Proper notation for integration on manifolds?

Question

If I have an integral over a manifold $\mathcal{M}\subset\mathbb{R}^n$, and I have an invertible continuously differentiable map $\varphi:\mathcal{M}\to\mathcal{D}\subseteq \mathbb{R}^d$, and $\mathcal{M} = \varphi^{-1}(\mathcal{D})$, what is the proper way to write an integral on $\mathcal{M}$ w.r.t. $\varphi^{-1}$?

One way I could write it is I guess in the following manner:

$$\int_{\mathcal{M}}f\,d\mu = \int_{\varphi^{-1}(\mathcal{D})}f d\varphi^{-1} = \int_{\mathcal{D}}f(\varphi^{-1}(x))\sqrt{\det[ (J_{\varphi^{-1}}(x))^TJ_{\varphi^{-1}}(x)]}\,dx.$$

This matches the notation typically used in u-substitution, in the sense that the second integral is still over the domain $\mathcal{M}$ instead of $\mathcal{D}$. Here $J_{{\varphi}^{-1}}(x)\in\mathbb{R}^{n\times d}$ is the Jacobian matrix of $\varphi^{-1}$.

On the other hand I could also write the following: $$\int_{\mathcal{M}}f\,d\mu = \int_{\mathcal{D}}f\circ\varphi^{-1}d\mu_{\varphi^{-1}} =\int_{\mathcal{D}}f(\varphi^{-1}(x))d\varphi^{-1}(x) =\int_{\mathcal{D}}f(\varphi^{-1}(x))\sqrt{\det[ (J_{\varphi^{-1}}(x))^TJ_{\varphi^{-1}}(x)]}\,dx.$$

Here the second integral uses the Lebesgue-Stieltjes measure, and the third integral is written as a Riemann-Stieltjes integral.

Are both variants correct or am I misunderstanding something? What bothers me is the discrepancy between $\int_{\varphi^{-1}(\mathcal{D})}f d\varphi^{-1}$ and $\int_{\mathcal{D}}f(\varphi^{-1}(x))d\varphi^{-1}(x)$. Is the former just abuse of notation? Or is it alright?

Edit:

I dug some more details on Riemann-Stieltjes for multivariable functions. These notes by Anevski are relevant. Also in the book Essential Mathematics for Applied Fields Meyer defines a Riemann-Stieltjes integral in the case where $f:\mathbb{R}^m\to\mathbb{R}$ and $G:\mathbb{R}^m\to\mathbb{R}$:

$$I = \int_{\mathcal{D}}f(x_1,\ldots,x_m)dG(x_1,\ldots,x_m), \quad \mathcal{D} = [a_1,b_1)\times[a_2,b_2)\times\ldots\times [a_m,b_m).$$

There he requires that $G$ is a $m$-CDF. This is formalized in the book, but as a special if we have a probability density function $g$ then we get the $m$-CDF:

$$G(x_1,\ldots,x_m) = \int_{[-\infty,x_1)\times \ldots \times [-\infty,x_m)}g(t_1,\ldots,t_m)\,dt_m\ldots dt_1.$$

Now if I have an invertible and continuously differentiable map $\psi:\mathcal{M}\to\mathcal{D}$, and $\psi^{-1}:\mathcal{D}\to\mathcal{M}$, with absolute value of the Jacobian determinant equal to $g$ up to a consant: $$|\det[J_{\psi^{-1}}(x_1,\ldots,x_m]| \propto g(x_1,\ldots,x_m),$$ then I believe the equivalent notation for differential forms is what Ted Shifrin mentioned: $$(\psi^{-1})_{*}dV_{\mathbb{R}^m} = \psi^*dV_{\mathbb{R}^m} = |\det[J_{\psi^{-1}}(\vec{x})]|dx_1\ldots dx_m = g(\vec{x})dx_1\ldots dx_m = dG(\vec{x}).$$

It is a bit unfortunate that the notation $dG$ doesn't seem to agree with the notation for the exterior derivative, since whenever $G$ is continuously differentiable, the symbol $dG(\vec{x})$ in the Riemann-Stieltjes integral, really just expands to the following: $$dG(\vec{x}) = \left|\frac{\partial^mG}{\partial x_1\ldots \partial x_m}(\vec{x})\right|dx_1\ldots dx_m.$$ Instead of what one would expect from the exterior derivative: $dG = \sum_i \partial_iG dx_i$.

But this does clarify that $d\psi^{-1}(\vec{x})$ is abuse of notation when used as the symbol in the Riemann-Stieltjes integral (at least in higher dimensions than one). What $d\psi^{-1}(\vec{x})$ was supposed to mean in the Riemann-Stieltjes sense was $dG(\vec{x})$ where $g(\vec{x}) \propto |\det[J_{\psi^{-1}}]|$. However the notation using the pushforward of the Lebesgue measure $(\psi^{-1})_{*}\lambda = \psi^*\lambda$ should be proper I believe.

When we have a parametrization $\varphi^{-1}$ of an embedded manifold with a dimensionality lower than the ambient space, e.g.\ $\mathcal{M}$ is an $m$-manifold embedded in $\mathbb{R}^n$ where $m<n$ then the $|\det J_{\psi^{-1}}|$ becomes the square root of the Gramian formed by $\langle\partial_i \varphi^{-1}, \partial_j \varphi^{-1}\rangle$ (those agree for square matrices). There I guess the proper way to write this is to again just stick to either $\varphi^*dV$ or $d\varphi^*\lambda$.

Answer 1

I think I figured out where my confusion stemmed from. I found this writeup on differential forms by Tao useful on the distinction between the different approaches which clarified some things for me. I also found this answer.

The abuse of notation $d\varphi^{-1}(\vec{x})$ that I am used to seems to stem from how one uses $d\vec{x}$ to denote $d\lambda(\vec{x})$ for the $m$-dimensional Lebesgue measure $\lambda$. The measure in question for my case is the measure $\mu$ which I used for my initial integral. I also had the bijective map $\varphi:\mathcal{M}\to\mathcal{D}$ which allowed me to push forward the measure $\mu$ to $\mathcal{D}$ as $\varphi_{*}\mu = \mu \circ \varphi^{-1}$. So $d\varphi^{-1}$ in the above really just stands for $d\varphi_{*}\mu$ as one would expect. That is, the proper way to write what I did is: $$\int_{\mathcal{M}}f\,d\mu = \int_{\mathcal{D}}f \circ h^{-1} \,d\varphi_{*}\mu = \int_{\mathcal{D}}f(h^{-1}(\vec{x}))\,d\mu(\varphi^{-1}(x)).$$

However the measure $\mu$ can be computed w.r.t. a chart as (at least locally as noted in the linked answer): $$\mu_{\varphi}(A) = \int_{\varphi(A)} \sqrt{\det G_{\varphi^{-1}}}\,d\lambda,$$ where $G_{\varphi^{-1}} = J_{\varphi^{-1}}^TJ_{\varphi^{-1}}$ is the Gramian formed by the product of the Jacobian transposed with the Jacobian (here it's a special case because I consider $\mathbb{R}^n$). But then I have that: $$d\mu_{\varphi}(\varphi^{-1}(\vec{x})) = \sqrt{\det G_{\varphi^{-1}}}d\lambda(\vec{x}).$$

The corresponding Lebesgue-Stieltjes measure ought to be $\mu_{\varphi} \circ \varphi^{-1}$ (at least as far as I am aware the Lebesgue-Stieltjes measure should be from $\mathcal{D}$ to $[0,\infty]$ and not from $\mathcal{M}$ to $[0,\infty]$, since we are trying to transfer the Lebesgue measure $\lambda$ from $\mathcal{D}$ to $\mathcal{M}$ using the map $\varphi^{-1}$). This is actually the function that would show up after the $d$ in a multivariable Riemann-Stieltjes integral, i.e. $d(\mu \circ \varphi^{-1})(\vec{x})$ so it's nice and consistent with the definitions of the Riemann-Stieltjes and Lebesgue-Stieltjes integrals.

The moral of the story is that I should keep my measures explicit in the differential notation $d\mu(\vec{x})$ and $d\mu(\varphi^{-1}(\vec{x}))$ instead of writing $d\vec{x}$ and $d\varphi^{-1}(\vec{x})$, otherwise it may read as nonsense, and would confuse me also. Then there's also no way that $d\mu(\varphi^{-1}(\vec{x}))$ would be misinterpreted as $d\varphi^{-1} = \sum_i \partial_i \varphi^{-1} dx_i$.