# Wedge Product of Differentials in Integration

• Jul 8th 2013, 07:01 AM
Phantasma
Wedge Product of Differentials in Integration
Can someone explain the difference between $\displaystyle \iint f(x,y) \, dx\wedge dy$ and $\displaystyle \iint f(x,y) \, dx \, dy$?

In general, is $\displaystyle \iint\limits_{D} f(x,y) \, dx \, dy = \iint\limits_{D} f(x,y) \, dy \, dx = \iint\limits_{D} f(x,y) \, dx\wedge dy$?

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.
• Jul 8th 2013, 04:13 PM
HallsofIvy
Re: Wedge Product of Differentials in Integration
There is no difference except, as you suggest, orientation. $\displaystyle \int\int f(x,y)dx\wedge dy$ is the integral on a surface with the normal oriented in the direction of $\displaystyle dx \times dy$. $\displaystyle \int\int f(x,y) dy\wedge dx$ is the integral on a surface with the normal oriented in the direction of $\displaystyle dy\times dx$. In "Calculus III" integration, the orientation is assumed to be given as part of the limits of integration.