Applet: Macroscopic and microscopic circulation in three dimensions
The relationship between the macroscopic circulation of a vector field $\dlvf$ around a curve (red boundary of surface) and the microscopic circulation of $\dlvf$ (illustrated by small green circles) along a surface in three dimensions must hold for any surface whose boundary is the curve. No matter which surface you choose (change by dragging the green point on the top slider), the total microscopic circulation of $\dlvf$ along the surface must equal the circulation of $\dlvf$ around the curve (We assume that the vector field $\dlvf$ is defined everywhere on the surface.)
You can change the curve to a more complicated shape by dragging the blue point on the bottom slider, and the relationship between the macroscopic and total microscopic circulation still holds.
The definition of microscopic circulation depends on the orientation of the surface, which is specified by the choice of the normal vector (shown in cyan). The definition of the macroscopic circulation depends on the orientation of the curve, which is specified by the red arrow on the curve. Since the orientation of these normal vectors have the proper relationship for Stokes' theorem to apply, the macroscopic and microscopic circulation values will match.
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