Some question about this proof about Riemannian volume form In these lecture notes lemma 2.3. is given as
$\omega_g = \sqrt{\det g_{ij}} \, dx^1 \wedge \cdots \wedge dx^n$ is independent of the choice of coordinate charts.
I am trying to understand the proof. The proof is this:

Could it be that there is a typo? Concretely, is $J = d\varphi$? If not, what is $J$?

Yes, $J$ here is definitely meant to be $d\varphi$.
In particular, the equality
$$
   dy^1 \wedge \cdots \wedge dy^n = \det(d\varphi) dx^1 \wedge \cdots \wedge dx^n
$$
holds, as you can check just by substituting
$$
  dy^i = \sum_j \frac{\partial y^i}{\partial x^j} dx^j
$$
and noting that the matrix $d\varphi$ has $ij$-th entry $\partial y^i / \partial x^j$.
(I'll also remark that it's important in the proof you've referenced that the chart $y^1, \dots, y^n$ have the same orientation as $x^1, \dots, x^n$, so that $\det(d\varphi)$ is positive.)
