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I want D[Subscript[a, k],Subscript[a,j]] to return something like the Kronecker delta. Is this possible?

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Of course, if you don't instist on logical correctness.

      Subscript /: D[Subscript[a_, b_], Subscript[a_, c_]] := 
                   KroneckerDelta[b, c]
     Subscript /: D[ Subscript[a_, _], Subscript[b_, _]] := 0 /; a != b

      D[ Subscript[a,k], Subscript[a,j]]
       Subscript[\[Delta], j,k]

      D[ Subscript[a,k], Subscript[1,k]]
      0
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    $\begingroup$ I am not sure why Subscript[\[Delta], j,k] appears in the code $\endgroup$
    – userrandrand
    May 19 at 20:37
  • $\begingroup$ Probably you are able to Trace the answer easily. Evaluation algorithm produces terms Derivatives[1,0][Subscript] and Derivative[0,1][Subscript] . The rest is printers professional secret. $\endgroup$
    – Roland F
    May 20 at 4:56
  • $\begingroup$ I think I might have understood what you did to get the Subscript[\[Delta], j,k]. Maybe you used copy as then selected plain text rather than directly copy which would show KroneckerDelta[j, k]. $\endgroup$
    – userrandrand
    May 20 at 7:03

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