Volume of a Euclidean unit sphere ($\mathrm{vol}(S^{n-1})=n\mathrm{vol}(D^n)$)

The $(n-1)$-dimensional volume of the Euclidean unit sphere equals $n$ times the $n$-dimensional volume of its unit ball. If $R={\sum }_i{x}^i{\partial }_i$ is the radial vector field and $\omega =d{x}^1\wedge \cdots \wedge d{x}^n$, then $d({\iota }_R\omega )=n\omega$, and the Generalized Stokes theorem proves the formula.