Volume formula

Suppose \(q:\mathbb{R}^{d} \rightarrow \mathbb{R}\) is a positive definite quadratic form. For any \(R>0\), a set \[\begin{align} B_{q}(R)=\{ x \in \mathbb{R}^{d} \ | \ q(x) \le R^{2}\} \subseteq \mathbb{R}^{d} \end{align}\] is called a closed ellipsoid of radius \(R\) with respect to \(q\). The volume of such an ellipsoid is \[\begin{align} \mathop{\mathrm{vol}}(B_{q}(R)) = \frac{ \pi^{d/2} R^{d}}{\Gamma(\frac{d}{2}+1) \sqrt{\det[ q(e_{i},e_{j})])}} \end{align}\]

In terms of factorials, \[\begin{align} \mathop{\mathrm{vol}}(B_{q}(R)) = \begin{cases} \frac{ \pi^{d/2} }{ (d/2)! \sqrt{\det[ q(e_{i},e_{j})])}} R^{d}& d \text{ even} \\ \frac{ 2 (4 \pi)^{\frac{d-1}{2}}\left( \frac{d-1}{2} \right)! }{{d}! \sqrt{\det[ q(e_{i},e_{j})])}} R^{d}& d \text{ odd}\\ \end{cases} \end{align}\]

Surface area

To get the surface area (or rather surface hypervolume), one can simply differentiate the above formula with respect to \(R\).

\[\begin{align} \mathop{\mathrm{surf}}(B_{q}(R)) = d \cdot {\mathop{\mathrm{vol}}(B_{q}(1)) } \cdot {R}^{d-1}. \end{align}\]