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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?

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  • $\begingroup$ For your case, you might be interested in reading en.wikipedia.org/wiki/Coarea_formula $\endgroup$ – Student May 16 '12 at 20:42
  • $\begingroup$ I like the coarea formula, but I don't see how I can apply it here directly because I have an integral over the unit sphere which is not an open set in $\mathbb{R}^{n}$. Do you mean I have to represent my integral as a derivative of an integral over the ball? $\endgroup$ – Appliqué May 16 '12 at 20:52
  • $\begingroup$ Sorry, I realize that the wiki webpage does not state it in full generality. I've found this more complete reference nd.edu/~lnicolae/Coarea.pdf, (see Theorem 1.3 and corollaries below). I think this might help you concerning the formalism you look for the Jacobian computation. $\endgroup$ – Student May 18 '12 at 14:18

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