The higher dimensional situation is more complicated than 1D because you have more room and so images of sets are more difficult to characterize (in 1D, the intermediate value theorem tells us the image of an interval is an interval). Another reason why the 1D theorem is so easy to state and prove is that if $a\leq b$, then we define $\int_b^a:=-\int_a^b$, so we have built in an order into our notation (the whole purpose being to make the equation $\int_a^b=\int_a^c+\int_c^b$ hold for all $a,b,c$, not just those with $a\leq c\leq b$). This notational trick (together with the chain rule and FTC) allows us to phrase the theorem for $\phi$’s which are not necessarily increasing (which agrees with the comment by @Didier). So, as nice, clean and general the statement/proof are in 1D, I would say it is a fluke.
But still analysts have developed various ways of finessing the change-of-variables formula
\begin{align}
\int_{\phi[\Omega]}f=\int_{\Omega}f\circ\phi\cdot|\det D\phi|.\tag{$*$}
\end{align}
- Usually, $(*)$ is proven when $f\in L^1$ and $\phi$ is a $C^1$ diffeomorphism.
- Using Sard’s theorem, one can improve (see here) the result slightly by still requiring $\phi$ to be $C^1$, but allowing $\phi$ to have some critical points, but then we still require that $\phi$ be injective away from the critical set.
- Still assuming $\phi$ is $C^1$, if you now want to drop injectivity of $\phi$ away from the critical set as well, then $(*)$ is no longer true (see the polar coordinate counterexample at the end of the link above). So, you need to take that non-injectivity into account explicitly by counting the cardinality of each preimage $\phi^{-1}(\{y\})$. See Michael E. Taylor’s Measure Theory and Integration, Appendix F for the precise statement and proof of such a version.
- Another direction for generalization is the regularity on $\phi$; one can weaken this to $\phi$ being a Lipschitz map. By Rademacher’s theorem, Lipschitz maps are a.e differentiable, and since integrals shouldn’t care about measure-zero sets, it would be plausible that one can extend $(*)$ to the case of Lipschitz $\phi$ as well. Another thing one might wonder is whether is is possible to deal with Lipschitz maps $\phi:\Bbb{R}^n\to\Bbb{R}^m$ with perhaps $n\neq m$. Indeed, there are such generalizations; see the Area and Coarea theorems in Evans and Gariepy’s Measure Theory and Fine Properties of Functions, Chapter 3 on Area and Coarea formulas.
Despite these generalizations, note that in practice (i.e everyday calculations of volumes, moments of inertia), the first ‘baby version’ (which is already a monster) suffices. The second bullet point is a nice reassurance (but if I’m being honest, I’ve never used the full strength of it… in practical cases one can easily reduce to the previous case even without Sard’s theorem). The third bullet point I’ve never actually had to use, and the final one, I’ve also never had the need for the full strength of the coarea formula with general Lipschitz maps. So my point is that while the generalizations are nice to know, you can probably get away with the basic version for like 95% of “practical stuff”.
