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Re: Variable transformations

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  • Subject: [mg131587] Re: Variable transformations
  • From: Youngjoo Chung <ychung12 at>
  • Date: Sun, 8 Sep 2013 03:08:35 -0400 (EDT)
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Dear Alexei,

You can use SCTransDeriv in the SymbolicComputing package (, 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 about the package.


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

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:

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

Youngjoo Chung

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