I have a very simple question, but I could not find the answer, so I have to ask this here: Given is a multidimensional pdf $f(x_1, ..., x_n)$. $x_1, ..., x_n$ are Carthesian coordinates. We want to evaluate the integral
$\int_\mathbb{R^n} f(x_1, ..., x_n) \cdot \log(f(x_1, ..., x_n)) \; \mathrm{d}(x_1, ..., x_n) $
I want to transform this integral into spherical coordinates $(r, \phi_1, ..., \phi_{n-1})$. Lets assume the the determinant of the Jacobian is known and given as $|J|$.
How does the transformed integral look like? I think I is just
$\int_\mathbb{R^n} f(r(\vec x), ..., \phi_{n-1}(\vec x)) \cdot \log\left( f(r(\vec x), ..., \phi_{n-1}(\vec x)) \right) \; |J| \mathrm{d}(r, \phi_1, ..., \phi_{n-1})$
According to a paper I have here, this can be transformed into
$\int_\mathbb{R} \frac{f(r, ..., \phi_{n-1})}{|J|} \cdot \log\left( \frac{f(r, ..., \phi_{n-1})}{|J|} \right) |J| \mathrm{d}(r, \phi_1, ..., \phi_{n-1}) =\\ \int_\mathbb{R} f(r, ..., \phi_{n-1}) \cdot \log\left( \frac{f(r, ..., \phi_{n-1})}{|J|} \right) \mathrm{d}(r, \phi_1, ..., \phi_{n-1})$
But I just don't see this transformation? Could somebody explain the steps to me, where the denominator with the Jacobian determinant comes from? Note: the result should be valid for all pdfs, so it has to be a general derivation.