Let $M$ be an $n$ dimensional smooth manifold in $\mathbb{R}^N$, $n<N$. Let $V$ be $n$ dimensional Lebesgue volume measure. Let $f$ be a smooth injective map from $\mathbb{R}^N$ to $\mathbb{R}^N$ (note that it is $N$ not $n$) such that $f(M)$ is a transformed $n$ dimensional manifold in $\mathbb{R}^N$. Let $J_f(x)$ be the determinant of $(\frac{\partial f_i}{\partial x_j})_{1\leq i \leq N, 1 \leq j \leq N}$. Can we use the change of variable $z=f(x)$ to have the following integration on manifold: \begin{align} \int_{f(M)} g(z)dV(z) = \int_{M} |J_f(x)| g(f(x))dV(x), \end{align} for a smooth integrable function $g$?
In a simple case, suppose $M$ is a line in $\mathbb{R}^2$ connecting $(0,0)$ and $(1,1)$. Let $f(x_1,x_2)=(\alpha x_1, \beta x_2)^\prime$, i.e. $f$ rescales the x,y coordinate by $\alpha,\beta>0$ respectively. Then $J_f$ should be $\alpha\beta$. Since $f(M)$ is a line connecting $(0,0)$ and $(\alpha,\beta)$, we should have $\int_{f(M)} dV(z) = \sqrt{\alpha^2+\beta^2}$, while $ \int_{M} |J_f(x)| dV(x) = \sqrt{2}\alpha\beta$.
What should be the correct form of the above change of variables formula, using the ''Jacobian'' $J_f$?