Just like Green's Theorem has its 2D divergence equivalent, The 3D divergence theorem is the Stokes' Theorem equivalent for divergence in 3D.

In 2D divergence we re-applied green's theorem to find the total flow through a boundary curve by summing up all little bits of outwards flow in the region, and we used a surface integral.

In the 3D Divergence theorem, we use divergence to split into tiny bits of outwards flow everywhere, then integrate, ending with
The surface must be closed. If it is not, there will be outward flow we cannot account for, and the 3D divergence theorem does not hold.

We recall three definitions, the divergence,

Then, we remember that the unit normal vector to a 3D surface is defined Accompanied by the surface integral,