Show that under a coordinate transformation $left(x_{1}, ldots, x_{n} ight) ightarrowleft(y_{1}(vec{x}), ldots, y_{n}(vec{x}) ight)$ in $mathbb{R}^{n}$,

Show that under a coordinate transformation $\left(x_{1}, \ldots, x_{n}\right) \rightarrow\left(y_{1}(\vec{x}), \ldots, y_{n}(\vec{x})\right)$ in $\mathbb{R}^{n}$, the $n$-form transforms as

$$d y^{1} \wedge d y^{2} \wedge \cdots \wedge d y^{n}=J(\vec{y}, \vec{x}) d x^{1} \wedge d x^{2} \wedge \cdots \wedge d x^{n}$$

where $J$ is the Jacobian determinant

$$J(\vec{y}, \vec{x})=\operatorname{det}\left(\frac{\partial y^{i}}{\partial x^{j}}\right)$$

Use the definition of the determinant of an $n \times n$ matrix.