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I dont understand the following surface measure,

My space is

$\Omega_E=\{w\in \Omega | H(w)=E\}$

My lecture notes state that my surface measure is given as


Can anyone enlighten me please?

Edit: The Hamiltonaian is defined as

$H(w)=\sum w_i$


$w\in \Omega=\mathbb{R}^{+N}$

N is the particle number.

I still havent figured it out how to arrive at the given surface measure . : )

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A bit of context is necessary, just by the definition of $\Omega_E$ it is impossible to arrive at your second formula. I suppose $\Omega$ is a phase space on which some Hamiltonian $H$ is defined and that one fixes a subset by choosing a fixed energy level $E$. This set apparantly has phase space volume $E^N/N!$, therefore the surface will have an area proportional to the derivative of that. – Raskolnikov Oct 6 '11 at 10:43

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