How to calculate Stokes' determinant?

Questions
1

Hello,

I need to calculate the Stokes’ determinant in Mathematica, such as

Screenshot2024-04-21152648.png
Screenshot2024-04-21152648.png255×155 2.57 KB

What should I do in Mathematica? I can’t image the right code in Mathematica.

2

This is an attempt:

Det[( {
   {d[y, z], d[z, x], d[x, y]},
   {dd[x], dd[y], dd[z]},
   {y^2, x, z^2}
  } )]
% /. a_.*dd[var_]*d[vars__] :> D[a, var] d[vars]
3

Thanks. It seems a little complicate to do this.

4

Determinants are complicated. One more approach:

stokes[{f1_, f2_, f3_}, {x_, y_, z_}] :=
  Sum[{d[y, z], d[z, x], d[x, y]}[[perm[[1]]]]*
    D[{f1, f2, f3}[[perm[[3]]]], {x, y, z}[[perm[[2]]]]]*
    Signature[perm],
   {perm, Permutations[Range[3]]}];
stokes[{y^2, x, z^2}, {x, y, z}]