The volume integral of the divergence of a vector field over a volume is equal to the surface integral of the normal component of this vector over the surface bounding the volume.
The divergence of any vector A is given by:
Take the volume integral on both sides
Since dv = dx dy dz
Consider an element volume in x direction.
Substitute Equation 4 in Equation 3:
Where dy dz = dsx = x component of surface area ds.
Similarly the following integrands become:
Then, Substitute in Equation 2:
Hence Divergence Theorem Proved !!