So I was just brushing up on some calculus when I came across a problem. I was trying to perform a surface integral I found online through the more general formulation of a differential form on a manifold. This led to some trouble.
I'm considering the integral over the vector field $F(x,y,z) = (2x,2y,2z)$ along a cylinder of radius $1$ and height $5$ paramtrized by $S(\theta, t) = (\cos(\theta),\sin(\theta),t).$ Calc 2 would tell me to just take the dot product of the vector field and the normal vector to the surface, but I wanted to do this from a manifolds perspective, according to http://en.wikipedia.org/wiki/Differential_form's integration section. Following its lead, I identify $F(x,y,z) = 2x dx + 2y dy + 2z dz$ as my differential form. But now I immediately run into trouble as I'm trying to integrate a 1-form on a 2-manifold.
Is there something intrinsically different about a surface integral versus an integral along a manifold? It seems like a surface integral assigns to each 1-form a 2-form that measures flow through a manifold rather than actually integrating along its surface.