For example,I want to match a multiple partial derivative of a function $f^{(1,0,1)}[x,y,z]$ which is the same as$\frac{\partial^2{f}}{\partial{x}\partial{z}}$.The only information given is the list of the independent variables such as {x,y,z} in this example. So I try to construct a pattern something like u_^[(n__)][independ] to match an arbitrary partial differential of an unknown function. However, it turns out to be wrong. For a simple case that has only one independent variable like $f^{(3)}[x]$ I can construct a pattern like D[u_[independ],{independ,n_}] to correctly match the partial derivative.But i don't know how to proceed to the multivariable cases. Are there some good soulutions?
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2 Answers
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Tutorial: Introduction to Patterns
"...the structure the Wolfram Language uses in pattern matching is the full form of expressions printed by FullForm."
Consider
D[foo[x, y, z], {x, 2}, {y, 3}]
See the underlying expression using FullForm:
FullForm @ %
Derivative[2, 3, 0][foo][x, y, z]
So the pattern you need is Derivative[__][_][__]:
list = {g[x], D[foo[x, y, z], {x, 2}, {y, 3}], x + 4, w''[x]};
Cases[Derivative[__][_][__]] @ list
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Define some test cases, including 1 that we would like to leave unchanged
tests = Flatten[{f[u], f'[x], D[f[x, y], {{x, y}, 2}]}];
Examining the FullForm of these gives us a clue as to what we are trying to match
FullForm[tests] // OutputForm
(* List[f[u], Derivative[1][f][x], Derivative[2, 0][f][x, y], Derivative[1, 1][f][x, y],
Derivative[1, 1][f][x, y], Derivative[0, 2][f][x, y]] *)
Define a pattern to match this (and a rule, transforming the result)
pattern = Derivative[ns__][g_][xs__] -> g[xs, ns];
Demonstrate that it matches where desired
tests /. pattern
(* {f[u], f[x, 1], f[x, y, 2, 0], f[x, y, 1, 1], f[x, y, 1, 1], f[x, y, 0, 2]} *)
