Let $S$ be a (n-1) dimensional hypersurface in $\mathbb{R}^n$. I see something like this written:
$$\int_S f(s)d\sigma(s)$$ where $d\sigma$ means the surface measure.
Now if $\Phi\colon T \to S$ is a diffeomorphism where $T$ is also another hypersurface, can I simply use the change of variables like so, with $s = \Phi(t)$, we have $$\int_S f(s)d\sigma(s) = \int_T f(\Phi(t))|\text{det}D\Phi|d\sigma(t)$$
Is it right to write $d\sigma (t)$? Here $D\Phi$ is the matrix of partial derivatives of $\Phi.$
can I just ignore the differential and concentrate on manipulating the integral in whatever way I wish (like I would on usual integrals over intervals)?