First let me recall the usual change of variable formula:
Let $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ be a bijection which is Frechet differentiable, $U\in\mathrm{Open}(\mathbb{R}^n)$, and $f:\mathbb{R}^n\to\mathbb{R}$ be given. Then the change of variable formula says that $$ \int_U f\circ\varphi = \int_{\varphi(U)}f\,\cdot|\det(D\left(\varphi^{-1}\right))| \tag{CoV}$$ where $D\left(\varphi^{-1}\right)$ is the Frechet derivative of $\varphi^{-1}$.
Now let me present the context I am actually interested in:
Let now $G$ be a (matrix?) Lie group. Let us interpret $G^n$ as the Cartesian product (which is itself a Lie group I presume). Let $\varphi:G^n\to G^n$ be a given bijection, $U\in\mathrm{Open}(G^n)$ and $\mu$ be the Haar measure on $G$. If $f:G^n\to\mathbb{R}$ is given, then
My question:
What is the analog of (CoV) for my context? I.e., what is $$ \int_{(g_1,\dots,g_n)\in U} (f\circ\varphi)(g_1,\dots,g_n)\,\mathrm{d}\mu(g_1)\dots\mathrm{d}\mu(g_n) = ? $$