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The essence of Stokes' theorem is the integration formula
$\int {\int}_{S}\left(\nabla \times \mathbf{F}\right)\xb7\mathbf{N}\mathrm{dsigma;}$ = ${\u2233}_{C}\mathbf{F}\xb7\mathbf{dr}$ = ${\u2233}_{C}\mathbf{F}\xb7\mathbf{T}\mathrm{ds}$
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with sufficient hypotheses for making the formula valid. For example, the vector field F should have continuously differentiable components. The surface $S$ should be oriented, its unit normal field N consistent with the orientation. In addition, $S$ should be described by piecewisesmooth functions and bounded by a piecewisesmooth, simple closed curve $C$, itself oriented consistently with the orientation of $S$.
In general, the surface $S$ is not closed. As such, it is called a capping surface for the bounding curve $C$. For example, a hemisphere is the capping surface for the circle $C$, at the "equator" of the hemisphere.
The integrand of the double integral on the left is the flux, through the surface $S$, of the curl of F. On the right, the tangential component of F is integrated around the bounding curve $C$, yielding the circulation of F around $C$. Stokes' theorem balances the fields net vorticity (circulation) flowing through the surface $S$, and the average circulation of F around the bounding curve $C$. Vorticity (or twist, rotation) of F on the surface $S$ is measured by the curl of F. Integrating the normal component of the curl of F on the surface $S$ allows local swirls that oppose each other to cancel out, leaving just the uncanceled parts on the boundary curve $C$ to reckon with. This residual vorticity is along the bounding curve $C$ and accumulates along $C$ in the line integral on the right side of the Stokes' formula.

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Figure 9.9.1 Cancellation of local rotation






Figure 9.9.1 illustrates how local rotation of the normal components of $\nabla \times \mathbf{F}$ "cancel" where two such vectors are contiguous, leaving just the points along the boundary to accumulate circulation. Thus, the net circulation as measured by the flux of the curl field can be measured by summing the tangential component of F along the bounding curve $C$.${}$${}$
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The flux of the curl of F through a closed surface is necessarily zero. This can be seen by applying the Divergence theorem, written as
$\int {\int}_{S}\mathbf{A}\xb7\mathbf{N}\mathrm{dsigma;}$ = $\int \int {\int}_{R}\nabla \mathbf{\xb7}\mathbf{A}\mathbf{}\mathrm{dv}$
${\u222f}_{S}\left(\nabla \times \mathbf{F}\right)\xb7\mathbf{N}\mathrm{dsigma;}$, where the double integral is now taken over the closed surface $S$. Take $\nabla \times \mathbf{F}$ in this flux integral as A in the Divergence theorem, and let the interior of the closed surface $S$ be the region $R$. The result is
${\u222f}_{S}\left(\nabla \times \mathbf{F}\right)\xb7\mathbf{N}\mathrm{dsigma;}$ = $\int \int {\int}_{R}\nabla \xb7\left(\nabla \times \mathbf{F}\right)\mathrm{dv}$ = 0
The rightmost (triple) integral vanishes because the integrand is the divergence of a curl. But the divergence of a curl is necessarily zero because curls don't spread, the distillation of Identity 1 in Table 9.4.1.
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