In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$

Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $\mathbb{R}^{n+m}$. We write $$ (\xi,\eta)= G^{-1}(x,y) \ . $$

This is a local parameterization of $U$. We will think of $$ U_\eta: = G^{-1}(\mathbb{R}^n \times \{y\} \cap V) $$ as the "horizontal" fibers, which are $n$-dimensional $C^1$-manifolds. Similarly, we refer to $$ U_\xi = G^{-1}( \{x\} \times \mathbb{R}^m \cap V) $$ as the ``vertical" fibers, which are $m$-dimensional $C^1$-manifolds.

**Question:** Is there a version of the Fubini's theorem that equates the integral over $U$ to a double integral over fibers?

I am wondering if such a result exists in any textbook. In any other publication? It must be, because it is so natural to desire an integration over the original foliations.

This is a paraphrase of the question asked previously here.

I came up with the following:

Let$ DG_{|U_\xi} (\xi,\eta)$ be the derivative, at point $(\xi,\eta)$, of the map $G_{|U_\xi}$, i.e. the restriction of $G$ to $U_\xi$. This restriction is from an $m$-dimensional $C^1$-manifold to a subset of $\mathbb{R}^m$.} So its derivative and its Jacobian are defined.

Suppose for some fixed $\eta_0$, the union of vertical fibers along $U_{\eta_0}$ covers the set $U$.

**Lemma** Under the assumptions above, for any integrable function $f: U \to \mathbb{R} $,
$$ \int_U f = \int_{U_{\eta_0}}\left(\int_{U_\xi} f(\xi,\eta) \frac{|\det DG_{U_\xi} (\xi,\eta)| \cdot |\det DG_{U_{\eta_0}} (\xi,\eta_0)|}{|\det DG(\xi,\eta)|} \ d\mathcal{H}^m(\eta)\right) d\mathcal{H}^n(\xi) \cdot $$

That is saying that to integrate over $U$ simply integrate along the fibers and then sum over the "base", which is what Fubini's theorem says in the standard orthogonal coordinates of the Euclidean space. The correcting factor is can be memorized as $$ \frac{\text{ Jacobian along fibers} \times \text {Jacobian along base}}{\text{full Jacobian}} \ .$$

**Corollary:** Integration in spherical coordinates. There, the fibers are orthogonal to the base, and so, the full Jacobian also factors, cancelling the numerator. Therefore, the correcting factor disappears (=1), and we get the familiar
$$ \int_U f \ d \mathcal{L}^n = \int_0^\infty \left( \int_{\mathbb{S}^{n-1}(r)\cap U} f \ d \sigma \right) dr \ . $$

**Example:** Let $P$ be a parallelepiped in the plane, resulting from tilting a rectangle so that the acute angle between its edges $A$ and $B$ is $\theta$. Then, foliating $P$ by parallel copies of the edges $B$, indexed $B_x$, leads to
$$ \int_P f \ d \mathcal{L}^2 = \int_A \left( \int_{B_x} f \sin(\theta) \ d \mathcal{H}^1 \right) dx \ = \sin(\theta) \int_A \left( \int_{B_x} f \ d \mathcal{H}^1 \right) dx \ . $$

Again my questions are:

1) Is there some alternative formula known out there?

2) Are there references giving specifically this? (I am not interested in "it can follow"s and "it can be done"s!)