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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)
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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



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