Applet: Macroscopic and microscopic circulation in three dimensions
The relationship between the macroscopic circulation of a vector field $\dlvf$ around a curve 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 blue point on the slider), the total microscopic circulation of $\dlvf$ along the surface must equal the circulation of $\dlvf$ around the curve, as long as the vector field $\dlvf$ is defined everywhere on the surface.
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