Basic details of Divergence and Curl (I)

For those whe study tensor mathematics these concepts are already familiar. However, when should have to realize that they also have a physical meaning when applied to particular cases.

- Divergence is always related with normal component of a certain flow trespassing the surface of a certain volume (variation of a certain magnitude per unit area and time). A direct consequence of this imply that if we measure the income and outcome contributions of this magnitude through its surface, this balance will be directly related with the value of divergence: If divergence is zero the net contribution of magnitude in this volume is exactly zero (the same amount is incoming that outcoming). If divergence is positive more flux of magnitude is outcoming in the selected region that incoming, and negative divergence will imply the opposite.

A classic example would be the magnetic field, a vectorial field. If we take a control volume around a magnet it can be seen that divergence of the magnetic field is zero. Knowing that a magnet is in fact a source of the magnetic field, zero divergence tell us that "streamlines" of the field must be closed: streamlines are generated in a point of the magnet, but they must flow to the other pole of the magnet. 

Other well known example is fluid mechanics. Zero divergence around a control volume imply mass conservation, and flow conservation with almost incompressible fluids like water.

It is remarkable that the divergence of a vectorial field is a SCALAR.

Curl also have a specific meaning when we are talking about vectorial magnitudes. This operator is linked with tangential component. To say some examples, in fluid mechanics curl operator appear when we define the concepts "flow circulation " and "vorticity". The result of applying curl to two vectorial field is another vectorial field orthogonal to the previous ones.