Question about Stokes' Theorem for Flux of Vector Field

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I am a Calc III student and I'm a bit confused about Stokes: I know it's used to calculate flux of curl of vector field but what about flux of vector field in direction or normal vector? How can I use Stokes for that?

The integral would probably look like $\int_{C} \phi \cdot n dr$ where C is a boundary curve. I'm also a bit confused how this relates to $\int_{C} \phi \cdot dr$