I know how to integrate both scalar and vector functions over 2-dimensional surfaces. The way to do that is the same explained here https://en.wikipedia.org/wiki/Surface_integral The thing is that it uses cross product of the parametrization to get a normal vector of norm 1, so how should I generalise it for higher dimensions?
I have heard something about a so called "fundamental form" to generalise these kind of things, but I don't remember exactly