ChangeVariables Package for Mathematica

Summary

A package to change variables in symbolic (differential) equations in Mathematica.

Example: x df(x,y)/dx and transform x->r with the definition r(x) = 1/x:

ChangeVariables[x D[f[x,y],x], x, r, (1/# &)]

Output:

{-r (f^(1,0))[r,y]}

You can find examples in UseChangeVariables.wls or in Tests/TestChangeVariables.wls.

Installation

• Either load the package with Get[absoluteFilePathToPackageFile], where the file path is to ChangeVariables.wl
• Or install the package by clicking file -> install.

Features

• In some equation change the variable a to the variable b with definition toVarDef (Lambda/Pure function): ChangeVariables[equation, a, b, toVarDef].
• the functions in equation can have an arbitrary amount of arguments, e.g. f[x,y,z,a,b]
• arbitrary amounts of derivatives D[f[x,y], {x,1337}, {y, 42}].
• If your definition toVarDef has a non-unique inverse, e.g. x->y=Cos[x], then x=ArcCos[y]+2 Pi c, where c is an integer.The output will show this.

Known Bugs

• I had to blacklist internal functions like Plus and Times in order to not transform them. If your expression contains a function like that that I have not thought of, then you might get nonsense. I would appreciate a quick note if you have encountered such an example.
• ChangeVariables[Cos[x] f[x, y], x, u, (Cos[#] &)] still has a conditional statement as output with c an integer, but there is actually no ambiguity left.

ToDo

• Convert more than one function at a time.
• Option for explicitly stating a function that should be blacklisted, see Known Bugs.
• Option to give inverse function as argument, e.g. if x->y=Log[x] you could give Exp[#]& instead and let Mathematica invert.
• Clean up source code such that it is actually readable.
• Tests.

Author

Marco Knipfer
University of Alabama