Stokes' Theorem

Orientation is coupled by the right-hand rule: if the thumb points along the normal of \(S\), the curled fingers give the direction of travel along \(C\). One needs \(S\) piecewise smooth and \(\mathbf F\) with continuous partial derivatives on an open set containing \(S\). Reversing the normal flips the sign of the flux, just as reversing \(C\) flips the circulation.

A practical consequence is that the flux of the curl depends only on the boundary: two surfaces with the same curve \(C\) give the same value, so a flat disk can replace a hemisphere. A guiding example: for \(\langle -y,x,z\rangle\) around a horizontal circle of radius \(a\), the curl is \(\langle0,0,2\rangle\) and the circulation is \(2\pi a^2\), regardless of the circle's height.

When \(S\) lies in the \(xy\)-plane with normal \(\mathbf k\), the integrand reduces to \(Q_x-P_y\) and Green's theorem is recovered. If \(\nabla\times\mathbf F=\mathbf 0\) on a region without holes, the circulation around every closed curve vanishes and \(\mathbf F\) is conservative. Typical mistakes: forgetting that \(S\) and \(C\) need compatible orientations, or using a surface whose boundary is not \(C\).

Stokes' theorem links a curve to a surface, and the divergence theorem, which comes next, links a closed surface to the solid it encloses.