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The Divergence theorem, alternatively called the "Divergence theorem of Gauss," should not be confused the "Gauss' theorem," an entirely different theorem whose proof follows from the Divergence theorem but which cannot be used to prove the Divergence theorem.
The essence of the Divergence theorem is the integration formula
$\int \int {\int}_{R}\nabla \xb7\mathbf{F}\mathrm{dv}equals;\int {\int}_{S}\mathbf{F}\xb7\mathbf{N}\mathrm{dsigma;}$
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where F is a vector field whose components have continuous derivatives inside and on the surface $S$. The surface $S$, bounding the open connected set $R$, must be piecewise smooth, closed, and oriented. The vector N is the unit outward normal on $S$. The volume represented by $R$ is connected, but not necessarily simply connected. Thus, $R$ can be the interior of a sphere (simply connected) or of a torus (not simply connected).
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The Divergence theorem is a statement of a law of balance. The left side is the integral of the divergence of a field throughout a region $R$. This gives a measure of the total "spread" of the field in $R$. The right side is the flux of the field through the boundary of $R$. Any change in the content inside $R$ must be accounted for by the passage of the field through the boundary.
Consequently, the Divergence theorem basically says "The integral of the divergence over the interior of a region must equal the flux through the boundary of the region."
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