Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

While I am aware that when integrating over a ball in $\mathbb{R}^n$, we have

$\int_{B(0,R)}f(x)dx=\int_{S^{n-1}}\int_0^Rf(\gamma r)r^{n-1}drd\sigma(\gamma)$

I cannot figure why it is true that


I know this is very trivial but I keep thinking that on changing the variables, the Jacobian should be $r^n$ and not $r^{n-1}$, and that is clearly wrong.

Would be grateful for the help. I only require a vector calculus explanation.

share|cite|improve this question
up vote 1 down vote accepted

It is the surface integral, not the volume integral. Surface area is proportional to $r^{n - 1}$. If you want to use Jacobian determinant, I guess you might need to be able to formulate what $d\sigma$ is first. Iterated spherical coordinates (I don't know what it's actually called) should work.

share|cite|improve this answer

In $\mathbb R^3$, the area of a sphere is $4 \pi r^2$, so when you change scale from $R$ to $1$ you need to multiply by a factor $R^2$.

share|cite|improve this answer
But why a factor of $R^2$. It must be the Jacobian of some transformation. I just want to know what that transformation is. – user38404 Aug 22 '12 at 22:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.