Suppose $M$ is a surface embedded in $\mathbb{R}^3$, then it has the natural induced Euclidean metric, denoted by $\textbf{g}$. Suppose $\tilde{\textbf{g}}$ is another Riemannian metric on $M$, we say it is conformal to $\textbf{g}$, if the two metrics differ by a scalar function $u\colon M \to \mathbb{R}$, nameley $\tilde{\textbf{g}} = e^{2u} \textbf{g}$.
A simple question. How to derive $e^{2u}$?
Thanks in advance.
If $\tilde g = \phi g$ and both $g, \tilde g$ are Riemannian metrics, then the function $\phi$ relating them must be positive, and thus can be represented as $\phi = e^{2u}$ for $u=\frac12 \ln \phi$. We simply choose to work with the conformal factor in this form to make some computations simpler.