Volumen in Riemannian manifold, coordinates change

Question

I reading the following thread

Is orientability needed to define volumes on riemannian manifolds?

But I don't understand the following steps

$$\int_{\boldsymbol x^{-1}(R)}\sqrt{\det H(\boldsymbol y^{-1}\circ \boldsymbol x(p))}|J|\,dx_1\dots dx_n=\int_{\boldsymbol y^{-1}(R)}\sqrt{\det H(p)}\,dy_1\dots dy_n,$$

How can I do that $|J|$ don't appear in the right hand of the equation.

Thanks.

Answer 1

It is the classical integration by substitution $$ \int_{U} f\circ \varphi |\det (d\varphi)|dx_1\cdots dx_n = \int_{\varphi(U)} f dy_1\cdots dy_n $$ when $\varphi \colon U\to \varphi(U)$ is a diffeomorphism and $f\colon U \to \Bbb R$ is integrable. Here $f = \sqrt{\det H}$, $\varphi =y^{-1}\circ x$, and $U=x^{-1}(R)$. Notice that $J= \det \left(d(y^{-1}\circ x)\right)$.