If I want to integrate over a surface $S$ in $\Bbb R^3$ parametrized by $\sigma:\overline A\subset\Bbb R^2\rightarrow S$ then I use $$\int\limits_S Fds=\int\limits_A F(\sigma(u,v))\cdot(\sigma_u\wedge\sigma_v)dudv$$ for a vector function $F=(F_1,F_2,F_3)$ or $$\int\limits_Sfds=\int\limits_Af(\sigma(u,v))\|\sigma_u\wedge\sigma_v\|dudv$$ for a scalar function $f$, where $\wedge$ is the cross product and $\sigma_u$ is $\frac{\partial\sigma}{\partial u}$.
But what if I have a surface (in my case it's the unit sphere over $\mathbb C^2$ defined as $S^2:=\{(z_1,z_2):|z_1|^2+|z_2|^2=1\}$) which I parametrized by $\sigma(\theta_1,\theta_2,t):[0,2\pi)^2\times[-3\pi/2,-\pi/2]\rightarrow S^2$ .
What is the integration formula for such a surface?
Since $\sigma$ goes from $\Bbb R^3$ to $\Bbb C^2$ its Jacobian matrix is not square... so there is no determinant