Given a continuous multivariate random variable $\bf X$ with pdf $f_\mathbf X$, and an invertible transformation $g:\mathbb{R}^n \to \mathbb{R}^n$, it is known that the joint pdf of $\mathbf{Y}=g(\mathbf{X})$ is given by
$$ f_\mathbf Y(\mathbf y)=f_\mathbf X(g^{-1}(\mathbf y))\Bigl|J_g(\mathbf y)\Bigr|^{-1}, $$
where $J$ is the Jacobian determinant of the transformation.
But I am curious about the case when $g$ is not invertible.
In particular, when $ g:\mathbb{R}^m \to \mathbb{R}^n, m>n$, is there a similar relation? If yes, what are the differences?
I came across an alternative approach using differential algebra instead of Jacobians, which seems to be able to accomplish this goal. However, I encountered difficulty when trying to express the $dy_i$'s in terms of $dx_i$'s.
Any help is much appreciated.
