Calculus

Stokes' Theorem

Local curl → boundary circulation

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  • about 6 min
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Why Stokes' Theorem Works

↺↺↺↺↺↺↺↺↺↺↺↺↺↺↺↺– Boundary curve○ Local circulations (curl)

Controls

Curl strength1.00
Grid density4.00

Calculation

Local loops16
∫∫ curl(F)·dS16.0
∮ F·dr16.0
∮ F·dr = ∫∫ curl(F)·dS ✓

∮ F·dr = ∫∫ (∇×F)·dS

💡 Why it works

Each tiny loop shares edges with its neighbors. On shared edges, circulations cancel (opposite directions). Only the outermost boundary circulation survives. Toggle "Show cancellation" to see it.