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I have a InterpolatingFunction from a list of complex data $j(x,y)$ in two variables $x$ and $y$.

I need to evaluate this equation $( x ∂j/∂y - y ∂j/∂x )$.

x = {0.01, 0.02, 0.03, 0.04, 0.01, 0.02, 0.03, 0.04, 0.01, 0.02, 0.03,
0.04, 0.01, 0.02, 0.03, 0.04};
y = {0.05, 0.05, 0.05, 0.05, 0.06, 0.06, 0.06, 0.06, 0.07, 0.07, 0.07,
0.07, 0.08, 0.08, 0.08, 0.08};

j = {0.010561397860905564` + 0.0015976985131723536` I, 
  0.014020635474832743` + 0.0025977169091002837` I, 
   0.017487513072750305` + 0.003602592422820732` I, 
   0.020961981006940338` + 
   0.0046123543767775275` I, -0.01371285709417238` - 
    0.0049735813378507425` I, -0.010312644229723354` - 
    0.004004629742597602` I, -0.00690445946081937` - 
    0.0030310857543026193` I, -0.003488357390773193` - 
     0.002052913771772308` I, -0.00006438567799935371` - 
    0.001070077781411395` I, 
    0.0033674092402317087` - 0.00008254401950370003` I, 
    0.006806973184737428` + 0.0009097221180533224` I, 
    0.010254263771322686` + 0.0019067570422475653` I, 
    0.01370921819568967` + 0.0029085892815736082` I, 
    0.017171802982761625` + 0.003915256260194262` I, 
    0.020641953493563608` + 
   0.004926787666825977` I, -0.013998626378818357` - 
    0.0046779948653150735` I};

I used the interpolation:

    jj = Interpolation[Transpose[{x, y, j}]]

I tried to do the derivative as following:

     dd[a_, b_, d_, i_] :=  
    Evaluate[a[[i]]*D[0, 1][jj[[i]]][a[[i]], b[[i]]] - 
    b[[i]]*D[1, 0][jj[[i]]][a[[i]], b[[i]]]]


    m = Table[dd[x, y, jj, i], {i, Length[x]}]

I am a new in Mathematica, and I tried the similar questions here, but I couldn't find a solution. Please can anyone help me to calculate this derivative.

Thank you

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2 Answers 2

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jj = Interpolation[Transpose[{x, y, j}]];
der[xx_,yy_] := xx Derivative[1, 0][jj][xx, yy] - yy Derivative[0, 1][jj][xx, yy]

der[0.015, 0.055]
(* 0.12972 + 0.0351906 I *)

Plot3D[
 Evaluate[
  ReIm[der[xx, yy]]
  ]
 , {xx, 0.01, 0.04}, {yy, 0.05, 0.08}
 , PlotStyle -> Opacity[0.5]
 ]

Mathematica graphics

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z[u_, v_] := {-v, u}.Grad[jj[a, b], {a, b}] /. Thread[{a, b} -> {u, v}]

Plot3D[Evaluate[ReIm[z[u, v]]], {u,##& @@ MinMax[x]}, {v, ##& @@ MinMax[y]}, 
 PlotStyle -> Opacity[0.5], PlotLegends -> {HoldForm @ Re[z[u, v]], HoldForm@Im[z[u, v]]}]

enter image description here

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