$H^1(\Omega)$ in Euclidean space vs $H^1(\Sigma)$ on a compact surface

Question

For an open subset $\Omega \subset \mathbb R^n$ the Sobolev space $H^1(\Omega)=W^{1,2}(\Omega)$ is defined as \begin{equation} H^1(\Omega)=\{ u \in L^2(\Omega) \, \vert \, \partial^{\alpha}u \in L^2(\Omega) \} \end{equation} in distributional sense. I stumbled upon some literature on the construction of Sobolev spaces on manifolds that seem too technical for me to understand. Is there an "easy" derivation of the Sobolev space on a compact two-dimensional surface $\Sigma \subset \mathbb R^3$?