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Re: polar coordinates

• To: mathgroup at smc.vnet.net
• Subject: [mg13469] Re: polar coordinates
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sun, 26 Jul 1998 02:33:32 -0400
• Organization: University of Western Australia
• References: <6p6pba$5g7@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Toshiyuki Meshii wrote:

> Let me know the way to work with patial differentiation on polar
> coordinates.
> For examples, when
> x = r Cos[q] and y = r Sin[q]
> how can you obtain
> du[x,y]/dx = Cos[q] * (du[x,y]/dr) - Sin[q] / r * (du[x,y]/dq) by
> Mathematica?

I see from my copy of the attached Notebook that it may have been
corrupted (due to a problem with TraditionalForm notation).

I have attached another copy which uses StandardForm.

Cheers,
	Paul
 
____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul at physics.uwa.edu.au  AUSTRALIA                            
http://www.pd.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________


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Cell[CellGroupData[{
Cell["Polar Coordinates", "Section"],

Cell[TextData[{
  "For the change of variables, ",
  Cell[BoxData[
      \(TraditionalForm
      \`{x, y} \[Rule] {r\ \(cos(\[Theta])\)\ , r\
\(sin(\[Theta])\)}\)]],
  ", first compute and save the matrix of partial derivatives:" }],
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    \(Clear[Derivative]\)], "Input"],

Cell[BoxData[
    \(\(Evaluate[Outer[D, {r[x, y], \[Theta][x, y]}, {x, y}]] = 
      Simplify[Inverse[
            Outer[D, {r\ Cos[\[Theta]], r\ Sin[\[Theta]]}, {r,
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          {r \[Rule] r[x, y], \[Theta] \[Rule] \[Theta][x, y]}]; \)\)], 
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Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[
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          Literal[Dt[h_, t_]] \[RuleDelayed] \[PartialD]\_t h[x, y]\) /.

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        RowBox[{\(cos(\[Theta])\), " ", 
          RowBox[{
            SuperscriptBox["u", 
              TagBox[\((1, 0)\),
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              MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}],
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        FractionBox[
          RowBox[{\(sin(\[Theta])\), " ", 
            RowBox[{
              SuperscriptBox["u", 
                TagBox[\((0, 1)\),
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                MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}],
"r"]}
        ], TraditionalForm]], "Output"] }, Open  ]],

Cell["Similarly, the Laplacian reads", "Text"],

Cell[CellGroupData[{

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      RowBox[{
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            SuperscriptBox["u", 
              TagBox[\((0, 2)\),
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        FractionBox[
          RowBox[{
            SuperscriptBox["u", 
              TagBox[\((1, 0)\),
                Derivative],
              MultilineFunction->None], "(", \(r, \[Theta]\), ")"}],
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          SuperscriptBox["u", 
            TagBox[\((2, 0)\),
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            MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], 
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