Green's Theorem states that the line integral around a positively oriented, piecewise smooth, simple closed curve C is equal to the double integral of the curl over the planar region D it encloses: ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dx dy By breaking the boundary into manageable segments (C₁, C₂, C₃, C₄) and integrating over [a, b], we convert the macroscopic flow around the boundary into the total microscopic circulation (rotation) throughout the entire area.

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