Let vectors $u_1,\ u_2,\ \cdots u_n$ and $\mathcal{P}$ the parallelepiped determined by these vectors : $$\mathcal{P}=\left\{\lambda_1 u_1+\cdots +\lambda_nu_n/\ \lambda_k\in [0,1]\right\}$$ How we can prove that : $$V=\int_{\mathcal{P}}dx_1\cdots dx_n=\det\left(u_1, u_2, \cdots, u_n\right)$$ Thanks you very much.
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