Wedge Product of Differentials in Integration

Can someone explain the difference between and ?

In general, is ?

Does Fubini's Theorem disregard orientation, instead working solely with a measure space without assuming additional structure?

I've been trying to figure this out, and now I have a headache from trying to read stuff about it because it uses some weird physics prefix notation. (Worried)

Any help is greatly appreciated. Thank you.

Re: Wedge Product of Differentials in Integration

There is no difference except, as you suggest, orientation. is the integral on a surface with the normal oriented in the direction of . is the integral on a surface with the normal oriented in the direction of . In "Calculus III" integration, the orientation is assumed to be given as part of the limits of integration.