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so in some previous videos we built the notion of an integral of a differential to form over surface in RN and we also looked at some examples and here we want to generalize that to an integral of a differential M form on an M dimensional subspace of RN but before we do that lets recall what weve built so far so earlier we showed that if we have a differential to form on RN it can be written in the following way so its this sum over I 1 and I 2 where I want is strictly less than I 2 and then we have F sub I 1 I 2 so those are differential functions on RN and then weve got this symbol DXi one wedge DXi two and well talk about what that does a little bit later and then weve also got our parametrized surface so fee goes from D to RN and D is a subset of the plane in other words r2 and you can think about D as building our surface s so here weve got over here the UV plane were using U and V is our parameters and weve got our region D and thats mapped onto the surface s over here