Consider an example. Let $f(x)$ be a function on the unit sphere $S^{n-1}$. Consider an integral $ \int\limits_{S^{n-1}} f(x) \, dx $ I want to make a substitution $x = x_{0}t + \sqrt{1-t^2}y$, where $t \in [-1,1]$, $y \bot x_{0}$, $|y| = 1$. After change of variables the manifold of integration changes.
In general case if I have an integral over manifold $M$ and after change of variables I obtain an integral over manifold $M'$ how do I compute Jacobian of such change of variables (additional multiplier under integral sign)? If it is the usual Jacobian, how I can justify it?