_1}$, $R_2 = \frac{1}{\kappa_2}$? where $\kappa_1$ and $\kappa_2$ are the principal curvatures? If this is true, then is the area element simply $da = R_1d\theta R_2 d \phi$? I eventually wish to prove something about the electric field of a charged surface in terms of the divergence of the normal vector of the surface, which I know satisfies the relation $\nabla \cdot \hat{\mathbf{n}} = \frac{1}{2}\left(\kappa_1 + \kappa_2 \right)$. I am open to advanced responses in terms of exterior algebras, I am familiar with abstract differential geometry but have little practice applying it to physical situations. Thanks in advance.