Scalar and vector fields

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What is the relation between the flux through a given surface by a vector field? And how does stokes theorem relate to the line integral around a surface in that field
 

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HallsofIvy
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The flux of a vector field [itex]\vec{F}[/itex] through surface S is, by definition,
[tex]\int\int_S \vec{F}\cdot d\vec{S}[/tex]

And Stokes' theorem say, specifically, that that is equal to
[tex]\int_{c} \nabla\times\vec{F}\cdot d\vec{r}[/tex]
Where c is the closed path bounding surface S.
 

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