(* 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[ 7213, 216] NotebookOptionsPosition[ 6594, 196] NotebookOutlinePosition[ 7035, 214] CellTagsIndexPosition[ 6992, 211] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"\[Epsilon]r", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Epsilon]", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "\[Epsilon]", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "\[Epsilon]"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Epsilon]c", "=", RowBox[{"\[Epsilon]", "*", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Rho]", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"1", "/", "\[Rho]"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "\[Rho]"}], "}"}]}], "}"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"A", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"1", "/", "s"}], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"1", "/", "\[Gamma]"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"detA", "=", RowBox[{"Det", "[", "A", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[Epsilon]c2", "=", RowBox[{ RowBox[{"Dot", "[", RowBox[{ RowBox[{"Dot", "[", RowBox[{"A", ",", "\[Epsilon]c"}], "]"}], ",", "A"}], "]"}], "/", "detA"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Acr", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", RowBox[{ RowBox[{"-", "\[Rho]"}], "*", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Sin", "[", "\[Theta]", "]"}], ",", RowBox[{"\[Rho]", "*", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Acrt", "=", RowBox[{"Transpose", "[", "Acr", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"detAcr", "=", "\[Rho]"}], ";"}], "\[IndentingNewLine]", RowBox[{"\[Epsilon]r2", "=", RowBox[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Dot", "[", RowBox[{ RowBox[{"Dot", "[", RowBox[{"Acr", ",", "\[Epsilon]c2"}], "]"}], ",", "Acrt"}], "]"}], "/", "detAcr"}], "]"}], "//", "MatrixForm"}]}]}], "Input", CellChangeTimes->{{3.8784402217024097`*^9, 3.8784402453104086`*^9}, { 3.878440326058539*^9, 3.8784403965948486`*^9}, {3.8784404366522737`*^9, 3.878440456332716*^9}, {3.878594347639612*^9, 3.8785943929514537`*^9}, { 3.8785944282330933`*^9, 3.8785944782155066`*^9}}, CellLabel->"In[20]:=",ExpressionUUID->"0a0210ff-e47b-4200-860b-f3e9b65237a9"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ FractionBox[ RowBox[{"\[Gamma]", " ", "\[Epsilon]", " ", "\[Rho]"}], "s"], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", FractionBox[ RowBox[{"s", " ", "\[Epsilon]"}], RowBox[{"\[Gamma]", " ", "\[Rho]"}]], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", RowBox[{"s", " ", "\[Gamma]", " ", "\[Epsilon]", " ", "\[Rho]"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.878440326964399*^9, {3.878440381710183*^9, 3.878440397927933*^9}, { 3.8784404393473454`*^9, 3.878440459466902*^9}, 3.878594394555008*^9, 3.878594431126652*^9, 3.8785944825771284`*^9}, CellLabel->"Out[24]=",ExpressionUUID->"6094801b-f11b-4c7b-bc86-15fe3803222c"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ FractionBox[ RowBox[{"\[Gamma]", " ", "\[Epsilon]", " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"]}], "s"], "+", FractionBox[ RowBox[{"s", " ", "\[Epsilon]", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}], "\[Gamma]"]}], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", SuperscriptBox["s", "2"]}], "+", SuperscriptBox["\[Gamma]", "2"]}], ")"}], " ", "\[Epsilon]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}], " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], RowBox[{"s", " ", "\[Gamma]"}]], "0"}, { FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", SuperscriptBox["s", "2"]}], "+", SuperscriptBox["\[Gamma]", "2"]}], ")"}], " ", "\[Epsilon]", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}], " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], RowBox[{"s", " ", "\[Gamma]"}]], RowBox[{ FractionBox[ RowBox[{"s", " ", "\[Epsilon]", " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"]}], "\[Gamma]"], "+", FractionBox[ RowBox[{"\[Gamma]", " ", "\[Epsilon]", " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}], "s"]}], "0"}, {"0", "0", RowBox[{"s", " ", "\[Gamma]", " ", "\[Epsilon]"}]} }, 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", CellChangeTimes->{ 3.878440326964399*^9, {3.878440381710183*^9, 3.878440397927933*^9}, { 3.8784404393473454`*^9, 3.878440459466902*^9}, 3.878594394555008*^9, 3.878594431126652*^9, 3.8785944826359067`*^9}, CellLabel-> "Out[28]//MatrixForm=",ExpressionUUID->"fb56cc87-39eb-4552-bc2a-\ 7ffcf5c67922"] }, Open ]] }, WindowSize->{999, 959}, WindowMargins->{{68, Automatic}, {Automatic, 6}}, Magnification:>1.4 Inherited, FrontEndVersion->"12.1 for Microsoft Windows (64-bit) (2020\:5e743\:670814\ \:65e5)", StyleDefinitions->"Default.nb", ExpressionUUID->"a4b3b212-6991-446e-81e9-1060d51d7038" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 2899, 84, 377, "Input",ExpressionUUID->"0a0210ff-e47b-4200-860b-f3e9b65237a9"], Cell[3482, 108, 813, 21, 65, "Output",ExpressionUUID->"6094801b-f11b-4c7b-bc86-15fe3803222c"], Cell[4298, 131, 2280, 62, 193, "Output",ExpressionUUID->"fb56cc87-39eb-4552-bc2a-7ffcf5c67922"] }, Open ]] } ] *)