n-dimensional differential

Question

I'm trying to show that $$d^n\vec{r} = \text{Vol}(S^{n - 1})r^{n - 1}dr$$ where $r = (x_1, \cdots, x_n)$. I started with $n = 2$ where $$d^2\vec{r} = dx_1dx_2 = 2\pi rdr.$$ Then I worked the $n = 3$ case by integrating the latitudinal circles on a sphere $$d^3r = dx_1dx_2dx_3 = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}dx_3(d^2\vec{r})r\cos(x_3) = 4\pi r^2dr$$ which works. However, I can't find an inductive step that would get me the general formula. Probably because my n-dimensional geometry intuition is lacking. Can someone help me out?

Answer 1

Thanks once more to Svyatoslav. Here is the complete answer based on the wiki page that was suggested. $$\int_{r = 0}^{r = 1}d^n\vec{r} = \int_{B^n}dV = \text{Vol}(B^n) = \int_0^1 \text{Vol}(S^{n - 1})r^{n - 1}dr.$$ Thus $$d^n\vec{r} = \text{Vol}(S^{n - 1})r^{n - 1}dr,$$ as was required.