The flux of fluid through the surface is determined by the component of F that is in the direction of n, i.e. by F⋅n. Note that F⋅n will be zero if F and n are perpendicular, positive if F and n are pointing the same direction, and negative if F and n are pointing in opposite directions.
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Find the flux of F=yˆj−zˆk through the paraboloid S=y=x2+z2,y≤1. The flux can be described by ∬SF⋅ndσ with n=2xˆi−ˆj+2zˆk√1+4x2+4z2. Substitute x2+z2=y to simplify n to −1+2z2y. The total flux through the surface is 0.
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