Statistical Mechanics -- partition function, change to polar coords

[binbagsss]

Homework Statement

Hi I have the following definition for the partition function of ##N## particles in ##s## dimensions:

I am looking at computing the partition function for this Hamiltonian:

The solution is here:

Homework Equations

above

The Attempt at a Solution

I don't understand the top line of this solution? The definition is to integrate over ##\Pi_{i=1}^{i=n} \Pi_{\alpha=1}^{\alpha=2}dP_{i,a}## not ##\Pi_{i=1}^{i=2N} d\vec P_i##

I can change to polar coordinates to get the integration over ##d\vec P_i## (just looking at the integration over ##P## and I have the same result, however in the first line of the solution there are no factors of ##2\pi## from the change of variables to polar coordinates, so how has the definition of the partition function been used? (or is it just by chance I've got the same answer)i.e. ##dP_x dP_y \neq d^2\vec P## does it?:

 

[lippyka]

$$Z = \int\Pi_{i=1}^{i=N}\Pi_{\alpha=1}^{\alpha=2}dP_{i,a}e^{-\beta \sum_{i=1}^N \frac{P_i^2}{2m}}$$$$= \int\Pi_{i=1}^{i=2N} d\vec P_i e^{-\beta \sum_{i=1}^N \frac{\vec P_i^2}{2m}}$$