# Impose relation between partial derivatives of functions

I have very long expressions with functions f[x,y] and g[x,y] which satisfy

Derivative[1, 0][f][x, y] == - Derivative[0, 1][g][x, y]


I want to simplify expressions such as

Derivative[2, 0][f][x, y] - Derivative[1, 1][g][x, y]


which should evaluate to zero.

Ideally, the relation between the functions would only be imposed locally (comparable to the local definitions of variables inside a Block or similar).

## What I have tried so far

I have found a solution which is non-local and also works only in the simplest cases. Using the function SetDelayed

Derivative[1, 0][f][x, y] := Derivative[0, 1][g][x, y]


correctly evaluates expressions like

Derivative[1, 0][f][x, y] - Derivative[0, 1][g][x, y]


to zero, but not

Derivative[2, 0][f][x, y] - Derivative[1, 1][g][x, y]


Interestingly, with the above SetDelayed the following evaluates to zero:

D[f[x, y], {x, 2}] - D[g[x, y], x, y]

• So why not add Derivative[k_Integer?Positive, 0][f][x_, y_] := Derivative[k - 1, 1][g][x, y] as a definition? – J. M.'s discontentment Dec 18 '16 at 8:25
• @J.M. This does indeed work for the examples I have provided but misses, for example, Derivative[1, 1][f][x, y] - Derivative[0, 2][g][x, y]. – user137217 Dec 18 '16 at 8:36
• Yes, because the previous definition fixed the order of the partial with respect to the second variable. I was nudging you to think carefully about the general rule for implementing Derivative[j, k][f][x, y]. – J. M.'s discontentment Dec 18 '16 at 8:38
• I see. Derivative[j_Integer?Positive, k_Integer?Positive][f][x_, y_] := Derivative[j - 1, k + 1][g][x, y]. Thanks a lot. Do you think it is ok to put ImposeRelation[expr_] := Block[{}, Derivative[j_Integer?Positive, k_Integer?Positive][f][x_, y_] := Derivative[j - 1, k + 1][g][x, y]; expr]? – user137217 Dec 18 '16 at 8:49
• If you want to account for 0, use NonNegative instead as the condition. I don't know about OK; why not just try it out? You can always start in a fresh kernel if things go south for some reason. – J. M.'s discontentment Dec 18 '16 at 8:51

## 1 Answer

Following the comments by J. M. the definition using SetDelayed can be extended to an arbitrary number of partial derivatives:

Derivative[j_Integer?Positive, k_Integer?NonNegative][f][x_, y_] :=
Derivative[j - 1, k + 1][g][x, y]


To satisfy the requirement of locality the following may be used:

ImposeRelation[expr_] :=
Block[{fL, gL},
Derivative[j_Integer?Positive, k_Integer?NonNegative][fL][x_, y_] :=
Derivative[j - 1, k + 1][gL][x, y];
expr /. {f -> fL, g -> gL} /. {fL -> f, gL -> g}]