Re: Variable transformations

*To*: mathgroup at smc.vnet.net*Subject*: [mg131587] Re: Variable transformations*From*: Youngjoo Chung <ychung12 at gmail.com>*Date*: Sun, 8 Sep 2013 03:08:35 -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*References*: <kt7b9o$dkf$1@smc.vnet.net>

Dear Alexei, You can use SCTransDeriv in the SymbolicComputing package (http://symbcomp.gist.ac.kr, WTC 2011) for the variable transformation of derivatives as shown below. All functions with the prefix SC are package functions. It is still a beta version, but it might help. The result of the variable transformation is shown in Out[236] below. Out[242] shows verification by inverse transformation. Copying the textual form of the input into Mathematica may not work since it does not faithfully reproduce the standard form of the input. Sorry I cannot explain all the details here. Please refer to the homepage http://symbcomp.gist.ac.kr about the package. <<SymbolicComputing` In[235]:= SCTotalDeriv[y, {x, 4}] SCMAF[%, SCTransDeriv, {All, TransVar -> {x, z, x == Sin[z]}}, SCCollectDerivs, {All, y, Apply -> {PowerExpand, Simplify}}, SCTrigToHalf, {Cos[2 z], Cos}, SCCollectDerivs, {All, y, Apply -> {Factor, Expand}}, SCFactor, {All, Sec[z]^4}] Out[235]= \[DifferentialD]^4y/\[DifferentialD]x^4 During evaluation of In[235]:= Sec[z]^4 \[DifferentialD]^4y/\[DifferentialD]z^4+Sec[z]^4 (-11+15 Sec[z]^2) \[DifferentialD]^2y/\[DifferentialD]z^2+6 Sec[z]^4 Tan[z] \[DifferentialD]^3y/\[DifferentialD]z^3+3 Sec[z]^4 (-2+5 Sec[z]^2) Tan[z] \[DifferentialD]y/\[DifferentialD]z Out[236]= Sec[z]^4 (\[DifferentialD]^4y/\[DifferentialD]z^4 + (-11 + 15 Sec[z]^2) \[DifferentialD]^2y/\[DifferentialD]z^2 + 6 Tan[z] \[DifferentialD]^3y/\[DifferentialD]z^3 + 3 (-2 + 5 Sec[z]^2) Tan[z] \[DifferentialD]y/\[DifferentialD]z) Verification by inverse transformation: In[241]:= Sec[z]^4 (\[DifferentialD]^4y/\[DifferentialD]z^4 + (-11 + 15 Sec[z]^2) \[DifferentialD]^2y/\[DifferentialD]z^2 + 6 Tan[z] \[DifferentialD]^3y/\[DifferentialD]z^3 + 3 (-2 + 5 Sec[z]^2) Tan[z] \[DifferentialD]y/\[DifferentialD]z); SCMAF[%, SCTransDeriv, {All, TransVar -> {z, x, z == ArcSin[x]}}, SCCollectDerivs -> {y, Apply -> Simplify}] Out[242]= \[DifferentialD]^4y/\[DifferentialD]x^4 Sincerely, Youngjoo Chung