Is there any "formula" that allows us make change of variables in surface integrals? For example, here (wikipedia) there are some "formulas" (or better stated "theorems") that allows us make change of variables in some integrals. I need an analogous for surface integrals. Could someone help me?
Thanks.
 A: Please note that by the time you are actually computing a surface integral, you've already parameterized it -- so, you're no longer dealing with an integral over the surface, but rather an integral over a planar region that parameterizes the surface.  So, anything that you could normally do with a double integral (including change of variables) can be done at that point.
For that matter, the standard parameterization formula allows you to choose ANY parameterization -- you can often build a change of variables right in by choosing a smart parameterization of your surface.
A: If the surface is expressed as a level set $\{x|\phi(x)=c\}$ bounding a solid $\{x|\phi(x)\le c\}$, a change of variables formula can be obtained from the coarea formula
$$\frac{\partial}{\partial c}\int_{\phi(x)\le c} f(x)\,dx = \int_{\phi(x)=c} f(x)\frac{da(x)}{|\nabla\phi(x)|}.$$ 
Make a change of variables $x=p(y)$ in the $dx$ integral, use coarea twice, and write $g=f/|\nabla\phi|$ to make it look better; I find for the surface integral, if I didn't mess up, that
$$ \int_{\phi=c} g\,da = \int_{\phi\circ p=c} g\circ p \frac{|(\nabla\phi)\circ p|}{|\nabla(\phi\circ p)|}\,da.$$
