(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 129971, 3544]*) (*NotebookOutlinePosition[ 130617, 3566]*) (* CellTagsIndexPosition[ 130573, 3562]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(\(\(<< Calculus`VectorAnalysis`\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(SetCoordinates[Cartesian[x, y, z]]\)], "Input"], Cell[BoxData[ \(Cartesian[x, y, z]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CoordinateSystem\)], "Input"], Cell[BoxData[ \(Cartesian\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ksi = {ksix, ksiy, ksiz} E^\((I\ \((kx\ x\ + \ ky\ y\ + \ kz\ z)\))\)\)], "Input"], Cell[BoxData[ \({\[ExponentialE]\^\(\[ImaginaryI]\ \((kx\ x + ky\ y + kz\ z)\)\)\ ksix, \ \[ExponentialE]\^\(\[ImaginaryI]\ \((kx\ x + ky\ y + kz\ z)\)\)\ ksiy, \ \[ExponentialE]\^\(\[ImaginaryI]\ \((kx\ x + ky\ y + kz\ z)\)\)\ ksiz}\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B = {0, 0, B0}\)], "Input"], Cell[BoxData[ \({0, 0, B0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Cross[Curl[Curl[Cross[ksi, B]]], B]/ E^\((I\ \((kx\ x\ + \ ky\ y\ + \ kz\ z)\))\) // Simplify\)], "Input"], Cell[BoxData[ \({\(-B0\^2\)\ \((ksiy\ kx\ ky + ksix\ \((kx\^2 + kz\^2)\))\), \(-B0\^2\)\ \((ksix\ kx\ ky + ksiy\ \((ky\^2 + kz\^2)\))\), 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(vA = B0^2\ {{kx^2 + kz^2, kx\ ky, \ 0}, {kx\ ky, \ ky^2\ + \ kz^2\ , 0}, {0, 0, 0}}\)], "Input"], Cell[BoxData[ \({{B0\^2\ \((kx\^2 + kz\^2)\), B0\^2\ kx\ ky, 0}, {B0\^2\ kx\ ky, B0\^2\ \((ky\^2 + kz\^2)\), 0}, {0, 0, 0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[vA]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(B0\^2\ \((kx\^2 + kz\^2)\)\), \(B0\^2\ kx\ ky\), "0"}, {\(B0\^2\ kx\ ky\), \(B0\^2\ \((ky\^2 + kz\^2)\)\), "0"}, {"0", "0", "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Eigenvalues\ [vA]\)], "Input"], Cell[BoxData[ \({0, B0\^2\ kz\^2, B0\^2\ kx\^2 + B0\^2\ ky\^2 + B0\^2\ kz\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(-Grad[gamma\ p\ Div[ksi]]\)/ E^\((I\ \((kx\ x\ + \ ky\ y\ + \ kz\ z)\))\) // Simplify\)], "Input"], Cell[BoxData[ \({gamma\ kx\ \((ksix\ kx + ksiy\ ky + ksiz\ kz)\)\ p, gamma\ ky\ \((ksix\ kx + ksiy\ ky + ksiz\ kz)\)\ p, gamma\ kz\ \((ksix\ kx + ksiy\ ky + ksiz\ kz)\)\ p}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cs = gamma\ p {{kx\ kx, \ kx\ ky, \ kx\ kz}, {kx\ ky, \ ky\ ky, \ ky\ kz}, \ {kz\ kx, \ kz\ ky, \ kz\ kz}}\)], "Input"], Cell[BoxData[ \({{gamma\ kx\^2\ p, gamma\ kx\ ky\ p, gamma\ kx\ kz\ p}, {gamma\ kx\ ky\ p, gamma\ ky\^2\ p, gamma\ ky\ kz\ p}, {gamma\ kx\ kz\ p, gamma\ ky\ kz\ p, gamma\ kz\^2\ p}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[cs]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(gamma\ kx\^2\ p\), \(gamma\ kx\ ky\ p\), \(gamma\ kx\ kz\ \ p\)}, {\(gamma\ kx\ ky\ p\), \(gamma\ ky\^2\ p\), \(gamma\ ky\ kz\ \ p\)}, {\(gamma\ kx\ kz\ p\), \(gamma\ ky\ kz\ p\), \(gamma\ kz\^2\ p\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Eigenvalues\ [cs]\)], "Input"], Cell[BoxData[ \({0, 0, gamma\ kx\^2\ p + gamma\ ky\^2\ p + gamma\ kz\^2\ p}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(roots = Eigenvalues[vA + cs]\)], "Input"], Cell[BoxData[ \({B0\^2\ kz\^2, 1\/2\ \((B0\^2\ kx\^2 + B0\^2\ ky\^2 + B0\^2\ kz\^2 + gamma\ kx\^2\ p + gamma\ ky\^2\ p + gamma\ kz\^2\ p - \[Sqrt]\((\((\(-B0\^2\)\ kx\^2 - B0\^2\ ky\^2 - \ B0\^2\ kz\^2 - gamma\ kx\^2\ p - gamma\ ky\^2\ p - gamma\ kz\^2\ p)\)\^2 - 4\ \((B0\^2\ gamma\ kx\^2\ kz\^2\ p + B0\^2\ gamma\ ky\^2\ kz\^2\ p + B0\^2\ gamma\ kz\^4\ p)\))\))\), 1\/2\ \((B0\^2\ kx\^2 + B0\^2\ ky\^2 + B0\^2\ kz\^2 + gamma\ kx\^2\ p + gamma\ ky\^2\ p + gamma\ kz\^2\ p + \[Sqrt]\((\((\(-B0\^2\)\ kx\^2 - B0\^2\ ky\^2 - \ B0\^2\ kz\^2 - gamma\ kx\^2\ p - gamma\ ky\^2\ p - gamma\ kz\^2\ p)\)\^2 - 4\ \((B0\^2\ gamma\ kx\^2\ kz\^2\ p + B0\^2\ gamma\ ky\^2\ kz\^2\ p + B0\^2\ gamma\ kz\^4\ p)\))\))\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[roots /. p \[Rule] 0, \ Assumptions \[Rule] {Element[B0, Reals], Element[kz, Reals], Element[ky, Reals], Element[kx, Reals]}]\)], "Input"], Cell[BoxData[ \({B0\^2\ kz\^2, 0, B0\^2\ \((kx\^2 + ky\^2 + kz\^2)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[roots /. 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