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NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 34571, 662]*) (*NotebookOutlinePosition[ 35277, 686]*) (* CellTagsIndexPosition[ 35233, 682]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(Off[General::"\"]; Off[General::"\"];\)], "Input"], Cell[CellGroupData[{ Cell[TextData[StyleBox["1. The functions describing the potential \ distribution in and around a spherical cell with a concentric organelle", FontWeight->"Bold"]], "Subsubtitle"], Cell[BoxData[{ \(\(\[CapitalPsi]1i[r_, \[Theta]_] := A1i\ r\ Cos[\[Theta]];\)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ 0\ \ \[LessEqual] \ r\ \[LessEqual] \ Ra\ *) \), "\[IndentingNewLine]", \(\(\[CapitalPsi]1m[r_, \[Theta]_] := \((A1m\ r + B1m\/r\^2)\) Cos[\[Theta]];\)\ \ \ \ \ \ \ (*\ Ra\ \[LessEqual] \ r\ \[LessEqual] \ Rb\ *) \), "\[IndentingNewLine]", \(\(\[CapitalPsi]2i[r_, \[Theta]_] := \((A2i\ r\ + B2i\/r\^2)\) Cos[\[Theta]];\)\ \ \ \ \ \ \ (*\ Rb\ \[LessEqual] \ r\ \[LessEqual] \ Rc\ *) \), "\[IndentingNewLine]", \(\(\[CapitalPsi]2m[r_, \[Theta]_] := \((A2m\ r + B2m\/r\^2)\) Cos[\[Theta]];\)\ \ \ \ \ \ (*\ Rc\ \[LessEqual] \ r\ \[LessEqual] \ Rd\ *) \), "\[IndentingNewLine]", \(\(\(\(\[CapitalPsi]e[ r_, \[Theta]_]\ \ \ \ := \((\(-E0\)\ r + Be\/r\^2)\) Cos[\[Theta]]\) \);\)\ \ \ \ \ \ \ (*\ Rd\ \[LessEqual] \ r\ \[LessEqual] \ \[Infinity]\ \ *) \), "\[IndentingNewLine]", \(\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["2. The constants in the functions describing the \ potential distribution", FontWeight->"Bold"]], "Subsubtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(ConstantsSolution = Flatten[FullSimplify[\[IndentingNewLine]Solve[{\ A1i\ Ra == A1m\ Ra + B1m\/Ra\^2, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ A1m\ Rb + B1m\/Rb\^2 \[Equal] A2i\ Rb + B2i\/Rb\^2, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ A2i\ Rc + B2i\/Rc\^2 \[Equal] A2m\ Rc + B2m\/Rc\^2, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ A2m\ Rd + B2m\/Rd\^2 == \(-E0\)\ Rd + Be\/Rd\^2, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \[CapitalLambda]1i\ A1i \[Equal] \[CapitalLambda]1m\ \((A1m - \(2\ \ B1m\)\/Ra\^3)\), \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \[CapitalLambda]1m\ \((A1m - \(2 B1m\)\/Rb\^3)\) \[Equal] \[CapitalLambda]2i \ \((A2i - \(2 B2i\)\/Rb\^3)\), \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \[CapitalLambda]2i\ \((A2i - \(2 B2i\)\/Rc\^3)\) \[Equal] \[CapitalLambda]2m \ \((A2m - \(2 B2m\)\/Rc\^3)\), \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \[CapitalLambda]2m\ \((A2m - \(2 B2m\)\/Rd\^3)\) \[Equal] \[CapitalLambda]e \ \((\(-E0\) - \(2 Be\)\/Rd\^3)\)}, \[IndentingNewLine]{A1i, A1m, B1m, A2i, B2i, A2m, B2m, Be}]]]\)], "Input"], Cell[BoxData[ \({A1i \[Rule] \(-\(\((81\ E0\ Rb\^3\ Rc\^3\ Rd\^3\ \[CapitalLambda]1m\ \ \[CapitalLambda]2i\ \[CapitalLambda]2m\ \[CapitalLambda]e)\)/\((2\ Ra\^3\ \((\ \[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \((Rb\^3\ \((2\ \ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \((\ \[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\)\)\), Be \[Rule] 1\/2\ E0\ Rd\^3\ \((\(-1\) + \((3\ \[CapitalLambda]2m\ \((2\ Ra\^3\ \ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \((Rb\^3\ \((2\ \ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\) + Rc\^3\ \((2\ \[CapitalLambda]2i + \ \[CapitalLambda]2m)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \((\ \[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\) + Rc\^3\ \((2\ \[CapitalLambda]2i + \ \[CapitalLambda]2m)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\))\))\))\))\)/\ \((2\ Ra\^3\ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \((Rb\^3\ \((2\ \ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \((\ \[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\))\), A1m \[Rule] \(-\(\((27\ E0\ Rb\^3\ Rc\^3\ Rd\^3\ \((\[CapitalLambda]1i \ + 2\ \[CapitalLambda]1m)\)\ \[CapitalLambda]2i\ \[CapitalLambda]2m\ \ \[CapitalLambda]e)\)/\((2\ Ra\^3\ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\ \)\ \((Rb\^3\ \((2\ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((Rc\^3\ \ \((2\ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \((\ \[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\)\)\), B1m \[Rule] \((27\ E0\ Ra\^3\ Rb\^3\ Rc\^3\ Rd\^3\ \ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \[CapitalLambda]2i\ \ \[CapitalLambda]2m\ \[CapitalLambda]e)\)/\((2\ Ra\^3\ \((\[CapitalLambda]1i - \ \[CapitalLambda]1m)\)\ \((Rb\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \[CapitalLambda]2i + \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\), A2i \[Rule] \(-\(\((9\ E0\ Rc\^3\ Rd\^3\ \((2\ Ra\^3\ \((\ \[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\))\)\ \[CapitalLambda]2m\ \ \[CapitalLambda]e)\)/\((2\ Ra\^3\ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\ \)\ \((Rb\^3\ \((2\ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((Rc\^3\ \ \((2\ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \((\ \[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\)\)\), B2i \[Rule] \((9\ E0\ Rb\^3\ Rc\^3\ Rd\^3\ \((Rb\^3\ \ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\) + Ra\^3\ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \((2\ \ \[CapitalLambda]1m + \[CapitalLambda]2i)\))\)\ \[CapitalLambda]2m\ \ \[CapitalLambda]e)\)/\((2\ Ra\^3\ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\ \)\ \((Rb\^3\ \((2\ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((Rc\^3\ \ \((2\ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\), A2m \[Rule] \(1\/\(\[CapitalLambda]2m + 2\ \[CapitalLambda]e\)\) \((3\ E0\ \[CapitalLambda]e\ \((\(-1\ \) + \((2\ Rc\^3\ \((Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((Rc\^3\ \((\ \[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\) + Rb\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\)\ \((2\ \[CapitalLambda]2i + \[CapitalLambda]2m)\))\) + Ra\^3\ \((\[CapitalLambda]1i - \ \[CapitalLambda]1m)\)\ \((2\ Rc\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\)\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\) + Rb\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\)\ \((2\ \[CapitalLambda]2i + \[CapitalLambda]2m)\))\))\)\ \ \((\[CapitalLambda]2m - \[CapitalLambda]e)\))\)/\((2\ Ra\^3\ \((\ \[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \((Rb\^3\ \((2\ \ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \((\ \[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\))\))\), B2m \[Rule] \(-\(\((3\ E0\ Rc\^3\ Rd\^3\ \((\(-Rb\^3\)\ \((\ \[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((Rc\^3\ \((\ \[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\) + Rb\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\)\ \((2\ \[CapitalLambda]2i + \[CapitalLambda]2m)\))\) - Ra\^3\ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \ \((2\ Rc\^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\) + Rb\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\)\ \((2\ \[CapitalLambda]2i + \[CapitalLambda]2m)\))\))\)\ \ \[CapitalLambda]e)\)/\((2\ Ra\^3\ \((\[CapitalLambda]1i - \ \[CapitalLambda]1m)\)\ \((Rb\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \[CapitalLambda]2i + \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \ \((2\ Rc\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m - \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + Rb\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ Rb\^3\ \((\ \[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((Rc\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + Rc\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ Rc\^3\ \((\ \[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + Rd\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalDelta]\[CapitalPsi]Cell = FullSimplify[\[CapitalPsi]2m[Rc, \[Theta]] - \[CapitalPsi]2m[ Rd, \[Theta]] /. ConstantsSolution] /. {Ra \[Rule] \((R1 - d1)\), Rb \[Rule] R1, Rc \[Rule] \((R2 - d2)\), Rd \[Rule] R2}\)], "Input"], Cell[BoxData[ \(\(-\(\((3\ d2\ E0\ R2\ \((\((\(-d1\) + R1)\)\^3\ \ \((\(-\[CapitalLambda]1i\) + \[CapitalLambda]1m)\)\ \((2\ \((R2\^2 + R2\ \((\(-d2\) + R2)\) + \((\(-d2\) + R2)\)\^2)\)\ \ \[CapitalLambda]2i\ \((\((\(-d2\) + R2)\)\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\) + R1\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\))\) - d2\ \((\(-d2\) + 3\ R2)\)\ \((2\ \((\(-d2\) + R2)\)\^3\ \((\(-\ \[CapitalLambda]1m\) + \[CapitalLambda]2i)\) + R1\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\))\)\ \[CapitalLambda]2m)\) - R1\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((\((R2\^2 + R2\ \((\(-d2\) + R2)\) + \((\(-d2\) + R2)\)\^2)\)\ \ \[CapitalLambda]2i\ \((2\ R1\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\) + \((\(-d2\) + R2)\)\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\))\) + d2\ \((\(-d2\) + 3\ R2)\)\ \((R1\^3\ \((\(-\[CapitalLambda]1m\) + \ \[CapitalLambda]2i)\) + \((\(-d2\) + R2)\)\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\))\)\ \ \[CapitalLambda]2m)\))\)\ \[CapitalLambda]e\ Cos[\[Theta]])\)/\((2\ \ \((\(-d1\) + R1)\)\^3\ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \ \((R1\^3\ \((2\ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((\((\(-d2\) + \ R2)\)\^3\ \((2\ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\ \[CapitalLambda]2m - \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + \((\(-d2\) + R2)\)\ \^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((2\ \((\(-d2\) + \ R2)\)\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m \ + 2\ \[CapitalLambda]e)\))\))\) + R1\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ R1\^3\ \ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((\((\(-d2\) + R2)\)\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + \((\(-d2\) + R2)\)\ \^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ \((\(-d2\) + R2)\)\^3\ \ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m \ + 2\ \[CapitalLambda]e)\))\))\))\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalDelta]\[CapitalPsi]Org = FullSimplify[\[CapitalPsi]1m[Ra, \[Theta]] - \[CapitalPsi]1m[ Rb, \[Theta]] /. ConstantsSolution] /. {Ra \[Rule] \((R1 - d1)\), Rb \[Rule] R1, Rc \[Rule] \((R2 - d2)\), Rd \[Rule] R2}\)], "Input"], Cell[BoxData[ \(\((27\ E0\ R1\ R2\^3\ \((\(-d2\) + R2)\)\^3\ \((\(-3\)\ R1\^2\ \((\(-d1\ \) + R1)\)\ \[CapitalLambda]1m + \((\(-d1\) + R1)\)\^3\ \ \((\(-\[CapitalLambda]1i\) + \[CapitalLambda]1m)\) + R1\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\))\)\ \[CapitalLambda]2i\ \ \[CapitalLambda]2m\ \[CapitalLambda]e\ Cos[\[Theta]])\)/\((2\ \((\(-d1\) + \ R1)\)\^3\ \((\[CapitalLambda]1i - \[CapitalLambda]1m)\)\ \((R1\^3\ \((2\ \ \[CapitalLambda]1m + \[CapitalLambda]2i)\)\ \((\((\(-d2\) + R2)\)\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + \((\(-d2\) + \ R2)\)\^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((2\ \((\(-d2\) + \ R2)\)\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\) + R1\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ R1\^3\ \((\[CapitalLambda]1m \ - \[CapitalLambda]2i)\)\ \((\((\(-d2\) + R2)\)\^3\ \((2\ \[CapitalLambda]2i + \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + \((\(-d2\) + \ R2)\)\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ \((\(-d2\) + R2)\)\^3\ \ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\))\))\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["3. Verification of the \"manual\" simplifications", FontWeight->"Bold"]], "Subsubtitle"], Cell[CellGroupData[{ Cell["The Numerator of \[CapitalDelta]\[CapitalPsi]Cell", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\(-\((3\ d2\ E0\ R2\ \((\((\(-d1\) + R1)\)\^3\ \((\(-\ \[CapitalLambda]1i\) + \[CapitalLambda]1m)\)\ \((2\ \((R2\^2 + R2\ \((\(-d2\) + R2)\) + \((\(-d2\) + R2)\)\^2)\)\ \ \[CapitalLambda]2i\ \((\((\(-d2\) + R2)\)\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\) + R1\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\))\) - d2\ \((\(-d2\) + 3\ R2)\)\ \((2\ \((\(-d2\) + R2)\)\^3\ \((\(-\ \[CapitalLambda]1m\) + \[CapitalLambda]2i)\) + R1\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\))\)\ \[CapitalLambda]2m)\) - R1\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((\((R2\^2 + R2\ \((\(-d2\) + R2)\) + \((\(-d2\) + R2)\)\^2)\)\ \ \[CapitalLambda]2i\ \((2\ R1\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\) + \((\(-d2\) + R2)\)\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\))\) + d2\ \((\(-d2\) + 3\ R2)\)\ \((R1\^3\ \((\(-\[CapitalLambda]1m\) \ + \[CapitalLambda]2i)\) + \((\(-d2\) + R2)\)\^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\))\)\ \ \[CapitalLambda]2m)\))\)\ \[CapitalLambda]e\ )\)\)]\ == Expand[3\ E0\ R2\ d2 \((\(\((R1 - d1)\)\^3\) \((\[CapitalLambda]1i - \ \[CapitalLambda]1m)\) \((2 \((3\ R2\^2 - 3\ d2\ R2 + d2\^2)\) \((\((R2 - d2)\)\^3\ \((\[CapitalLambda]1m \ - \[CapitalLambda]2i)\) + R1\^3\ \((2 \[CapitalLambda]1m + \ \[CapitalLambda]2i)\))\) \[CapitalLambda]2i + \((3\ R2\ d2 - d2\^2)\)\ \((2 \((R2 - d2)\)\^3\ \ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\) - R1\^3\ \((2 \[CapitalLambda]1m + \ \[CapitalLambda]2i)\))\)\ \[CapitalLambda]2m)\) + \(R1\^3\) \((\ \[CapitalLambda]1i + 2 \[CapitalLambda]1m)\) \((\((3\ R2\^2 - 3\ d2\ R2 + d2\^2)\)\ \((\((R2 - d2)\)\^3\ \ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\) + 2\ R1\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\))\) \[CapitalLambda]2i + \((3\ R2\ d2 - d2\^2)\)\ \((\((R2 - d2)\)\^3\ \ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\) - R1\^3\ \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\))\)\ \[CapitalLambda]2m)\))\) \[CapitalLambda]e]\)], \ "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The Numerator of \[CapitalDelta]\[CapitalPsi]Org", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((27\ E0\ R1\ R2\^3\ \((\(-d2\) + R2)\)\^3\ \((\(-3\)\ R1\^2\ \ \((\(-d1\) + R1)\)\ \[CapitalLambda]1m + \((\(-d1\) + R1)\)\^3\ \ \((\(-\[CapitalLambda]1i\) + \[CapitalLambda]1m)\) + R1\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\))\)\ \[CapitalLambda]2i\ \ \[CapitalLambda]2m\ \[CapitalLambda]e\ Cos[\[Theta]])\)] \[Equal] Expand[27 E0\ R1\ \(R2\^3\) d1 \(\((R2 - d2)\)\^3\) \((3 \( R1\^2\) \[CapitalLambda]1i - d1 \((3 R1 - d1)\) \((\[CapitalLambda]1i - \[CapitalLambda]1m)\))\) \ \[CapitalLambda]2i\ \[CapitalLambda]2m\ \[CapitalLambda]e\ Cos[\[Theta]]]\)], \ "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The Denominator", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((2\ \((\(-d1\) + R1)\)\^3\ \((\[CapitalLambda]1i - \ \[CapitalLambda]1m)\)\ \((R1\^3\ \((2\ \[CapitalLambda]1m + \ \[CapitalLambda]2i)\)\ \((\((\(-d2\) + R2)\)\^3\ \((2\ \[CapitalLambda]2i + \ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + \((\(-d2\) + R2)\)\ \^3\ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((2\ \((\(-d2\) + \ R2)\)\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m \ + 2\ \[CapitalLambda]e)\))\))\) + R1\^3\ \((\[CapitalLambda]1i + 2\ \[CapitalLambda]1m)\)\ \((2\ R1\^3\ \ \((\[CapitalLambda]1m - \[CapitalLambda]2i)\)\ \((\((\(-d2\) + R2)\)\^3\ \((2\ \ \[CapitalLambda]2i + \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \ \((\[CapitalLambda]2m + 2\ \[CapitalLambda]e)\))\) + \((\(-d2\) + R2)\)\ \^3\ \((\[CapitalLambda]1m + 2\ \[CapitalLambda]2i)\)\ \((2\ \((\(-d2\) + R2)\)\^3\ \ \((\[CapitalLambda]2i - \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m - \ \[CapitalLambda]e)\) + R2\^3\ \((\[CapitalLambda]2i + 2\ \[CapitalLambda]2m)\)\ \((\[CapitalLambda]2m \ + 2\ \[CapitalLambda]e)\))\))\))\)] \[Equal] Expand[2 \(\((R1 - d1)\)\^3\) \((\[CapitalLambda]1i - \[CapitalLambda]1m)\) \ \((\(\((R2 - d2)\)\^3\) \((\[CapitalLambda]1m - \ \[CapitalLambda]2i)\) \((2 \(\((R2 - d2)\)\^3\) \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\) \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + \ \(R2\^3\) \((\[CapitalLambda]2i + 2 \[CapitalLambda]2m)\) \((\[CapitalLambda]2m + 2 \[CapitalLambda]e)\))\) + \(R1\^3\) \((2 \ \[CapitalLambda]1m + \[CapitalLambda]2i)\) \((\(\((R2 - d2)\)\^3\) \((2 \[CapitalLambda]2i + \ \[CapitalLambda]2m)\) \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + \ \(R2\^3\) \((\[CapitalLambda]2i - \[CapitalLambda]2m)\) \((\[CapitalLambda]2m \ + 2 \[CapitalLambda]e)\))\))\) + \(R1\^3\) \((\[CapitalLambda]1i + 2 \[CapitalLambda]1m)\) \((\(\((R2 - d2)\)\^3\) \((\[CapitalLambda]1m + 2 \[CapitalLambda]2i)\) \((2 \(\((R2 - d2)\)\^3\) \((\[CapitalLambda]2i - \ \[CapitalLambda]2m)\) \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + \ \(R2\^3\) \((\[CapitalLambda]2i + 2 \[CapitalLambda]2m)\) \((\[CapitalLambda]2m + 2 \[CapitalLambda]e)\))\) + 2 \( R1\^3\) \((\[CapitalLambda]1m - \[CapitalLambda]2i)\) \ \((\(\((R2 - d2)\)\^3\) \((2 \[CapitalLambda]2i + \ \[CapitalLambda]2m)\) \((\[CapitalLambda]2m - \[CapitalLambda]e)\) + \ \(R2\^3\) \((\[CapitalLambda]2i - \[CapitalLambda]2m)\) \((\[CapitalLambda]2m \ + 2 \[CapitalLambda]e)\))\))\)]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1152}, {0, 819}}, WindowSize->{1021, 318}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic} ] (******************************************************************* Cached data follows. 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