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I have the following generic set of variables:

x = {x1, x2, ..., xn}
nu = {0, 1, 2, ..., n - 1}
order(nu) := Total @ nu;

I would like to calculate:

$$ \frac{\partial^{\operatorname{order}(v)}}{\partial_1^{\nu_1} \partial_2^{\nu_2} \dots \partial_n^{\nu_n}} $$

With a generic $x$. Indeed, I would like a function of this sort:

MyMixedDerivative[f,x_?VectorQ]:= ...

Suggestions?

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  • $\begingroup$ You might need Derivative. $\endgroup$ – Αλέξανδρος Ζεγγ Jul 31 at 7:58
  • $\begingroup$ I tried this: but it does not seam to work out: testx = {x1, x2, x3}; testf[x1_, x2_, x3_] := x1^3 x2^2 x3^2 Derivative[Table[1, {i, 1, 3}]][testf[x1, x2, x3]] $\endgroup$ – Mirko Aveta Jul 31 at 8:07
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    $\begingroup$ How about Derivative[Sequence @@ Table[1, {i, 1, 3}]][testf][Sequence @@ testx]? $\endgroup$ – Αλέξανδρος Ζεγγ Jul 31 at 8:12
  • $\begingroup$ It doesn't perform the derivative. I really can't figure out why. $\endgroup$ – Mirko Aveta Jul 31 at 8:14
  • $\begingroup$ It does. See my new comment. $\endgroup$ – Αλέξανδρος Ζεγγ Jul 31 at 8:16
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OK. If one looks at the documentation of Derivative, it should be found that Derivative accepts two arguments. The first is a Sequence of orders of derivatives to be taken and the second is a Function or a Symbol representing a function, say f. So when the order argument is passed to Derivative, it should be destructured from a List to a Sequence:

testx = {x1, x2, x3}; testf[x1_, x2_, x3_] := x1^3 x2^2 x3^2;
Derivative[Sequence @@ Table[1, {i, 1, 3}]][testf][Sequence @@ testx]

which gives

12 x1^2 x2 x3

Moreover, if one digs deeper in the "Details" part of the documentation, s/he should find out that the above description of requirement is not quite accurate. The true requirement is to achieve a structure consistency between the order argument and the argument of f. So below codes work as well

testf2[{x1_, x2_, x3_}] := x1^3 x2^2 x3^2;
Derivative[Table[1, {i, 1, 3}]][testf2][testx]

testf3[{x1_, x2_}, x3_] := x1^3 x2^2 x3^2;
Derivative[{1, 1}, 1][testf3][{x1, x2}, x3]

and other combinations of Sequence and List as the argument.

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  • $\begingroup$ The second part of your comment is inspiring. Thank you a lot. $\endgroup$ – Mirko Aveta Jul 31 at 8:38
  • $\begingroup$ @MirkoAveta You are welcome. Actually, it is a new point to me, too :) $\endgroup$ – Αλέξανδρος Ζεγγ Jul 31 at 8:50
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    $\begingroup$ In short, Derivative[] is supposed to inherit the argument structure of the function it is operating on. To give yet another example: Derivative[{1, 0}, {0}, 0][HypergeometricPFQ][{-2, 1/2}, {1}, 1/2] is equivalent to Derivative[1, 0, 0, 0][Hypergeometric2F1][-2, 1/2, 1, 1/2], since HypergeometricPFQ[{a, b}, {c}, x] == Hypergeometric2F1[a, b, c, x]. $\endgroup$ – J. M. will be back soon Aug 1 at 12:42
  • $\begingroup$ @J.M.isaway Thx for your comments. Yes, "to inherit" is an accurate and concise description. $\endgroup$ – Αλέξανδρος Ζεγγ Aug 1 at 12:48

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