Re: Variable transformations

*To*: mathgroup at smc.vnet.net*Subject*: [mg131464] Re: Variable transformations*From*: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>*Date*: Mon, 29 Jul 2013 23:21:08 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-outx@smc.vnet.net*Delivered-to*: mathgroup-newsendx@smc.vnet.net

Say I have some differential equation in terms of the independent variable x and dependent variable f(x). Now say I want to do a coordinate/variable transformation where x=x(u) and I want to write my equation in terms of only u and f(u). By pencil & paper, you can do this by applying the chain rule, but I have a complicated equation and I want to check if I've made an algebra slip-up (it involves several dependent and independent variables and up to 4th order derivatives). Is there any way to automate something like this in Mathematica? I've messed around with replacement rules, but I can't get Mathematica to do things quite how I want. Thanks! Hi, Nicholas, This problem has been discussed few times, and I copy-paste here one of the answers, the one I like most. I do not remember, however, who gave it: The rule has the form: y->(y[g[#]]&) the round parentheses are important. It does not work without. (* The task is to substitute x->z^2 *) y'[x] /. y -> (y[#^2] &) /. x -> z y''[x] /. y -> (y[#^2] &) /. x -> z 2 z Derivative[1][y][z^2] 2 Derivative[1][y][z^2] + 4 z^2 (y^\[Prime]\[Prime])[z^2] (* The task is to substitute x->Sin[z] into the fourth order derivative *) y''''[x] /. y -> (y[Sin[#]] &) /. x -> z Sin[z] Derivative[1][y][Sin[z]] - 4 Cos[z]^2 (y^\[Prime]\[Prime])[Sin[z]] + 2 Sin[z]^2 (y^\[Prime]\[Prime])[Sin[z]] - 5 Cos[z]^2 Sin[z] \!\(\*SuperscriptBox[\(y\), TagBox[ RowBox[{"(", "3", ")"}], Derivative], MultilineFunction->None]\)[Sin[z]] - Sin[z] (-Sin[z] (y^\[Prime]\[Prime])[Sin[z]] + Cos[z]^2 \!\(\*SuperscriptBox[\(y\), TagBox[ RowBox[{"(", "3", ")"}], Derivative], MultilineFunction->None]\)[Sin[z]]) + Cos[z]^4 \!\(\*SuperscriptBox[\(y\), TagBox[ RowBox[{"(", "4", ")"}], Derivative], MultilineFunction->None]\)[Sin[z]] This method applies also to substitute a function into an expression containing derivatives, such that these derivatives would be calculated as well as the function itself: Clear[expr, rule]; expr = (r[\[CurlyPhi]]^2 + 2 Derivative[1][r][\[CurlyPhi]]^2 - r[\[CurlyPhi]] ( r^\[Prime]\[Prime])[\[CurlyPhi]])^2/(r[\[CurlyPhi]]^2 + Derivative[1][r][\[CurlyPhi]]^2)^2; rule = r -> (r0 + r1*Cos[#] &); expr /. rule (r1 Cos[\[CurlyPhi]] (r0 + r1 Cos[\[CurlyPhi]]) + (r0 + r1 Cos[\[CurlyPhi]])^2 + 2 r1^2 Sin[\[CurlyPhi]]^2)^2/((r0 + r1 Cos[\[CurlyPhi]])^2 + r1^2 Sin[\[CurlyPhi]]^2)^2 Have fun, Alexei Alexei BOULBITCH, Dr., habil. IEE S.A. ZAE Weiergewan, 11, rue Edmond Reuter, L-5326 Contern, LUXEMBOURG Office phone : +352-2454-2566 Office fax: +352-2454-3566 mobile phone: +49 151 52 40 66 44 e-mail: alexei.boulbitch at iee.lu