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MathGroup Archive 1998

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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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Notebook[{

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:" }],
"Text"],

Cell[BoxData[
    \(Clear[Derivative]\)], "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,
\[Theta]}]] /. 
          {r \[Rule] r[x, y], \[Theta] \[Rule] \[Theta][x, y]}]; \)\)], 
  "Input"],

Cell["The operator you are computing becomes", "Text"],

Cell[CellGroupData[{

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

        {r[x, y] \[Rule] r, \[Theta][x, y] \[Rule] \[Theta]}]\)],
"Input"],

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

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

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[
      \(Dt[u[r, \[Theta]], {x, 2}] + Dt[u[r, \[Theta]], {y, 2}] /. 
          Literal[Dt[h_, t_]] \[RuleDelayed] \[PartialD]\_t h[x, y]\) /.

        {r[x, y] \[Rule] r, \[Theta][x, y] \[Rule] \[Theta]}]\)],
"Input"],

Cell[BoxData[
    FormBox[
      RowBox[{
        FractionBox[
          RowBox[{
            SuperscriptBox["u", 
              TagBox[\((0, 2)\),
                Derivative],
              MultilineFunction->None], "(", \(r, \[Theta]\), ")"}], 
          \(r\^2\)], "+", 
        FractionBox[
          RowBox[{
            SuperscriptBox["u", 
              TagBox[\((1, 0)\),
                Derivative],
              MultilineFunction->None], "(", \(r, \[Theta]\), ")"}],
"r"], 
        "+", 
        RowBox[{
          SuperscriptBox["u", 
            TagBox[\((2, 0)\),
              Derivative],
            MultilineFunction->None], "(", \(r, \[Theta]\), ")"}]}], 
      TraditionalForm]], "Output"]
}, Open  ]]
}, Open  ]]
},
FrontEndVersion->"3.5 for Macintosh", ScreenRectangle->{{0, 800}, {0,
580}}, WindowSize->{520, 485},
WindowMargins->{{60, Automatic}, {Automatic, 28}},
MacintoshSystemPageSetup->"\<\
00<0001804P00000=j8W4?m0oeThGBNb0dL5N`?P0080001804P000000`d26P01
0000I00000400 at 410?l00BL?00400@0000000000000000P801T1T0000000H000
00000000004000000000000000000000\>"
]


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