Change of variables between spaces with different dimensionality in using the Riemann integral

Question

On wikipedia (https://en.wikipedia.org/wiki/Integration_by_substitution) I find that, \begin{equation} \int_{\phi(U)} f(v)dv = \int_U f(\phi(u))|\det(D\phi(u))|du \end{equation} for the Riemann integral under a number of conditions. I'm looking for a more general formulation, where $f:R^m\rightarrow R$, $M$ is a mapping $M:V \rightarrow U$, where $V\subset R^n$ is open and $U \subset R^m$ is some n dimensional surface, so that: \begin{equation} \int_{V} f(M(v))dv = \int_U f(u) \cdot \text{something} \cdot du \end{equation} Already in the same wikipedia article, conditions for doing this using Lebesgue integration is given, but I want to know when and how this can be done using the Riemann integral, because I'm writing a short text that should be easy to understand, so I don't want to make it too abstract. A literature reference would be appreciated.