In vector calculus, the "'divergence theorem "', also known as "'Gauss's theorem "'or "'Ostrogradsky's theorem "', is a result that relates the flow ( that is, flux ) of a vector field through a surface to the behavior of the vector field inside the surface.
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However, as a consequence of the divergence theorem, because the region away from the origin is vacuum ( source-free ) it is only the homology class of the surface in that matters in computing the integral, so it can be replaced by any convenient surface in the same homology class, in particular a sphere, where spherical coordinates can be used to calculate the integral.
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For such objects, the integral may be taken over the entire surface ( A ) by taking the absolute value of the integrand ( so that the " top " and " bottom " of the object do not subtract away, as would be required by the Divergence Theorem applied to the constant vector field \ mathbf { \ hat { r } } ) and dividing by two:
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Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily a conservative one, and it is free from sources or sinks, having net flux equal to zero through any open domain without holes . ( These two observations combine as real and imaginary parts in Cauchy's integral theorem . ) In fluid dynamics, such a vector field is a potential flow.
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