How one can proof that the area of the an n-sphere is $r^{n-1} \omega_n$?

Asked Mar 28 '17 at 14:43

Active Mar 28 '17 at 14:43

Viewed 57 times

Let $\omega_n$ be the area of the unit sphere $S^{n-1} = \left \{ y \in \mathbb R^n: \|y\| = 1 \right \}$.

If $B_r(x) = \left \{ y \in \mathbb R^n: \|y - x\| \leq r \right \}$ and $S_r(x) = \partial B_r(x) = \left \{ y \in \mathbb R^n: \|y-x\| = 1 \right \}.$

How one can proof that the area of $S_r(x)$ is $A(r) = r^{n-1}\omega_n$ and the volume of $B_r(x)$ is $V = \frac{r^n}{n} \omega_n$?

Since the $V = \int_{0}^r A(\gamma)\, d\gamma $, it's sufficient showing that $A(r) = r^{n-1}\omega_n$ .

Help?

asked Mar 28 '17 at 14:43

user 242964

1,898

2

To see that volume of an $n$-ball scales as claimed, apply the change of variables theorem to the mapping $x \mapsto rx$. For the relationship between the $n$-volume of a ball and the $(n-1)$-volume of a sphere, see the many answers to Why is the derivative of a circle's area its perimeter (and similarly for spheres)? – Andrew D. Hwang Mar 28 '17 at 15:30

How one can proof that the area of the an n-sphere is $r^{n-1} \omega_n$?

0 Answers0