What is the physical significance of divergence theorem?

It describes how fields from many infinitesimally small point sources add together to get a macroscopic affect along the surface of a material

Suppose you have an object like a shirt that carries some net . The charges themselves carry a little electric field that "diverges" in all directions. Positive charges carry a positive divergence (field lines that spread out in all directions) and negative charges carry a negative divergence (field lines that come in from all directions). Now suppose you would like to know what what surrounding all those charges looks like. Or more accurately, you would like to know how much electric field is penetrating the shirt to affect other objects and things around the shirt.

There are two ways you can do this: you can look at all the fields from all those little charges and add them all together, or you can look at the electric field along the surface of the shirt. The divergence theorem says that either way, you will end up with the same amount of penetration from the total electric field coming from the charges.

Or using another crude example, imagine you have several water facets pouring at a constant rate into the same sink. There are two ways to measure the total amount of water that is pouring into the sink: you can measure the water pouring from each facet and add it all up, or you can measure how much water drains out of the sink. The divergence theorem says you will end up with the same number for both measurements

Mathematically the divergence theorem is:

##int"div F "dV=intF*dA##

The left side of the equation is mathematically equivalent to adding all the divergence from all the point sources found in the the volume V. This is like adding all the fields from all the charges in the shirt or adding up all the water pouring from each facet.

The right side of the equation is mathematically equivalent to adding up all the field passing through the surface that surrounds the volume. This is like looking at the field coming from the surface of the shirt or measuring the water water pouring down the sink from all the pouring water facets.