Stokes' Theorem

Calculus 3 · Axiom Academy

Circulation around a closed curve equals the flux of curl through any surface it bounds — Green's Theorem, lifted into 3D. 1. The Theorem: Circulation = Curl-Flux Let S be an oriented surface whose boundary is the closed curve C . For a vector field with continuous partial derivatives, the circulation of around C equals the flux of its curl through S . A 1-D line integral around the boundary equals a 2-D surface integral over the region. 2. Curl: The Field's Microscopic Rotation For , the curl measures how much the field rotates at each point — its circulation density . Drop a tiny paddle wheel in the field: the curl is how fast (and which way) it spins. For the curl is constant, — pointing straight out of the plane. 3. Orientation: The Right-Hand Rule For the two sides to agree, S and C must be oriented consistently . The right-hand rule fixes the pairing. The dot product keeps only the part of the curl that pierces the surface. Reverse C and flips with it — sending to . A wrong-handed pairing is a sign error. 4. Any Surface Works — and Where It Shows Up Because is divergence-free, the curl-flux depends only on the boundary C : every surface sharing that boundary gives the same value. So you may deform S to whatever shape makes the integral easiest. This surface-independence is what makes Stokes' Theorem so useful across physics and engineering: The circulation of the electric field around a loop equals minus the rate of change of magnetic flux through it: .

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