Reconciling the classical formulation of a surface integral with a general integral over a manifold 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.

Vector fields correspond to either one-forms or two-forms in three dimensions. I usually use the notation that follows: if $\vec{F} = \langle a,b,c \rangle$ then the corresponding work form  is
$$ \omega_{\vec{F}} = adx +bdy+cdz $$
whereas the corresponding flux form is
$$ \Phi_{\vec{F}} = ady \wedge dz +bdz \wedge dx+cdx \wedge dy $$
surface integrals measure flux so you should use the flux form.
