I'm trying to compute the Laplace-Beltrami Operator in local coordinates of some Riemannian manifold $M$. By definition, Laplace-Beltrami Operator
$$\Delta=d\delta+\delta d$$ acts in p-form on $M$, where is defined by $$\delta=(-1)^{n(p+1)+1}\star d\star$$ and $\star$ is the Hodge operator.
So, I started with the $M=\mathbb{R}^n$ and by direct calculation is simple to verifie that, in coordinates $(x_1,\dots,x_n)$ if $\omega=fdx_{i_1}\wedge\dots\wedge dx_{i_p}$ is a p-form on $\mathbb{R}^n$ then
$$\Delta\omega=-\sum_{s=1}^n \frac{\partial^2f}{\partial x_s ^2}dx_{i_1}\wedge\dots\wedge dx_{i_p}.$$
The question is if I have a local coordinates of $\mathbb{S}^n$, for example, $F:\mathbb{S}^n \setminus{(0,\dots,0,1)}\subset\mathbb{R}^{n+1}\to \mathbb{R}^n$ defined by $$F(x_1,\dots,x_{n+1})=\frac{1}{x_{n+1}-1}(x_1,\dots,x_n)$$ how can I write the Laplace-Beltrami operator of the p-form $\omega=fdx_{i_1}\wedge\dots\wedge dx_{i_p}$?
Any idea? Thanks so much.