Transform probability density from low to high dimension with Jacobian

If $f$ is differentiable as you assume, then its image $f(\mathbb R^n)$ will be supported on an (at most $n$-dimensional) submanifold of $\mathbb R^m$. Since the ($m$-dimensional) Lebesgue measure of this submanifold will equal zero, $Y$ will never possess a probability density (with respect to the Lebesgue measure).

As far as I know, there is no general method to address this problem, since probability measures (or random variables) supported on submanifolds of $\mathbb R^m$ are complicated for exactly the above reason. In particular cases, solutions can be found which describe the distribution of $\vec Y$ in other ways than through a probability density.

Update: In your example, I do not see what $f$, $\vec Y$, $\rho_{\vec X}$ and $\rho_{\vec Y}$ are supposed to be.