Important Properties

The curl of a gradient is the zero vector, curl(grad f)=(0,0,0)

The divergence of a curl is zero, div(curl F)=0


  Clear[f]
  Simplify[curl3[grad3[f]]]
  Simplify[curl2[grad2[f]]]


  Simplify[div3[curl3[f]]]

Up to Divergence and Curl