1. Show that under a coordinate transformation $(x_1, \dots, x_n) \to (y_1(\vec{x}), \dots, y_n(\vec{x}))$ in $\mathbb{R}^n$
the $n$-form
$dy^1 \wedge dy^2 \wedge \dots \wedge dy^n = J(\vec{y}, \vec{x})dx^1 \wedge dx^2 \wedge \dots \wedge dx^n$,
where $J$ is the Jacobian determinant
$J(\vec{y}, \vec{x}) = \det \left(\frac{\partial y^i}{\partial x^j}\right)$.
Hint: you may need the following determinant formula
$\det \left(\frac{\partial y^i}{\partial x^j}\right) = \epsilon_{j_1 \dots j_n} \frac{\partial y^1}{\partial x^{j_1}} \dots \frac{\partial y^n}{\partial x^{j_n}}$ (1)
which uses the completely antisymmetric $\epsilon$-symbol. This is the general determinant
formula
$\det M = \det(M_{ij}) = \epsilon_{j_1 \dots j_n} M_{1j_1} \dots M_{nj_n}$ (2)
