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- Jan 26, 2012

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**Problem**: Let $(M,g)$ be an oriented Riemannian manifold with the standard Euclidean metric, and let $\omega\in\mathcal{A}^k(M)$ be a $k$ form. We define the Laplace-Beltrami operator as a map $\Delta:\mathcal{A}^k(M)\rightarrow\mathcal{A}^k(M)$ defined by $\Delta = dd^{\ast}+d^{\ast}d$, where

\[d^{\ast}\omega = (-1)^{n(k+1)+1}\ast d\ast \omega\]

with $\ast:\bigwedge^k T^{\ast}M\rightarrow\bigwedge^{n-k} T^{\ast}M$ denoting the Hodge star operator. When $k=0$, show that $\Delta$ agrees with the typical Laplacian on real valued functions: $\displaystyle\Delta u = -\text{div}\,(\text{grad}\,u)= -\sum_{i=1}^n\frac{\partial^2u}{(\partial x^i)^2}$.

**Remark**: Note for $k=0$, $\mathcal{A}^0(M)=C^{\infty}(M)$, the set of continuous infinitely differentiable functions on the manifold $M$.

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