Divergence Theorem - Explanation and Proof

Divergence Theorem

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.

Proof:

The divergence of any vector A is given by:

  -----> Equation (1)

Take the volume integral on both sides

  ------> Equation (2)

Since dv = dx dy dz

Consider an element volume in x direction.

 ------> Equation (3)

But,

        -------> Equation (4)

Substitute Equation 4 in Equation 3:

  

Where dy dz = ds_x = x component of surface area ds.

Similarly the following integrands become:

Then, Substitute in Equation 2:

Hence Divergence Theorem Proved !!