I have an exercise and I have no idea how to do it:
Let be $U$ and $V$ open sets of $\mathbb{R}^{n}$ and $f:U\rightarrow V$ an orientation-preserving diffeomorphism, then
$$f^{*}(\operatorname{vol}_{V}) = \sqrt{\det(g_{ij}(x))} \operatorname{vol}_{U}$$
Where $\operatorname{vol}_{V}$ and $\operatorname{vol}_{U}$ are the volume forms of $U$ and $V$ $f^{*}$ the pullback of $f$, and $g_{ij}(x)=D_xf(e_i)\cdot D_xf(e_j)$
