Any one know how to prove four nonlinear discretization has different properties?
 

In the context of incompressible fluid dynamics.

The four nonlinear discretizations are
1. rotational form (momentum + kinetic energy conservation)
2. divergence form (momentum conservation)
3. convective form (non of the two are conserved)
4. skew-symmetric (both are conserved)

How to prove? I just read some paper that keep saying those point again and again but I don't know how to prove them.

Any paper or books that you have seen, is helpful!

Have a look here
JOURNAL OF COMPUTATIONAL PHYSICS 143, 90–124 (1998)