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

In[211]:=

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

Out[211]=

        (1,0,1)               (1,1,0)
  {0, -f       [#1, #2, #3], f       [#1, #2, #3]} &

Out[212]=

0 &

In[213]:=

  Simplify[div3[curl3[f]]]

Out[213]=

0 &