Let $g_1:\mathbb{R}^{d_1} \rightarrow \mathbb{R}^{d_2}$ and $g_2:\mathbb{R}^{d_2} \rightarrow \mathbb{R}^{d_3}$ be injective and differentiable with $d_1 < d_2 < d_3$. Let $g = g_2 \circ g_1$ and $A \in \mathbb{R}^{d_1}$. I am interested in computing the volume of $A$: \begin{equation} \displaystyle \int_A dx \end{equation} using the change of variable formula, and derived this in two different ways, arriving at two different results which I do not think match in general. I would appreciate any help pointing out what I am doing wrong.

First way: By the change of variable formula and then applying the chain rule, \begin{align} \displaystyle \int_A dx & = \int_{g(A)}|\det J_g^\top(g^{-1}(z))J_g(g^{-1}(z))|^{-1/2}dz\\ & = \int_{g(A)} |\det J_{g_1}^\top(g^{-1}(z))J_{g_2}^\top(g_2^{-1}(z))J_{g_2}(g_2^{-1}(z))J_{g_1}(g^{-1}(z))|^{-1/2}dz \end{align} where $J_g(x)$ denotes the Jacobian matrix of $g$ evaluated at $x$, and $g^{-1}$ is well defined over $g(A)$.

Second way: Again, by the change of variable formula, but this time applying it twice, \begin{align} \displaystyle \int_A dx & = \int_{g_1(A)}|\det J_{g_1}^\top(g_1^{-1}(y))J_{g_1}(g_1^{-1}(y))|^{-1/2}dy \\ & = \int_{g_2(g_1(A))} |\det J_{g_1}^\top(g_1^{-1}(g_2^{-1}(z)))J_{g_1}(g_1^{-1}(g_2^{-1}(z)))|^{-1/2} |\det J_{g_2}^\top(g_2^{-1}(z))J_{g_2}(g_2^{-1}(z))|^{-1/2}dz\\ & = \int_{g(A)} |\det J_{g_1}^\top(g^{-1}(z))J_{g_1}(g^{-1}(z))|^{-1/2} |\det J_{g_2}^\top(g_2^{-1}(z))J_{g_2}(g_2^{-1}(z))|^{-1/2}dz\\ \end{align}

Now, because this holds for any $A$, it would imply the integrands are equal almost surely and thus: \begin{equation} |\det J_1^\top J_2^\top J_2 J_1| = |\det J_1^\top J_1||\det J_2^\top J_2| \end{equation} where I simplified notation for clarity. I don't think the above equation is true for arbitrary $J_1 \in \mathbb{R}^{d_2 \times d_1}$ and $J_2 \in \mathbb{R}^{d_3 \times d_2}$ (or even if $J_1$ and $J_2$ are full rank as would be the case due to injectivity). Any help is very appreciated!


The problem is in the "second way", the second equal sign. The term $\text{det}J_2^TJ_2$ contains too much information, e.g. how $d_2$-dimensional volume is being deformed by map $g_2$ from ${\mathbb R}^{d_2}$ to $ {\mathbb R}^{d_3}$. But in reality we only need to check how the $d_1$-dimensional volume in the direction of $g_1(A)$, which is a $d_1$-dimensional submanifold of $ {\mathbb R}^{d_2}$, is being deformed by $g_2$.

$\text{det}J_2^TJ_2$ brings in information like, among other things, how the geometry in the direction perpendicular to $g_1(A)$, is being deformed by $g_2$; this information is totally irrelevant for our purpose.


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