- #1

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Surface area =

[tex]\int_{S} \hat{n} \cdot \hat{n} dA = \int_{V} \nabla \cdot \hat{n} dV[/tex]

where S is the closed surface that bounds a volume V. Since the normal n is defined on S, how do you interpret div n in the interior region? Do you just extend the field n on S to a field N on V in such a way that it is continuously differentiable and satisfies N = n on S?

I am assuming that there's nothing wrong with applying Stokes'/Divergence theorem when the vector field being integrated depends on the region of integration.