Volume of n-simplex | n维单纯形的体积

An is an n-simplex which is supported by n vectors (a1,a2,…,an) of same origin.
Vn is the signed volume of An.

After we apply Gram–Schmidt orthogonalization to (a1,a2,…,an), we will get (a1,a2’,…,an’), the orthogonal basis of n-dimensional vector space.
Transpose of a1 =(h1,0,…,0)
Transpose of a2’=(0,h2,…,0)
…
Transpose of an’=(0,0,…,hn)

① V1=h1
② t = h2-h

$$V_{2}=\int_{0}^{h_{2}} V_{1}\left ( \frac{t}{h_{2}} \right )\,{\rm d}t$$
$$V_{3}=\int_{0}^{h_{3}}V_{2}\left ( \frac{t}{h_{3}} \right )^{2}\,{\rm d}t$$
$$...$$

Vn=∫hn0Vn−1(thn)