Does anyone know of a modern proof of the First Minkowski Formula for a compact embedded hypersurface $\psi \colon \mathcal{M}^n \hookrightarrow \mathbb{R}^{n+1}$ ? The integral formula is 
$$ \int\_{\mathcal{M}} H \langle \psi , \nu \rangle \mathrm{d}A +A = 0$$
where $A$ is the area of $\mathcal{M}$, $H$ is the mean curvature of $\mathcal{M}$ and $\nu$ is the outward normal field on $\mathcal{M}$.


Many thanks for any suggestions.



First Minkowski Formula
Consider the following differential $(n-1)$-form $\omega$ on $M$: for $p\in M$ and $v\_1,\dots,v\_{n-1}\in T\_pM$, define
$$
 \omega(v\_1,\dots,v\_{n-1}) = [ \psi(p), \nu(p), d\psi(v\_1),\dots,d\psi(v\_{n-1})]
$$
where the square brackets denote the standard volume form in $\mathbb R^{n+1}$ (in other words, the determinant).


A simple computation shows that
$$
 d\omega = n(1+H\langle\psi,\nu\rangle)dA .
$$
Now the formula follows from the fact that $\int\_M d\omega=0$ by Stokes.


This works for any immersed orientable hypersurface. For a non-orientable one, consider the oriented double cover. Note that in the embedded case one may assume that $M$ is a submanifold and $\psi$ is the inclusion, and then the formula for $\omega$ gets simpler: just remove all $\psi$ and $d\psi$.


Unfortunately I don't remember where I have read this proof, it was too long ago. (And I think that that text was only for $n=2$ anyway.)


