Averaged Lagrangian and High-Frequency Gravitational Waves by MacCallum & Taub.nb

(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 12.1' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       158,          7]
NotebookDataLength[     47033,       1156]
NotebookOptionsPosition[     40748,       1052]
NotebookOutlinePosition[     41215,       1069]
CellTagsIndexPosition[     41172,       1066]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["\<\
Notebook for :   
Averaged Lagrangian and High-Frequency Gravitational Waves by MacCallum & Taub\
\>", "Title",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.8217294718661222`*^9, 3.821729502043838*^9}, 
   3.8217295683553457`*^9, 3.821732221863006*^9, {3.821811203295624*^9, 
   3.8218112036548367`*^9}, {3.822041768762134*^9, 3.822041781290854*^9}, {
   3.822048009752109*^9, 3.8220480569607773`*^9}, {3.822504877727007*^9, 
   3.8225048874039307`*^9}, {3.822565447028533*^9, 3.822565457908236*^9}, {
   3.822822107930128*^9, 3.822822113128089*^9}, {3.823176459993638*^9, 
   3.823176470345335*^9}, {3.823359338529508*^9, 3.823359361520629*^9}, {
   3.823437107739418*^9, 3.823437114378701*^9}, {3.823437155219769*^9, 
   3.823437162159172*^9}, {3.823454321027132*^9, 3.823454332946658*^9}, 
   3.8554995723903637`*^9, 3.855504967182631*^9},
 TextAlignment->Center,ExpressionUUID->"c20da370-3896-41d0-9717-1bb2c5220b1c"],

Cell["\<\
Geoff Cope
University of Utah
\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\|01040f\|01042d\|01044c\|010432\|010442\|010432\
\|010449\|010445\|01042e\|01043b\|010428 \|010432\|010442 \|01040f\|01042d\
\|01043b\|01042b
February 26, 2021\
\>", "Author",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.821729507652328*^9, 3.821729529657844*^9}, 
   3.8220480097522783`*^9, {3.822048070235491*^9, 3.822048084689715*^9}, {
   3.8225654346519403`*^9, 3.8225654361641483`*^9}, {3.8231764728567343`*^9, 
   3.823176482313031*^9}, {3.8233640113399878`*^9, 3.823364023347041*^9}, {
   3.8234229506702557`*^9, 3.823422956381791*^9}, {3.827953252419902*^9, 
   3.827953254868108*^9}, {3.851065213363542*^9, 3.8510652255731983`*^9}, {
   3.8510652620952*^9, 3.851065307413348*^9}, {3.851099255534511*^9, 
   3.851099264486588*^9}, {3.851252093025189*^9, 3.851252093273655*^9}, 
   3.851252176997463*^9, 3.8512537234246607`*^9, 3.85125469290289*^9, 
   3.851256927304118*^9, {3.855405234504258*^9, 3.855405238736866*^9}, 
   3.855405272664227*^9, {3.855407872416093*^9, 3.855407879483934*^9}, 
   3.8554963948316936`*^9, {3.855504241007119*^9, 3.855504245254983*^9}, 
   3.8555049671827374`*^9},
 TextAlignment->Center,ExpressionUUID->"7155b898-d8cc-45d1-a4a8-b77f156a8efc"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Hyperlink To Documentation", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{{3.820944675605734*^9, 
  3.8209446931410646`*^9}},ExpressionUUID->"053b918f-2261-4176-962e-\
30961c5fe519"],

Cell[BoxData[
 RowBox[{"Hyperlink", "[", 
  RowBox[{
  "\"\<GeneralRelativityTensors Documentation and Download\>\"", ",", 
   "\[IndentingNewLine]", 
   "\"\<https://github.com/BlackHolePerturbationToolkit\>\""}], 
  "]"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{{3.820944619768764*^9, 3.820944681268063*^9}},
 CellLabel->
  "In[118]:=",ExpressionUUID->"22aa5f92-723b-4052-832d-cd1f0f007d2a"],

Cell[BoxData[
 TemplateBox[{
  "\"GeneralRelativityTensors Documentation and Download\"", 
   "https://github.com/BlackHolePerturbationToolkit"},
  "HyperlinkURL"]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{{3.820944653562152*^9, 3.820944681268403*^9}, 
   3.821729905862431*^9, 3.821732641380642*^9, 3.8218069931429567`*^9, 
   3.821811425912986*^9, 3.822042652654992*^9, {3.822048793766124*^9, 
   3.8220488009116488`*^9}, 3.82250502887521*^9, 3.822505578829629*^9, 
   3.822565490613895*^9, 3.8225662097813597`*^9, 3.822606717894772*^9, 
   3.822739433713368*^9, 3.822822210464344*^9, 3.8233594284652433`*^9, 
   3.8233607803938913`*^9, 3.823360880234103*^9, 3.823361224481102*^9, 
   3.82336209870968*^9, 3.823364030542309*^9, 3.823437204902042*^9, 
   3.823454358677424*^9, 3.855532294082947*^9},
 CellLabel->
  "Out[118]=",ExpressionUUID->"a92ab7e0-ddc5-43b4-89e1-0a1d787fdf6d"]
}, Closed]],

Cell[CellGroupData[{

Cell["Hyperlink  To Article ", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{
  3.820937794883616*^9, {3.820938266275806*^9, 3.82093829464357*^9}, 
   3.820943883376583*^9, {3.8210324492405148`*^9, 3.821032453951701*^9}, {
   3.8217297279218817`*^9, 3.8217297313782063`*^9}, {3.822593613363134*^9, 
   3.822593616181035*^9}, {3.822736144359599*^9, 3.8227361450547543`*^9}, {
   3.823359417697976*^9, 3.823359420672374*^9}, 
   3.82621089806524*^9},ExpressionUUID->"6aa87865-d9d1-440a-b010-\
a6ae8bfe0b3c"],

Cell[BoxData[
 RowBox[{"Hyperlink", "[", 
  RowBox[{
  "\"\<Averaged Lagrangian and High-Frequency Gravitational Waves by \
MacCallum & Taub\>\"", ",", "\[IndentingNewLine]", 
   "\"\<https://projecteuclid.org/download/pdf_1/euclid.cmp/1103858809\>\""}],
   "]"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.8218867746254797`*^9, 3.821886810805148*^9}, 
   3.823176507207197*^9, {3.823472496118194*^9, 3.8234725120275393`*^9}, {
   3.826210891060401*^9, 3.8262108939307327`*^9}},
 CellLabel->
  "In[119]:=",ExpressionUUID->"85bd99cb-0440-4d61-a3eb-c95a2e4adef3"],

Cell[BoxData[
 TemplateBox[{
  "\"Averaged Lagrangian and High-Frequency Gravitational Waves by MacCallum \
& Taub\"", "https://projecteuclid.org/download/pdf_1/euclid.cmp/1103858809"},
  "HyperlinkURL"]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{3.821886813557355*^9, 3.8231765072073307`*^9, 
  3.823359428608782*^9, 3.8233607804338617`*^9, 3.823360880363216*^9, 
  3.823361224626254*^9, 3.823362098931829*^9, 3.823364030688095*^9, 
  3.8234372051552353`*^9, 3.823454358796459*^9, 3.8234725126509457`*^9, 
  3.855532294127578*^9},
 CellLabel->
  "Out[119]=",ExpressionUUID->"87e9decd-7b98-4d8c-b9ea-025a155541be"],

Cell[BoxData[
 RowBox[{"Hyperlink", "[", 
  RowBox[{
  "\"\<Colliding Plane Waves by Griffiths - Book\>\"", ",", 
   "\"\<ftp://nozdr.ru/biblio/kolxoz/P/PGr/Griffiths%20J.B.%20Colliding%\
20plane%20waves%20in%20general%20relativity%20(OUP,%201991)(ISBN%200198532091)\
(254s)_PGr_.pdf\>\""}], "]"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.821971700891346*^9, 3.821971727288177*^9}, 
   3.8220418523671618`*^9, {3.8220419016581783`*^9, 3.822041910552957*^9}},
 CellLabel->
  "In[120]:=",ExpressionUUID->"1eed751f-ebd0-40b9-960d-53879f3276ec"],

Cell[BoxData[
 TemplateBox[{
  "\"Colliding Plane Waves by Griffiths - Book\"", 
   "ftp://nozdr.ru/biblio/kolxoz/P/PGr/Griffiths%20J.B.%20Colliding%20plane%\
20waves%20in%20general%20relativity%20(OUP,%201991)(ISBN%200198532091)(254s)_\
PGr_.pdf"},
  "HyperlinkURL"]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{
  3.8219717866333103`*^9, 3.822041852367313*^9, 3.822041913511463*^9, 
   3.8220426527016563`*^9, {3.822048793815063*^9, 3.8220488009576597`*^9}, 
   3.822505029060484*^9, 3.822505578941113*^9, 3.82256549066683*^9, 
   3.822566209832062*^9, 3.822606717963243*^9, 3.822739433796803*^9, 
   3.822822210610302*^9, 3.8233594286355753`*^9, 3.8233607804460297`*^9, 
   3.8233608803853903`*^9, 3.823361224649728*^9, 3.823362098967597*^9, 
   3.823364030715418*^9, 3.823437205194394*^9, 3.8234543588186893`*^9, 
   3.855532294141523*^9},
 CellLabel->
  "Out[120]=",ExpressionUUID->"ac648e2a-435f-4baa-828e-981f32db640a"]
}, Closed]],

Cell[CellGroupData[{

Cell["Utilities and Package Load", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{{3.820937646765551*^9, 
  3.8209376725449123`*^9}},ExpressionUUID->"d9f48d09-18d6-4716-8349-\
81a622ec457e"],

Cell[BoxData[{
 RowBox[{"ClearAll", "[", "\"\<Global`*\>\"", "]"}], "\[IndentingNewLine]", 
 RowBox[{"<<", "Utilities`CleanSlate`"}], "\[IndentingNewLine]", 
 RowBox[{"CleanSlate", "[", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"ClearInOut", "[", "]"}], 
  "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"pdConv", "[", "f_", "]"}], ":=", 
  RowBox[{"TraditionalForm", "[", 
   RowBox[{"f", "/.", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"Derivative", "[", "inds__", "]"}], "[", "g_", "]"}], "[", 
      "vars__", "]"}], "\[RuleDelayed]", 
     RowBox[{"Apply", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"Defer", "[", 
         RowBox[{"D", "[", 
          RowBox[{
           RowBox[{"g", "[", "vars", "]"}], ",", "##"}], "]"}], "]"}], "&"}], 
       ",", 
       RowBox[{
        RowBox[{"Transpose", "[", 
         RowBox[{"{", 
          RowBox[{
           RowBox[{"{", "vars", "}"}], ",", 
           RowBox[{"{", "inds", "}"}]}], "}"}], "]"}], "/.", 
        RowBox[{"{", 
         RowBox[{
          RowBox[{
           RowBox[{"{", 
            RowBox[{"var_", ",", "0"}], "}"}], "\[RuleDelayed]", 
           RowBox[{"Sequence", "[", "]"}]}], ",", 
          RowBox[{
           RowBox[{"{", 
            RowBox[{"var_", ",", "1"}], "}"}], "\[RuleDelayed]", 
           RowBox[{"{", "var", "}"}]}]}], "}"}]}]}], "]"}]}]}], 
   "]"}]}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{
  3.8209376549030037`*^9, {3.821032517164062*^9, 3.821032518258361*^9}},
 CellLabel->
  "In[121]:=",ExpressionUUID->"b6d8af22-2efe-47f7-81cb-20865cad30c8"],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"  (CleanSlate) Contexts purged: \"\>", "\[InvisibleSpace]", 
   RowBox[{"{", "\<\"Global`\"\>", "}"}]}],
  SequenceForm["  (CleanSlate) Contexts purged: ", {"Global`"}],
  Editable->False]], "Print",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{
  3.820932754769992*^9, 3.820933427630714*^9, 3.820937654903143*^9, 
   3.821729906190301*^9, 3.821732641685704*^9, 3.8218069933778133`*^9, 
   3.82181142618342*^9, 3.8220426528487463`*^9, {3.822048793940605*^9, 
   3.822048801089246*^9}, 3.822505029191699*^9, 3.822505579049889*^9, 
   3.822565490838564*^9, 3.822566210008327*^9, 3.822606718180009*^9, 
   3.8226371008270607`*^9, 3.822739433887854*^9, 3.822761800089258*^9, 
   3.8228172997642097`*^9, 3.822822210725956*^9, 3.822988687208267*^9, 
   3.823359428732101*^9, 3.823360780582996*^9, 3.823360880477087*^9, 
   3.823361224742215*^9, 3.8233620990886593`*^9, 3.8233640308219624`*^9, 
   3.823437205329228*^9, 3.823454358888331*^9, 3.8555322943489733`*^9},
 CellLabel->
  "During evaluation of \
In[121]:=",ExpressionUUID->"d835a44a-82f3-4465-87e9-a6a2294eb2b1"],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"  (CleanSlate) Approximate kernel memory recovered: \"\>", 
   "\[InvisibleSpace]", "\<\"1389 Kb\"\>"}],
  SequenceForm[
  "  (CleanSlate) Approximate kernel memory recovered: ", "1389 Kb"],
  Editable->False]], "Print",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{
  3.820932754769992*^9, 3.820933427630714*^9, 3.820937654903143*^9, 
   3.821729906190301*^9, 3.821732641685704*^9, 3.8218069933778133`*^9, 
   3.82181142618342*^9, 3.8220426528487463`*^9, {3.822048793940605*^9, 
   3.822048801089246*^9}, 3.822505029191699*^9, 3.822505579049889*^9, 
   3.822565490838564*^9, 3.822566210008327*^9, 3.822606718180009*^9, 
   3.8226371008270607`*^9, 3.822739433887854*^9, 3.822761800089258*^9, 
   3.8228172997642097`*^9, 3.822822210725956*^9, 3.822988687208267*^9, 
   3.823359428732101*^9, 3.823360780582996*^9, 3.823360880477087*^9, 
   3.823361224742215*^9, 3.8233620990886593`*^9, 3.8233640308219624`*^9, 
   3.823437205329228*^9, 3.823454358888331*^9, 3.855532294352951*^9},
 CellLabel->
  "During evaluation of \
In[121]:=",ExpressionUUID->"21677a79-9854-4048-94fd-0348ac460784"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"\<\"Utilities`CleanSlate`\"\>", ",", "\<\"VariationalMethods`\"\>",
    ",", "\<\"GeneralRelativityTensors`\"\>", 
   ",", "\<\"GeneralRelativityTensors`CommonTensors`\"\>", 
   ",", "\<\"GeneralRelativityTensors`TensorDerivatives`\"\>", 
   ",", "\<\"GeneralRelativityTensors`TensorManipulation`\"\>", 
   ",", "\<\"GeneralRelativityTensors`TensorDefinitions`\"\>", 
   ",", "\<\"GeneralRelativityTensors`Utils`\"\>", 
   ",", "\<\"DocumentationSearch`\"\>", ",", "\<\"ResourceLocator`\"\>", 
   ",", "\<\"NaturalLanguageProcessingLoader`\"\>", ",", "\<\"System`\"\>", 
   ",", "\<\"Global`\"\>"}], "}"}]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{
  3.820932754773623*^9, 3.820933427635436*^9, 3.8209376549033947`*^9, 
   3.821729906193787*^9, 3.821732641691049*^9, 3.821806993386058*^9, 
   3.821811426187071*^9, 3.822042652852829*^9, {3.822048793944077*^9, 
   3.822048801092909*^9}, 3.822505029195156*^9, 3.8225055790570173`*^9, 
   3.8225654908422747`*^9, 3.8225662100125237`*^9, 3.8226067181902227`*^9, 
   3.822637100839163*^9, 3.8227394339100733`*^9, 3.822761800103311*^9, 
   3.822817299776285*^9, 3.822822210739685*^9, 3.822988687217742*^9, 
   3.823359428736412*^9, 3.823360780587545*^9, 3.8233608804810534`*^9, 
   3.823361224746982*^9, 3.823362099093968*^9, 3.823364030838743*^9, 
   3.8234372053364553`*^9, 3.8234543588917103`*^9, 3.855532294355199*^9},
 CellLabel->"Out[0]=",ExpressionUUID->"41925857-1790-4280-89ea-8a8cc3782863"],

Cell[BoxData[
 RowBox[{"<<", "GeneralRelativityTensors`"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{3.8209376549035053`*^9},
 CellLabel->"In[2]:=",ExpressionUUID->"ba39d9fc-63f4-4084-9b25-3b2b2660e10d"]
}, Closed]],

Cell[CellGroupData[{

Cell["Custom Notation", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.8225114042235403`*^9, 3.822511435665856*^9}, 
   3.823176577809782*^9},ExpressionUUID->"cdcf7fc4-5b5a-4d86-936f-\
704c66c3f570"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "dtReplace", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"dtReplace", " ", "=", " ", 
   RowBox[{"{", "\[IndentingNewLine]", 
    RowBox[{
     RowBox[{
      RowBox[{"Dt", "[", "r", "]"}], "\[Rule]", " ", "dr"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "t", "]"}], "\[Rule]", " ", "dt"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "u", "]"}], "\[Rule]", " ", "du"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "v", "]"}], "\[Rule]", " ", "dv"}], " ", ",", " ", 
     "\[IndentingNewLine]", " ", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Theta]", "]"}], " ", "\[Rule]", " ", 
      "d\[Theta]"}], " ", ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Phi]", "]"}], " ", "\[Rule]", " ", "d\[Phi]"}], 
     "  ", ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Eta]", "]"}], " ", "\[Rule]", " ", "d\[Eta]"}], 
     "  ", ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Chi]", "]"}], "\[Rule]", " ", "d\[Chi]"}], " ", 
     ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Rho]", "]"}], " ", "\[Rule]", " ", "d\[Rho]"}], 
     " ", ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[CurlyPhi]", "]"}], " ", "\[Rule]", " ", 
      "d\[CurlyPhi]"}], " ", ",", " ", "\[IndentingNewLine]", " ", 
     RowBox[{
      RowBox[{"Dt", "[", "\[ScriptT]", "]"}], " ", "\[Rule]", " ", 
      "d\[ScriptT]"}], " ", ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[ScriptZ]", "]"}], " ", "\[Rule]", " ", 
      "d\[ScriptZ]"}], ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "U", "]"}], "\[Rule]", " ", "dU"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "V", "]"}], "\[Rule]", " ", "dV"}], " ", ",", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "z", "]"}], "\[Rule]", " ", "dz"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[ScriptR]", "]"}], "\[Rule]", " ", "d\[ScriptR]"}],
      "  ", ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[CapitalPhi]", "]"}], "\[Rule]", " ", 
      "d\[CapitalPhi]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"dtReplace", " ", "//", " ", "TableForm"}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.82317656063256*^9, 3.823176577809904*^9}, {
  3.823437608803*^9, 3.823437618694414*^9}},
 CellLabel->"In[3]:=",ExpressionUUID->"24d6d47b-fc02-4b2a-a506-ebb60170cc3c"],

Cell[BoxData[
 TagBox[
  TagBox[GridBox[{
     {
      RowBox[{
       RowBox[{"Dt", "[", "r", "]"}], "\[Rule]", "dr"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "t", "]"}], "\[Rule]", "dt"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "u", "]"}], "\[Rule]", "du"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "v", "]"}], "\[Rule]", "dv"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Theta]", "]"}], "\[Rule]", "d\[Theta]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Phi]", "]"}], "\[Rule]", "d\[Phi]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Eta]", "]"}], "\[Rule]", "d\[Eta]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Chi]", "]"}], "\[Rule]", "d\[Chi]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Rho]", "]"}], "\[Rule]", "d\[Rho]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[CurlyPhi]", "]"}], "\[Rule]", "d\[CurlyPhi]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[ScriptT]", "]"}], "\[Rule]", "d\[ScriptT]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[ScriptZ]", "]"}], "\[Rule]", "d\[ScriptZ]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "U", "]"}], "\[Rule]", "dU"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "V", "]"}], "\[Rule]", "dV"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "z", "]"}], "\[Rule]", "dz"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[ScriptR]", "]"}], "\[Rule]", "d\[ScriptR]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[CapitalPhi]", "]"}], "\[Rule]", 
       "d\[CapitalPhi]"}]}
    },
    GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
    GridBoxSpacings->{"Columns" -> {
        Offset[0.27999999999999997`], {
         Offset[0.5599999999999999]}, 
        Offset[0.27999999999999997`]}, "Rows" -> {
        Offset[0.2], {
         Offset[0.4]}, 
        Offset[0.2]}}],
   Column],
  Function[BoxForm`e$, 
   TableForm[BoxForm`e$]]]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.8231765606326637`*^9, 3.82317657781*^9}, 
   3.8233594287980833`*^9, 3.823360780614534*^9, 3.8233608805437098`*^9, 
   3.823361224810658*^9, 3.823362099159925*^9, 3.8233640309198627`*^9, 
   3.823437205444152*^9, 3.823437619154374*^9, 3.823454358956871*^9, 
   3.855532294391087*^9},
 CellLabel->
  "Out[5]//TableForm=",ExpressionUUID->"4bd01435-aefe-4ee3-999f-4b882ab80e3d"]
}, Closed]],

Cell[CellGroupData[{

Cell["Line Element and Metric Functions", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.},
 CellChangeTimes->{{3.820937548919169*^9, 3.8209375728447866`*^9}, 
   3.8209383129401073`*^9, 3.8220481344968433`*^9, {3.8225115057038183`*^9, 
   3.822511507783984*^9}, 
   3.822763734568725*^9},ExpressionUUID->"f062add6-b0a7-4986-8a1d-\
8f6eafb02689"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "lineToMetric", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"lineToMetric", "[", 
    RowBox[{"lineelement_", " ", ",", " ", "differentials_"}], "]"}], ":=", 
   " ", "\[IndentingNewLine]", 
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"If", "[", 
      RowBox[{
       RowBox[{"i", "\[Equal]", "j"}], " ", ",", 
       RowBox[{"Coefficient", "[", 
        RowBox[{"lineelement", ",", " ", 
         RowBox[{
          RowBox[{"differentials", "[", 
           RowBox[{"[", "i", "]"}], "]"}], " ", 
          RowBox[{"differentials", "[", 
           RowBox[{"[", "j", "]"}], "]"}]}]}], "]"}], ",", 
       "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{"1", "/", "2"}], ")"}], "*", 
        RowBox[{"Coefficient", "[", 
         RowBox[{"lineelement", ",", " ", 
          RowBox[{
           RowBox[{"differentials", "[", 
            RowBox[{"[", "i", "]"}], "]"}], " ", 
           RowBox[{"differentials", "[", 
            RowBox[{"[", "j", "]"}], "]"}]}]}], "]"}]}]}], "]"}], ",", 
     "\[IndentingNewLine]", 
     RowBox[{"{", 
      RowBox[{"i", ",", "1", ",", 
       RowBox[{"Length", "[", "differentials", "]"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"j", ",", "1", ",", 
       RowBox[{"Length", "[", "differentials", "]"}]}], "}"}]}], "]"}]}], 
  "  ", ";"}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.},
 CellChangeTimes->{3.821567412203915*^9, 3.821641945174967*^9, 
  3.821641981470141*^9, 3.8220481344970007`*^9, 3.822763734569355*^9},
 CellLabel->"In[6]:=",ExpressionUUID->"788fc160-dcff-4444-9960-a4033035a0de"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "metricToLine", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"metricToLine", "[", 
   RowBox[{"metric_", ",", " ", "differentials_"}], "]"}], ":=", " ", 
  "\[IndentingNewLine]", 
  RowBox[{"Sum", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"metric", "[", 
      RowBox[{"[", 
       RowBox[{"i", ",", "j"}], "]"}], "]"}], " ", 
     RowBox[{"differentials", "[", 
      RowBox[{"[", "i", "]"}], "]"}], " ", 
     RowBox[{"differentials", "[", 
      RowBox[{"[", "j", "]"}], "]"}]}], " ", ",", " ", 
    RowBox[{"{", 
     RowBox[{"i", ",", "1", ",", 
      RowBox[{"Length", "[", "differentials", "]"}]}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"j", ",", "1", ",", 
      RowBox[{"Length", "[", "differentials", "]"}]}], "}"}]}], 
   "]"}]}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.},
 CellChangeTimes->{{3.822762557092309*^9, 3.8227626322008677`*^9}, 
   3.8227637345698223`*^9, 3.822988709740047*^9},
 CellLabel->"In[8]:=",ExpressionUUID->"5238ea2a-5832-4be1-bd00-5ea6e3219e50"]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Line Element and Metric 4.1 (Appears to be misprint in paper... Upper case K \
in metric and lower case k as 2D Laplacian )  Come back and redo this.. \
\>", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.843085739976366*^9, 3.843085756774782*^9}, {
  3.8555319912178307`*^9, 3.855532022746344*^9}, {3.855532479225672*^9, 
  3.85553248913765*^9}},ExpressionUUID->"cf29f344-6614-4cb9-aad8-\
9a3b1ae660fa"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "eq4pt1", "]"}], "\[IndentingNewLine]", 
 RowBox[{"eq4pt1", " ", "=", " ", "\[IndentingNewLine]", 
  RowBox[{
   RowBox[{
    FractionBox[
     SuperscriptBox["\[CurlyRho]", "2"], 
     SuperscriptBox["p", "2"]], 
    RowBox[{"(", " ", 
     RowBox[{
      SuperscriptBox["d\[Xi]", "2"], "+", " ", 
      SuperscriptBox["d\[Eta]", "2"]}], ")"}]}], " ", "-", " ", 
   RowBox[{"2", " ", "d\[CurlyRho]", " ", "d\[Sigma]"}], " ", "-", " ", 
   RowBox[{
    RowBox[{"(", " ", 
     RowBox[{"K", " ", "-", " ", 
      FractionBox[
       RowBox[{"2", " ", "m"}], "\[CurlyRho]"], "-", " ", 
      RowBox[{"2", " ", "H", " ", "\[CurlyRho]"}]}], ")"}], " ", 
    SuperscriptBox["d\[Sigma]", "2"]}]}]}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.823454380839529*^9, 3.8234544542920637`*^9}, 
   3.843085749370772*^9},
 CellLabel->"In[10]:=",ExpressionUUID->"16283699-9076-4668-8929-b40f2fa8cc04"],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"-", "2"}], " ", "d\[CurlyRho]", " ", "d\[Sigma]"}], "+", 
  FractionBox[
   RowBox[{
    RowBox[{"(", 
     RowBox[{
      SuperscriptBox["d\[Eta]", "2"], "+", 
      SuperscriptBox["d\[Xi]", "2"]}], ")"}], " ", 
    SuperscriptBox["\[CurlyRho]", "2"]}], 
   SuperscriptBox["p", "2"]], "-", 
  RowBox[{
   SuperscriptBox["d\[Sigma]", "2"], " ", 
   RowBox[{"(", 
    RowBox[{"K", "-", 
     FractionBox[
      RowBox[{"2", " ", "m"}], "\[CurlyRho]"], "-", 
     RowBox[{"2", " ", "H", " ", "\[CurlyRho]"}]}], ")"}]}]}]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{3.82345445554453*^9, 3.843085749371073*^9, 
  3.855532294443372*^9},
 CellLabel->"Out[11]=",ExpressionUUID->"fa5e624a-7eda-4220-82fe-e805bedabbec"],

Cell[BoxData[
 RowBox[{
  RowBox[{"lineToMetric", "[", " ", 
   RowBox[{"eq4pt1", " ", ",", " ", 
    RowBox[{"{", 
     RowBox[{
     "d\[Xi]", ",", "d\[Eta]", ",", "d\[CurlyRho]", ",", "d\[Sigma]"}], 
     "}"}]}], "]"}], " ", "//", " ", "MatrixForm"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.823454530579596*^9, 3.823454553985614*^9}, 
   3.843085749371477*^9},
 CellLabel->"In[12]:=",ExpressionUUID->"f4485dbd-7a3f-4d0c-bcb5-ae55928e6b67"],

Cell[BoxData[
 TagBox[
  RowBox[{"(", "\[NoBreak]", GridBox[{
     {
      FractionBox[
       SuperscriptBox["\[CurlyRho]", "2"], 
       SuperscriptBox["p", "2"]], "0", "0", "0"},
     {"0", 
      FractionBox[
       SuperscriptBox["\[CurlyRho]", "2"], 
       SuperscriptBox["p", "2"]], "0", "0"},
     {"0", "0", "0", 
      RowBox[{"-", "1"}]},
     {"0", "0", 
      RowBox[{"-", "1"}], 
      RowBox[{
       RowBox[{"-", "K"}], "+", 
       FractionBox[
        RowBox[{"2", " ", "m"}], "\[CurlyRho]"], "+", 
       RowBox[{"2", " ", "H", " ", "\[CurlyRho]"}]}]}
    },
    GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}},
    GridBoxSpacings->{"Columns" -> {
        Offset[0.27999999999999997`], {
         Offset[0.7]}, 
        Offset[0.27999999999999997`]}, "Rows" -> {
        Offset[0.2], {
         Offset[0.4]}, 
        Offset[0.2]}}], "\[NoBreak]", ")"}],
  Function[BoxForm`e$, 
   MatrixForm[BoxForm`e$]]]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.823454549711*^9, 3.8234545542823887`*^9}, 
   3.843085749371735*^9, 3.855532294455756*^9},
 CellLabel->
  "Out[12]//MatrixForm=",ExpressionUUID->"09f8ae0c-745d-4a08-b507-\
5c0ccebe5205"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "\[CapitalDelta]", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"\[CapitalDelta]", "[", "u_", "]"}], ":=", " ", 
  "\[IndentingNewLine]", 
  RowBox[{"Expand", "[", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"p", "[", 
      RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "2"], "*", 
    RowBox[{"(", " ", 
     RowBox[{
      RowBox[{"D", "[", 
       RowBox[{"u", ",", 
        RowBox[{"{", 
         RowBox[{"\[Xi]", ",", "2"}], "}"}]}], "]"}], " ", "+", "  ", 
      RowBox[{"D", "[", 
       RowBox[{"u", ",", 
        RowBox[{"{", 
         RowBox[{"\[Eta]", ",", "2"}], "}"}]}], "]"}]}], " ", ")"}]}], 
   "]"}]}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.855531598452751*^9, 3.855531605003587*^9}, {
  3.855531653074009*^9, 3.855531798971002*^9}, {3.855531846107283*^9, 
  3.855531864187183*^9}},
 CellLabel->"In[13]:=",ExpressionUUID->"9e6ec828-0136-4844-bc10-be9fa0961d14"],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Clear", "[", "kReplace", "]"}], " ", 
  RowBox[{"(*", " ", 
   RowBox[{
   "See", " ", "paper", " ", "about", " ", "upper", " ", "case", " ", "K", 
    " ", "and", " ", "lower", " ", "case", " ", "k"}], " ", "*)"}], 
  " "}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"kReplace", " ", "=", " ", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{"K", "[", 
     RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], " ", "->", 
    "  ", 
    RowBox[{"\[CapitalDelta]", "[", 
     RowBox[{"Log", "[", 
      RowBox[{"p", "[", 
       RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], " ", "]"}], 
     " ", "]"}]}]}], " ", ";"}], "\[IndentingNewLine]", 
 RowBox[{"kReplace", " ", "//", " ", "pdConv", " "}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.85553174747125*^9, 3.855531768167197*^9}, {
  3.855531868099104*^9, 3.855531871698926*^9}, {3.855532043035149*^9, 
  3.8555320633625307`*^9}, {3.855532107566736*^9, 3.8555321230104628`*^9}, {
  3.855532317427063*^9, 
  3.855532331618638*^9}},ExpressionUUID->"0565ece6-7d70-49d5-aea1-\
6f6175a91406"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{"K", "[", 
    RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "\[Rule]", 
   RowBox[{
    RowBox[{
     RowBox[{"p", "(", 
      RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}], " ", 
     FractionBox[
      RowBox[{
       SuperscriptBox["\[PartialD]", "2"], 
       RowBox[{"p", "(", 
        RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}]}], 
      RowBox[{"\[PartialD]", 
       SuperscriptBox["\[Eta]", "2"]}],
      MultilineFunction->None]}], "+", 
    RowBox[{
     RowBox[{"p", "(", 
      RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}], " ", 
     FractionBox[
      RowBox[{
       SuperscriptBox["\[PartialD]", "2"], 
       RowBox[{"p", "(", 
        RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}]}], 
      RowBox[{"\[PartialD]", 
       SuperscriptBox["\[Xi]", "2"]}],
      MultilineFunction->None]}], "-", 
    SuperscriptBox[
     RowBox[{"(", 
      FractionBox[
       RowBox[{"\[PartialD]", 
        RowBox[{"p", "(", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}]}], 
       RowBox[{"\[PartialD]", "\[Eta]"}],
       MultilineFunction->None], ")"}], "2"], "-", 
    SuperscriptBox[
     RowBox[{"(", 
      FractionBox[
       RowBox[{"\[PartialD]", 
        RowBox[{"p", "(", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}]}], 
       RowBox[{"\[PartialD]", "\[Xi]"}],
       MultilineFunction->None], ")"}], "2"]}]}], TraditionalForm]], "Output",\

 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.855531768894503*^9, 3.8555318243166113`*^9}, {
   3.8555318564226437`*^9, 3.855531872039763*^9}, 3.855532068612615*^9, {
   3.855532114266714*^9, 3.855532123469819*^9}, {3.855532294499299*^9, 
   3.855532320325905*^9}},
 CellLabel->
  "Out[28]//TraditionalForm=",ExpressionUUID->"f1e5013c-88ee-499d-ace8-\
c4358c9702c2"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "hReplace", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"hReplace", " ", "=", " ", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{"H", "[", 
     RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], " ", "->", 
    "  ", 
    RowBox[{"D", "[", 
     RowBox[{
      RowBox[{"Log", "[", 
       RowBox[{"p", "[", 
        RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "]"}], ",", 
      "\[Sigma]"}], "]"}]}]}], " ", ";"}], "\[IndentingNewLine]", 
 RowBox[{"hReplace", "  ", "//", " ", "pdConv", " "}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.855531909275144*^9, 3.855531978474676*^9}, {
  3.855532125243307*^9, 3.8555321265331287`*^9}},
 CellLabel->"In[18]:=",ExpressionUUID->"c7cbdcdf-7c0b-40c3-a2d2-101ca85aeddd"],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{"H", "(", 
    RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}], "\[Rule]", 
   FractionBox[
    FractionBox[
     RowBox[{"\[PartialD]", 
      RowBox[{"p", "(", 
       RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}]}], 
     RowBox[{"\[PartialD]", "\[Sigma]"}],
     MultilineFunction->None], 
    RowBox[{"p", "(", 
     RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}]]}], 
  TraditionalForm]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.855531936649707*^9, 3.855531978830381*^9}, 
   3.855532126899234*^9, 3.855532258547847*^9, 3.855532294511821*^9},
 CellLabel->
  "Out[20]//TraditionalForm=",ExpressionUUID->"707ff051-5e03-4518-8a8f-\
0270220128cb"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "eq4pt1b", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"eq4pt1b", " ", "=", " ", 
   RowBox[{"{", "\[IndentingNewLine]", 
    RowBox[{
     RowBox[{"p", "\[Rule]", " ", 
      RowBox[{"p", "[", 
       RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]}], " ", ",", 
     " ", "\[IndentingNewLine]", 
     RowBox[{"m", "\[Rule]", " ", 
      RowBox[{"m", "[", "\[Sigma]", "]"}]}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{"K", " ", "->", " ", 
      RowBox[{"K", "[", 
       RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]}], " ", ",", 
     " ", "\[IndentingNewLine]", 
     RowBox[{"H", "->", " ", 
      RowBox[{"H", "[", 
       RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]}]}], " ", 
    "\[IndentingNewLine]", "}"}]}], " ", ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"eq4pt1b", " ", "//", " ", "TableForm"}], " ", "//", " ", "pdConv", 
  " "}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.8234544722280283`*^9, 3.823454527817523*^9}, 
   3.8430857493719378`*^9, {3.855532145155263*^9, 3.855532150538891*^9}, {
   3.8555322632350683`*^9, 3.8555322648664007`*^9}, {3.855532356363476*^9, 
   3.8555323951879473`*^9}},
 CellLabel->"In[33]:=",ExpressionUUID->"ad4e246e-62eb-4117-9c8d-9d4ded7857e7"],

Cell[BoxData[
 FormBox[
  TagBox[
   TagBox[GridBox[{
      {
       RowBox[{"p", "\[Rule]", 
        RowBox[{"p", "(", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}]}]},
      {
       RowBox[{"m", "\[Rule]", 
        RowBox[{"m", "(", "\[Sigma]", ")"}]}]},
      {
       RowBox[{"K", "\[Rule]", 
        RowBox[{"K", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]}]},
      {
       RowBox[{"H", "\[Rule]", 
        RowBox[{"H", "(", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], ")"}]}]}
     },
     GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
     GridBoxSpacings->{"Columns" -> {
         Offset[0.27999999999999997`], {
          Offset[0.5599999999999999]}, 
         Offset[0.27999999999999997`]}, "Rows" -> {
         Offset[0.2], {
          Offset[0.4]}, 
         Offset[0.2]}}],
    Column],
   Function[BoxForm`e$, 
    TableForm[BoxForm`e$]]], TraditionalForm]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{
  3.8234545281910677`*^9, 3.8430857493721867`*^9, {3.8555322602913437`*^9, 
   3.855532294537327*^9}, 3.855532335539019*^9, 3.855532397260092*^9},
 CellLabel->
  "Out[35]//TraditionalForm=",ExpressionUUID->"7601a5ee-2907-4a83-9673-\
6a85582f52f4"],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"lineToMetric", "[", " ", 
    RowBox[{"eq4pt1", " ", ",", " ", 
     RowBox[{"{", 
      RowBox[{
      "d\[Xi]", ",", "d\[Eta]", ",", "d\[CurlyRho]", ",", "d\[Sigma]"}], 
      "}"}]}], "]"}], "  ", "/.", " ", "eq4pt1b"}], "  ", "//", " ", 
  "MatrixForm"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.855532303218793*^9, 3.855532304049955*^9}, 
   3.8555324986546*^9},
 CellLabel->"In[36]:=",ExpressionUUID->"85cace78-0533-4e72-84c6-3ea2c29ea00f"],

Cell[BoxData[
 TagBox[
  RowBox[{"(", "\[NoBreak]", GridBox[{
     {
      FractionBox[
       SuperscriptBox["\[CurlyRho]", "2"], 
       SuperscriptBox[
        RowBox[{"p", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "2"]], "0",
       "0", "0"},
     {"0", 
      FractionBox[
       SuperscriptBox["\[CurlyRho]", "2"], 
       SuperscriptBox[
        RowBox[{"p", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "2"]], "0",
       "0"},
     {"0", "0", "0", 
      RowBox[{"-", "1"}]},
     {"0", "0", 
      RowBox[{"-", "1"}], 
      RowBox[{
       RowBox[{"2", " ", "\[CurlyRho]", " ", 
        RowBox[{"H", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]}], "-", 
       RowBox[{"K", "[", 
        RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "+", 
       FractionBox[
        RowBox[{"2", " ", 
         RowBox[{"m", "[", "\[Sigma]", "]"}]}], "\[CurlyRho]"]}]}
    },
    GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}},
    GridBoxSpacings->{"Columns" -> {
        Offset[0.27999999999999997`], {
         Offset[0.7]}, 
        Offset[0.27999999999999997`]}, "Rows" -> {
        Offset[0.2], {
         Offset[0.4]}, 
        Offset[0.2]}}], "\[NoBreak]", ")"}],
  Function[BoxForm`e$, 
   MatrixForm[BoxForm`e$]]]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.85553229454873*^9, 3.855532304662757*^9}, 
   3.855532340444972*^9, 3.855532402067403*^9, 3.855532498655118*^9},
 CellLabel->
  "Out[36]//MatrixForm=",ExpressionUUID->"487a4a22-7390-4200-8089-\
b210c31d1591"],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"lineToMetric", "[", " ", 
      RowBox[{"eq4pt1", " ", ",", " ", 
       RowBox[{"{", 
        RowBox[{
        "d\[Xi]", ",", "d\[Eta]", ",", "d\[CurlyRho]", ",", "d\[Sigma]"}], 
        "}"}]}], "]"}], "  ", "/.", " ", "eq4pt1b"}], "    ", "/.", " ", 
    "kReplace"}], " ", "/.", " ", "hReplace"}], " ", "//", " ", 
  "MatrixForm"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{{3.85553241762607*^9, 3.8555324235941343`*^9}, 
   3.855532498655353*^9},
 CellLabel->"In[37]:=",ExpressionUUID->"e8f7cd0b-f315-4a77-942f-fbac98f0e489"],

Cell[BoxData[
 TagBox[
  RowBox[{"(", "\[NoBreak]", GridBox[{
     {
      FractionBox[
       SuperscriptBox["\[CurlyRho]", "2"], 
       SuperscriptBox[
        RowBox[{"p", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "2"]], "0",
       "0", "0"},
     {"0", 
      FractionBox[
       SuperscriptBox["\[CurlyRho]", "2"], 
       SuperscriptBox[
        RowBox[{"p", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "2"]], "0",
       "0"},
     {"0", "0", "0", 
      RowBox[{"-", "1"}]},
     {"0", "0", 
      RowBox[{"-", "1"}], 
      RowBox[{
       FractionBox[
        RowBox[{"2", " ", 
         RowBox[{"m", "[", "\[Sigma]", "]"}]}], "\[CurlyRho]"], "+", 
       FractionBox[
        RowBox[{"2", " ", "\[CurlyRho]", " ", 
         RowBox[{
          SuperscriptBox["p", 
           TagBox[
            RowBox[{"(", 
             RowBox[{"0", ",", "0", ",", "1"}], ")"}],
            Derivative],
           MultilineFunction->None], "[", 
          RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]}], 
        RowBox[{"p", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]], "+", 
       SuperscriptBox[
        RowBox[{
         SuperscriptBox["p", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"0", ",", "1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "2"], "-", 
       
       RowBox[{
        RowBox[{"p", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], " ", 
        RowBox[{
         SuperscriptBox["p", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"0", ",", "2", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]}], "+", 
       SuperscriptBox[
        RowBox[{
         SuperscriptBox["p", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "0", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], "2"], "-", 
       
       RowBox[{
        RowBox[{"p", "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}], " ", 
        RowBox[{
         SuperscriptBox["p", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"2", ",", "0", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"\[Xi]", ",", "\[Eta]", ",", "\[Sigma]"}], "]"}]}]}]}
    },
    GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}},
    GridBoxSpacings->{"Columns" -> {
        Offset[0.27999999999999997`], {
         Offset[0.7]}, 
        Offset[0.27999999999999997`]}, "Rows" -> {
        Offset[0.2], {
         Offset[0.4]}, 
        Offset[0.2]}}], "\[NoBreak]", ")"}],
  Function[BoxForm`e$, 
   MatrixForm[BoxForm`e$]]]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.},
 CellChangeTimes->{3.8555324240988894`*^9, 3.855532498655754*^9},
 CellLabel->
  "Out[37]//MatrixForm=",ExpressionUUID->"bf2a7bb9-c5d2-4a92-a1df-\
db7091294368"]
}, Closed]]
},
WindowSize->{1276, 675},
WindowMargins->{{Automatic, 1}, {Automatic, 80}},
FrontEndVersion->"13.0 for Mac OS X x86 (64-bit) (December 2, 2021)",
StyleDefinitions->FrontEnd`FileName[{"Report"}, "StandardReport.nb", 
  CharacterEncoding -> "UTF-8"],
ExpressionUUID->"52ce3f80-43c2-4be4-9b2b-5ee0b99b3a72"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[580, 22, 963, 15, 221, "Title",ExpressionUUID->"c20da370-3896-41d0-9717-1bb2c5220b1c",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[1546, 39, 1285, 22, 94, "Author",ExpressionUUID->"7155b898-d8cc-45d1-a4a8-b77f156a8efc",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}]
}, Open  ]],
Cell[CellGroupData[{
Cell[2868, 66, 228, 4, 69, "Section",ExpressionUUID->"053b918f-2261-4176-962e-30961c5fe519",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[3099, 72, 438, 10, 63, "Input",ExpressionUUID->"22aa5f92-723b-4052-832d-cd1f0f007d2a",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[3540, 84, 927, 16, 43, "Output",ExpressionUUID->"a92ab7e0-ddc5-43b4-89e1-0a1d787fdf6d",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}]
}, Closed]],
Cell[CellGroupData[{
Cell[4504, 105, 544, 9, 55, "Section",ExpressionUUID->"6aa87865-d9d1-440a-b010-a6ae8bfe0b3c",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[5051, 116, 606, 12, 63, "Input",ExpressionUUID->"85bd99cb-0440-4d61-a3eb-c95a2e4adef3",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[5660, 130, 657, 12, 43, "Output",ExpressionUUID->"87e9decd-7b98-4d8c-b9ea-025a155541be",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[6320, 144, 588, 11, 84, "Input",ExpressionUUID->"1eed751f-ebd0-40b9-960d-53879f3276ec",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[6911, 157, 967, 18, 43, "Output",ExpressionUUID->"ac648e2a-435f-4baa-828e-981f32db640a",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}]
}, Closed]],
Cell[CellGroupData[{
Cell[7915, 180, 228, 4, 55, "Section",ExpressionUUID->"d9f48d09-18d6-4716-8349-81a622ec457e",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[8146, 186, 1645, 45, 189, "Input",ExpressionUUID->"b6d8af22-2efe-47f7-81cb-20865cad30c8",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[9794, 233, 1143, 20, 24, "Print",ExpressionUUID->"d835a44a-82f3-4465-87e9-a6a2294eb2b1",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[10940, 255, 1162, 21, 24, "Print",ExpressionUUID->"21677a79-9854-4048-94fd-0348ac460784",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[12105, 278, 1529, 24, 84, "Output",ExpressionUUID->"41925857-1790-4280-89ea-8a8cc3782863",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[13637, 304, 246, 4, 41, "Input",ExpressionUUID->"ba39d9fc-63f4-4084-9b25-3b2b2660e10d",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}]
}, Closed]],
Cell[CellGroupData[{
Cell[13920, 313, 240, 4, 55, "Section",ExpressionUUID->"cdcf7fc4-5b5a-4d86-936f-704c66c3f570",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[14163, 319, 2714, 61, 441, "Input",ExpressionUUID->"24d6d47b-fc02-4b2a-a506-ebb60170cc3c",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[16880, 382, 2404, 74, 339, "Output",ExpressionUUID->"4bd01435-aefe-4ee3-999f-4b882ab80e3d",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}]
}, Closed]],
Cell[CellGroupData[{
Cell[19321, 461, 362, 6, 55, "Section",ExpressionUUID->"f062add6-b0a7-4986-8a1d-8f6eafb02689",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.}],
Cell[19686, 469, 1659, 41, 126, "Input",ExpressionUUID->"788fc160-dcff-4444-9960-a4033035a0de",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.}],
Cell[21348, 512, 1055, 26, 84, "Input",ExpressionUUID->"5238ea2a-5832-4be1-bd00-5ea6e3219e50",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.}]
}, Closed]],
Cell[CellGroupData[{
Cell[22440, 543, 455, 8, 93, "Section",ExpressionUUID->"cf29f344-6614-4cb9-aad8-9a3b1ae660fa",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[22898, 553, 970, 23, 111, "Input",ExpressionUUID->"16283699-9076-4668-8929-b40f2fa8cc04",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[23871, 578, 799, 22, 66, "Output",ExpressionUUID->"fa5e624a-7eda-4220-82fe-e805bedabbec",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[24673, 602, 491, 11, 41, "Input",ExpressionUUID->"f4485dbd-7a3f-4d0c-bcb5-ae55928e6b67",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[25167, 615, 1223, 36, 143, "Output",ExpressionUUID->"09f8ae0c-745d-4a08-b507-5c0ccebe5205",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[26393, 653, 987, 25, 86, "Input",ExpressionUUID->"9e6ec828-0136-4844-bc10-be9fa0961d14",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[27383, 680, 1142, 26, 105, "Input",ExpressionUUID->"0565ece6-7d70-49d5-aea1-6f6175a91406",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[28528, 708, 1905, 52, 76, "Output",ExpressionUUID->"f1e5013c-88ee-499d-ace8-c4358c9702c2",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[30436, 762, 825, 18, 105, "Input",ExpressionUUID->"c7cbdcdf-7c0b-40c3-a2d2-101ca85aeddd",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[31264, 782, 772, 20, 79, "Output",ExpressionUUID->"707ff051-5e03-4518-8a8f-0270220128cb",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[32039, 804, 1341, 29, 189, "Input",ExpressionUUID->"ad4e246e-62eb-4117-9c8d-9d4ded7857e7",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[33383, 835, 1295, 37, 100, "Output",ExpressionUUID->"7601a5ee-2907-4a83-9673-6a85582f52f4",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[34681, 874, 540, 13, 41, "Input",ExpressionUUID->"85cace78-0533-4e72-84c6-3ea2c29ea00f",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[35224, 889, 1636, 46, 143, "Output",ExpressionUUID->"487a4a22-7390-4200-8089-b210c31d1591",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[36863, 937, 647, 16, 41, "Input",ExpressionUUID->"e8f7cd0b-f315-4a77-942f-fbac98f0e489",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}],
Cell[37513, 955, 3219, 94, 147, "Output",ExpressionUUID->"bf2a7bb9-c5d2-4a92-a1df-db7091294368",
 CellGroupingRules->{"GroupTogetherGrouping", 10004.}]
}, Closed]]
}
]
*)