Defining line bundle of a properly embedded hypersurface

Local defining functions $f_{\alpha }$ for a properly embedded hypersurface $Y\subset X$ have nowhere-zero smooth ratios $f_{\alpha }/f_{\beta }$ on overlaps. These ratios define a real line bundle with a global transverse section whose local representatives are the $f_{\alpha }$ and whose zero set is $Y$. The bundle is naturally isomorphic to the normal bundle of $Y$.