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the first integration I want to understand why it looks like this:

enter image description here

And the first answer here Using the divergence theorem to prove that $\frac{1}{|B_R(0)|} \int_{B_R(0)} M \textbf{y} . \textbf{y} dy = \frac{R^2}{ n + 2} \text{trace}(M)$ suggested that the author was trying to show (inside the solution) the detailed proof of that the area of $n$-sphere of radius $R$ is $R^{n-1}$ times the area of the unit sphere, I do not understand where is this proof actually, could someone clarify this to me please?

Brain
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  • Your exponent is wrong. Remember that the ball is not its boundary sphere. – Ted Shifrin Aug 25 '22 at 03:47
  • @TedShifrin is the title better now? – Brain Aug 25 '22 at 04:07
  • Not completely sure what's being asked. Are you happy with the ideas that 1. The sphere in $(n-1)$-dimensional; 2. In Euclidean space, scaling a $k$-dimensional object by a factor $r$ multiplies its $k$-dimensional content by $r^k$? (E.g., tripling the scale of a curve multiplies the length by $3$; similarly scaling a surface multiplies the area by $9$; similarly scaling a solid multiplies the volume by $27$.) – Andrew D. Hwang Aug 25 '22 at 12:41
  • @AndrewD.Hwang I am asking about the detailed proof of that the area of $n$-sphere of radius $R$ is $R^{n-1}$ times the area of the unit sphere, I do not understand where is this proof actually in the link of the other question I posted and where specifically it is used in the 3 integrations I posted above in my current question? – Brain Aug 25 '22 at 13:51
  • In fact, it's rather proved that the volume of the $n-$ball of radius $R$ is $R^n$ time the radius of the unit $n-$sphere. – Surb Aug 25 '22 at 16:14
  • Does not the unit $n$ -sphere have radius 1? @Surb – Brain Aug 25 '22 at 16:22
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    Sorry it's a typo of course... the volume of a $n-$th ball of radius $R$ is $R^n$ times the volume of the unit $n-$ball. (of course unit sphere has radius 1). – Surb Aug 26 '22 at 19:43

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