Volume of a Euclidean unit sphere
= Volume of a Euclidean unit sphere
{title2=$\operatorname{vol}(S^{n-1})=n\operatorname{vol}(D^n)$}
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 $\omega=dx^1\wedge\cdots\wedge dx^n$, then $d(\iota_R\omega)=n\omega$, and the <Generalized Stokes theorem> proves the formula.