I want to solve a specific integral, by using substitution. As it is too specific to describe my situation and probably also not of general interest, let me give a toy example.

Let $\overline{\Omega} \subseteq \mathbb{R}^3$ and $\Omega \subseteq \mathbb{R}^2$ be two domains of different dimension. Note that the intrinsic dimension of $\overline{\Omega}$ is two, although the ambient space is three dimensional. Furthermore, I know a bijective map $\varphi: \overline{\Omega} \rightarrow \Omega$. I want to solve the following integral as follows:

$$\int_{x\in \Omega} f(x)dx = \int_{x\in \varphi(\overline{\Omega})} f(x)dx = \int_{y\in \overline{\Omega}} f(\varphi(y)) \ ? ? ? \ dy.$$

Where I have written the three question marks, there should be a dependence on $\varphi$. According to Wikipedia, if $\varphi$ would be a function from $\mathbb{R}^n$ to $\mathbb{R}^n$, I should take the absolute value of the determinant of the Jacobian. (https://en.wikipedia.org/wiki/Integration_by_substitution)

But $\varphi'$ is not a square matrix. So something else needs to be done. I feel there should be a general theorem that one can look up if one knows integrals better.

Question 1: What should be at the three question marks?

Question 2: Can someone give me a citeable source?

many thanks Till

I want to solve a specific integral, by using substitution. As it is too specific to describe my situation and probably also not of general interest, let me give a toy example.

Let $\overline{\Omega} \subseteq \mathbb{R}^3$ and $\Omega \subseteq \mathbb{R}^2$ be two domains of different dimension. Note that the intrinsic dimension of $\overline{\Omega}$ is two, although the ambient space is three dimensional. Furthermore, I know a bijective map $\varphi: \overline{\Omega} \rightarrow \Omega$. We may assume that all partial derivatives of $\varphi$ exist. I want to solve the following integral as follows:

$$\int_{x\in \Omega} f(x)dx = \int_{x\in \varphi(\overline{\Omega})} f(x)dx = \int_{y\in \overline{\Omega}} f(\varphi(y)) \ ? ? ? \ dy.$$

Where I have written the three question marks, there should be a dependence on $\varphi$. According to Wikipedia, if $\varphi$ would be a function from $\mathbb{R}^n$ to $\mathbb{R}^n$, I should take the absolute value of the determinant of the Jacobian. (https://en.wikipedia.org/wiki/Integration_by_substitution)

But $\varphi'$ is not a square matrix. So something else needs to be done. I feel there should be a general theorem that one can look up if one knows integrals better.

Question 1: What should be at the three question marks?

Question 2: Can someone give me a citeable source?

many thanks Till