I'm following the proof of the local expression for the Laplacian on a compact manifold and I'm having problems understanding how the integral on a manifold translates into an integral in $R^n$, in particular in the following calculation:
\begin{align*} (\Delta f,\varphi)&=(d f, d \varphi)= \int_M \! df \wedge *d\varphi =\int_M \!
after the fourth equal symbol I think it should go $ \sum_{\alpha} \int_{U_{\alpha}} \! \rho_\alpha\sum_{i,j} g^{ij} \frac{\partial f}{\partial x^i}\frac{\partial \varphi }{\partial x^j} \sqrt{g} dx^1 ... dx^n $ where $\rho_\alpha$ are the local expression of a partition of unity, but then in the last step I would have $ - \sum_{\alpha,i,j}\int_{U_{\alpha}} \! \frac{1}{\sqrt{g}}\frac{\partial}{\partial x^j}\left( \rho_{\alpha}\sqrt{g} g^{ij} \frac{\partial f}{\partial x^i} \right) \varphi \sqrt{g} dx^1 ... dx^n$ and I don't know how to continue.