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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?

Lifetime Beginner
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  • Searching the site for "volume sphere ball" (or the like) should turn up numerous helpful questions and answers, such as https://math.stackexchange.com/questions/625 – Andrew D. Hwang Oct 13 '22 at 02:56
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    https://en.wikipedia.org/wiki/N-sphere – Svyatoslav Oct 13 '22 at 03:01
  • Thank you Svyatoslav. I saw this in the wiki article at location https://en.wikipedia.org/wiki/N-sphere#Recurrences. It has the following formula: $$V_{n + 1} = \int_0^1S_nr^ndr.$$ This seems to work if I add in $$\int d^{n + 1}\vec{r} = \int dV = V_{n + 1}.$$ – Lifetime Beginner Oct 13 '22 at 14:27

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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.

Lifetime Beginner
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