I just meant the area. Even though there's no cross product, if we pretend there is we can still take its magnitude.
We could use the same formula to integrate a scalar field over the duocylinder, or any surface in 4D, and I think it would be easy to find formulae to integrate a scalar or vector field over a 3D or 4D region in 4D. But how could we go about integrating a vector field F
over a duocylinder? The integrand for a surface in 3D is F
So in general, do we just integrate the volume of the parallelepiped determined by F
<sub>u</sub> and r
If so, then that's all the integrals we can do in 4D
. The only other thing we need is a 4D version of Stoke's theorem or the Divergence theorem.