What is the general transformation law of the components of the metric tensor $g_{\alpha\beta}$ on a parametric surface $x(u,v) = (x_1(u,v), x_2(u,v), x_3(u,v))$ under an active transformation $x_{new}(u,v) =(f_1(x_{1},x_{2},x_{3}), f_2(x_{1},x_{2},x_{3}), f_3(x_{1},x_{2},x_{3}))$? The functions $f_1, f_2, f_3$ might be different. Note that we are not changing the curvilinear coordinates--we are changing the parametric surface itself.
Denoting $x'_{i} = f_{i}(\textbf{x})$ and using the chain rule, we can find the components of the new metric tensor using the equation:
\begin{equation} g'_{\alpha\beta} = \sum_{i=1}^{3} \frac{\partial x'_{i}}{\partial u^{\alpha}} \frac{\partial x'_{i}}{\partial u^{\beta}}. \end{equation}
Using the chain rule and some algebra, it's easy to rewrite the equation above in terms of the Jacobian matrix $\textbf{J}$ with components $J_{ij} = \frac{\partial f_{i}}{\partial x_{j}}$: \begin{equation} g'_{ij} = \sum_{k=1}^{3} J_{kl} J_{km} \left(\textbf{e}_{i}\right)^{l} \left(\textbf{e}_{j}\right)^{m} \end{equation} where we use the summation convention over indices $l$ and $m$.
It seems as if the transformation of the components of the metric cannot be written as a matrix equation. However, from GR, we can treat an active transformation as a passive transformation as well. Can this problem be solved and can one get a nice transformation law? I do not know much differential geometry so I am struggling to find an answer.
Would greatly appreciate anyone's help!