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Consider the local representation of the Laplace-Beltrami operator on a 2-dimesnsional manifold (immersed in $\mathbf R^3$) with Riemannian metric $(g_{ij})$. Please, I want help in showing that:

\begin{equation} \Delta f = \frac{1}{\sqrt{\text{det}(g)}} \sum_{i,j } \frac{\partial}{\partial x_i}f (g^{ij} \sqrt{\text{det}(g)}\frac{\partial}{\partial x_j}f) \end{equation}

I have looked on the web and found this but the proof used terms I'm not familiar with. I have some background in vector analysis though. I hope someone would be kind in enough to take me through the proof.

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