I have an integral that looks like $$ \int_{\zeta_1, \zeta_2 \in I} e^{\gamma F(\zeta_1, \zeta_2)} d\zeta_1 d\zeta_2, $$ where $I$ is some fixed interval. I want to apply the change of variable $t_1 = F(\zeta_1, \zeta_2)$ and $t_2 = \zeta_2$, and express it in terms of $t_1$ and $t_2$. I wasn't sure how to set up the Jacobian and I would greatly appreciate any help with this. Thank you very much.
PS I am assuming $F$ is "nice" so that it works... I would also appreciate some explanation of what I require for $F$ so that this actually works.
In order for this change of variable to make sense, you need at least that there exists an inverse to the function $$G_{\zeta_2}:\zeta_1\longrightarrow F(\zeta_1,\zeta_2)$$ for all $\zeta_2$.