Take the 2-minute tour ×
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.

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|improve this question

2 Answers 2

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|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|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. –  Vivek Aug 22 '12 at 22:56

Your Answer


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

Not the answer you're looking for? Browse other questions tagged or ask your own question.