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
$\int_{S(0,r)}f(x)d\sigma(x)=\int_{S(0,1)}f(y)r^{n-1}d\sigma(y)$
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.