I was wondering if a change of coordinate (e.g cylindrical change) could affect the variational formulation with respect to the metric. I mean, does the metric $\mathbf{dx}$ which appears in the following form, for any domain $\Omega \subset \mathbb{R}^3$, with solution $u$ and test function $v \in \mathbb{R}$ $\int_\Omega u(x, y, z) v(x, y, z) \mathbf{dx} = 0$ will be changed if one performs a cylindrical change of coordinate ? I guess we'll need to consider the jacobian determinant of the diffeomorphism matching the change of coordinates. Moreover, I'd like to know how to rewrite the local shape functions $\phi$ such that, if we consider a triangular mesh, over the element $T_i$, $u_i(x, y, z) = \sum_{j}\phi_{ij}(x, y, z)u_{ij}$, $j$ the $j^{th}$ degree of freedom on $T_i$.
Thx in advance
