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Divergence (Gauss-Ostrogradsky) theorem

Let be a smooth (differentiable) three-component vector field on the three dimensional space and is its divergence then the field divergence integral over the arbitrary three dimensional volume equals to the integral over the surface of the field itself projected onto the unite length vector field always perpendicular to the surface and pointing outside the surface which contains this volume or otherwise the inner values of the field divergence make virtually no contributions to the integral over the volume i.e.

where and the wraps the .

ProofEdit

We can approximate the integral of the divergence over the volume by the finite sum by dividing densely the space inside the volume   into small cubes with the edges   and the corners   as well as approximating three of the coordinate derivatives by their difference quotients. We will keep the edges coordinate names for the convenience even if they are equal. We get

 

Let us focus on a single contribution to this sum related to the derivative with respect to a chosen coordinate for example   i.e. for example  . For a fixed   we have

 

Note now that because of the alternating signs the vast majority of terms in the right sum vanish and we have

 

which reduces only to two terms or

 

where the bordering   and   with the first coordinate obviously depending on the choice of   and   are such that those points are the closed to the surface   containing the volume  .

Also note that while   is an infinitesimal (small) element of the surface parallel to the   plane and for the unite vector   perpendicular to it   and so for the second point the right side is an approximate to the growth   of the surface integral   i.e.

 ,

 ,

Repeating the estimate for the two other dimensions and coming back to the original sum we get

 

so the right side is the approximate surface integral (sum over the surfaces of the cubes closest to the surface  ) of the field itself projected on the outward unit vector field which proves the therem i.e.

 .