the following is drawn from a rather rough set of lecture notes and I am not sure I understand it.
the goal is to determine for which values of $p$ we have $ \int_{|x|\leq 1} \frac{1}{|x|^p} \,dx < \infty \qquad $
where $d\mathbf{x} = dx_1 \cdots dx_n$ denotes Lebesgue measure.
according to the notes, the first step is to the change variables to $\mathbf{x} = r\mathbf{w}$ where $r > 0$ and $\mathbf{w}$ lies on the unit sphere $\mathbb{S}^{n-1}$. we then have that
$d\mathbf{x} = r^{n-1}drd\mathbf{w}$
and this is where I get stuck - how do I compute this? I tried to look up spherical coordinates, but the formulas seem different. and if I try out an example, say in $\mathbb{R}^2$ then I have \begin{align*} d\mathbf{x} &= dx_1dx_2 \\ &= (\frac{\partial x_1}{\partial r} dr + \frac{\partial x_1}{\partial w_1} dw_1 + \frac{\partial x_1}{\partial w_2} dw_2) (\frac{\partial x_2}{\partial r} dr + \frac{\partial x_2}{\partial w_1} dw_1 + \frac{\partial x_2}{\partial w_2} dw_2) \\ &= (w_1 dr + r dw_1 )(w_2 dr + r dw_2) \\ &= w_1rdrdw_2 + rw_2dw_1dr + r^2dw_1dw_2 \\ &= (w_2 dw_1 - w_1dw_2)rdr + r^2d\mathbf{w} \end{align*} which does look different. Many thanks for help and hints on how to understand the above formula!