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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'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.

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up vote 4 down vote accepted

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.

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Makes perfect sense. Thanks! – MGN Aug 22 '12 at 4:29
@MGN And if you want a general version of this correspondence, look up Hodge star. – user31373 Aug 22 '12 at 11:34
Just looked up the wiki article. Beautiful. Thank you – MGN Aug 28 '12 at 4:13

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