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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[ 25258, 616]*) (*NotebookOutlinePosition[ 55629, 1681]*) (* CellTagsIndexPosition[ 55558, 1675]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The Maxwell equations in the Light gauge: QED?", "Subtitle", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["doug "]], "Subsubtitle", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Introduction\nThe E and B fields, and the gauge with no name\nThe Maxwell \ equations in the light gauge\nImplications"]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Introduction"]], "Subsection", CellMargins->{{0, 23}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "What makes a theory non-classical? Use an operational definition: a \ classical approach neatly separates the scalar and vector terms of a \ quaternion. Recall how the electric field was defined (where {A, B} is the \ even or symmetric product over 2, and [A, B] is the odd, antisymmetric \ product over two or cross product). "]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[{ RowBox[{"E", " ", "==", StyleBox[" ", FontWeight->"Bold"], RowBox[{ RowBox[{ StyleBox["vector", FontWeight->"Bold"], "[", \(evenop[{q[dt, \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\)]}, \n\t\t\tq[\[Phi][t], \(-\(A\& \[RightVector] \)[t]\)]]\), "]"}], ".", \({1, 0}\)}]}], \(E == {0, \(-\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[Phi][t]\ ) \)\) - \[PartialD]\(A\& \[RightVector] \)[t]\/\[PartialD]t}\), \(B\ == \ oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[\[Phi][t], \(A\& \[RightVector] \)[t]]].{1, 0}\), \(B == {0, \(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(A\& \[RightVector] \)[t]\ \)}\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ RowBox[{ RowBox[{"E", " ", "==", StyleBox[" ", FontWeight->"Bold"], RowBox[{ StyleBox["vector", FontWeight->"Bold"], RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], \(-A\&\[RightVector]\))\)\), StyleBox["}", FontWeight->"Bold"]}]}]}], "=", \((0, \(-\[EmptyDownTriangle]\&\[RightVector]\)\ \[Phi] - \[PartialD]A\&\[RightVector]\/\[PartialD]t)\)}], RowBox[{"B", " ", "=", " ", RowBox[{ RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], A\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}], "=", \((0, \ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]\ ) \)}]}]}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The scalar information is explicitly discarded from the E field quaternion. \ In this notebook, the scalar field that arises will be examined and shown to \ be the field which gives rise to gauge symmetry. The commutators and \ anticommutators of this scalar and vector field do not alter the homogeneous \ terms of the Maxwell equations, but may explain why light is a quantized, \ transverse wave. \ \>", "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ " The E and B fields, and the gauge with no name"]], "Subsection", CellMargins->{{0, 23}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "In the previous notebook, the electric field was generated differently from \ the magnetic field, since the scalar field was discard. This time that will \ not be done."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[{ \(E\ == \ evenop[{q[dt, \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\)]}, q[\[Phi][t], \(-\(A\& \[RightVector] \)[t]\)]].{1, 0}\), \(E == { \(-\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \(A\& \[RightVector] \)[t]\ \ )\)\) + \[PartialD]\[Phi][t]\/\[PartialD]t, \(-\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[Phi][t]\ ) \)\) - \[PartialD]\(A\& \[RightVector] \)[t]\/\[PartialD]t}\), \(B\ == \ oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[\[Phi][t], \(A\& \[RightVector] \)[t]]].{1, 0}\), \(B == {0, \(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(A\& \[RightVector] \)[t]\ \)}\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ RowBox[{"E", " ", "=", RowBox[{ RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], \(-A\&\[RightVector]\))\)\), StyleBox["}", FontWeight->"Bold"]}], "=", \((\(-\ \[EmptyDownTriangle]\&\[RightVector]\)\ \[CenterDot]\ A\&\[RightVector]\ + \[PartialD]\[Phi]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\)\ \[Phi]\ - \[PartialD]A\&\[RightVector]\/\[PartialD]t)\)}]}], RowBox[{"B", " ", "=", RowBox[{ RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], A\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}], "=", \((0, \ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]) \)}]}]}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "What is the name of the scalar field, d phi/dt - Del.A which looks like some \ sort of gauge? It is not the Lorenz or Landau gauge which has a plus sign \ between the two. It is none of the popular gauges: Coulomb (Del.A = 0), \ axial (Az = 0), temporal (phi = 0), Feynman, unitary..."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "The standard definition of a gauge starts with an arbitrary scalar function \ psi. The following substitutions do not effect the resulting equations."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[{ \(\[Phi]\ -> \ \[Phi]\ - \ \[PartialD]\[Psi]\/\[PartialD]t\), \(A\&\[RightVector]\ -> \ A\&\[RightVector]\ + \ \(\[EmptyDownTriangle]\&\[RightVector]\) \[Psi]\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(\[Phi]\ -> \ \[Phi]\ - \ \[PartialD]\[Psi]\/\[PartialD]t\), \(A\&\[RightVector]\ -> \ A\&\[RightVector]\ + \ \(\[EmptyDownTriangle]\&\[RightVector]\) \[Psi]\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "This can be written as one quaternion transformation."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[ \(q[\[Phi], A\&\[RightVector]]\ \ -> \ q[\[Phi], A\&\[RightVector]]\ + \ q[\(-\(\[PartialD]\[Psi]\/\[PartialD]t\)\), \(\[EmptyDownTriangle]\&\[RightVector]\) \[Psi]]\)], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\((\[Phi], A\&\[RightVector])\)\ \ -> \ \((\[Phi], A\&\[RightVector])\)\ + \ \((\(-\(\[PartialD]\[Psi]\/\[PartialD]t\)\), \(\[EmptyDownTriangle]\&\[RightVector]\) \[Psi])\)\)], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "The goal here is to find an arbitrary scalar and a 3-vector that does the \ same work as the scalar function psi. Let "]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[ \(p\ = \ \(\(-\(\[PartialD]\[Psi]\/\[PartialD]t\)\)\ \ \ \ \ \ \ \ \ and\ \ \ \[Alpha]\&\[RightVector]\ = \ \(\[EmptyDownTriangle]\&\[RightVector]\) \[Psi]\)\)], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(p\ = \ \(\(-\(\[PartialD]\[Psi]\/\[PartialD]t\)\)\ \ \ \ \ \ \ \ \ and\ \ \ \[Alpha]\&\[RightVector]\ = \ \(\[EmptyDownTriangle]\&\[RightVector]\) \[Psi]\)\)], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "Look at how the gauge symmetry changes by taking its derivative."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[{ \(op[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[\(-\(d\[Psi]\/dt\)[t]\), \(\[EmptyDownTriangle]\& \[RightVector] \) \[Psi][t]]].{1, 0}\), \({\(- \((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \((\(\[EmptyDownTriangle]\& \[RightVector] \)\ \[Psi][t]) \)\ \ )\)\) - \[PartialD]\(d\[Psi]\/dt\)[t]\/\[PartialD]t, \ \n\t\(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(( \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[Psi][t])\)\ - \(\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \(d\[Psi]\/dt\)[t]\ \) + \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[PartialD]\[Psi][t]\/\[PartialD]t}\n = \ {\(-\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \(\[Alpha]\& \[RightVector] \)\ \ )\)\) + \[PartialD]p\/\[PartialD]t, \ 0}\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\) \((\(-\(\[PartialD]\[Psi]\/\[PartialD]t\)\), \(\[EmptyDownTriangle]\&\[RightVector]\) \[Psi])\)\n = \(\((\(-\ \[EmptyDownTriangle]\&\[RightVector]\)\ \[CenterDot]\[EmptyDownTriangle]\&\[RightVector]\ \[Psi] - \[PartialD]\^2 \[Psi]\/\[PartialD]t\^2, \ \[EmptyDownTriangle]\&\[RightVector]\ X\ \[EmptyDownTriangle]\&\[RightVector]\ \[Psi]\ - \ \[EmptyDownTriangle]\&\[RightVector]\ \[PartialD]\[Psi]\/\[PartialD]t\ + \[EmptyDownTriangle]\&\[RightVector]\ \[PartialD]\[Psi]\/\[PartialD]t)\)\n = \ \((\(-\ \[EmptyDownTriangle]\&\[RightVector]\)\ \[CenterDot]\ \[Alpha]\&\[RightVector] + \[PartialD]p\/\[PartialD]t, \ 0) \)\)\)], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "This is the gauge with no name! Call it the \"light gauge\". That name was \ chosen because if the rate of change in the scalar potential phi is equal to \ the spatial change of the 3-vector potential A as should be the case for a \ photon, the distance is zero."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ " The Maxwell equations in the light gauge"]], "Subsection", CellMargins->{{0, 23}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "The homogeneous terms of The Maxwell equations are formed from the sum of \ both orders of the commutator and anticommutator."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[{ \({0, \ \(0\& \[RightVector] \)} == \((evenop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, \n\t\t\t oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[\[Phi][t], \(A\& \[RightVector] \)[t]]]]\ \n\t\t + \ oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, \n \t\t\t\tevenop[{ q[dt, \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\)]}, q[\[Phi][t], \(-\(A\& \[RightVector] \)[t]\)]]])\).{1, 0}\), RowBox[{\({0, \(0\& \[RightVector] \)}\), "==", RowBox[{"{", RowBox[{ \(-\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(A\& \[RightVector] \)[t]\ )\)\ \ )\)\), ",", " ", RowBox[{ \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\), " ", "X", " ", SuperscriptBox[\((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[Phi][t])\), TagBox["", Derivative], MultilineFunction->None]}]}], "}"}]}]}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], A\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}]}], StyleBox["}", FontWeight->"Bold"]}], " ", "\n", "+", RowBox[{ StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\))\), \((\[Phi], \(-A\&\[RightVector]\))\)\), StyleBox["}", FontWeight->"Bold"]}]}], StyleBox["]", FontWeight->"Bold"]}]}], "\n", "=", RowBox[{ RowBox[{"(", RowBox[{ \(\(-\[EmptyDownTriangle]\&\[RightVector]\)\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]\), " ", ",", RowBox[{ \(-\[EmptyDownTriangle]\&\[RightVector]\), " ", "X", " ", \(\[EmptyDownTriangle]\&\[RightVector]\), " ", SuperscriptBox["\[Phi]", TagBox["", Derivative], MultilineFunction->None]}]}], ")"}], "=", \((0, \ 0\&\[RightVector])\)}]}]], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "The source terms arise from of two commutators and two anticommutators. In \ the classical case discussed in the previous notebook, this involved a \ difference. Here a sum will be used because it generates a simpler \ differential equation."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[{ \({4\ \[Pi]\ \[Rho], 4\ \[Pi]\ \(J\& \[RightVector] \)} == \((oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, \n\t\t\t oddop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, q[\[Phi][t], \(A\& \[RightVector] \)[t]]]]\ \n\t\t + \ evenop[{q[dt, \(\[EmptyDownTriangle]\& \[RightVector] \)]}, \n \t\t\t\tevenop[{ q[dt, \(-\(\[EmptyDownTriangle]\& \[RightVector] \)\)]}, q[\[Phi][t], \(-\(A\& \[RightVector] \)[t]\)]]])\)\n \ \ \ \ \ \ \ \ \ .{1, 0}\), \({4\ \[Pi]\ \[Rho], 4\ \[Pi]\ \(J\& \[RightVector] \)} == {\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[Phi][t]\ + \[PartialD]\^2 \[Phi][t]\/\[PartialD]t\^2, \n\t\t\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ X\ \(A\& \[RightVector] \)[t]\ )\)\ - \ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \((\ \(\[EmptyDownTriangle]\& \[RightVector] \)\ \[CenterDot]\ \(A\& \[RightVector] \)[t])\)\ - \[PartialD]\^2\( A\& \[RightVector] \)[t]\/\[PartialD]t\^2}\n == {\ \(\[EmptyDownTriangle]\& \[RightVector] \)\^2\ \[Phi][t]\ + \[PartialD]\^2 \[Phi][t]\/\[PartialD]t\^2, \(-\ \(\[EmptyDownTriangle]\& \[RightVector] \)\^2\)\ \(A\& \[RightVector] \)[t]\ - \[PartialD]\^2\( A\& \[RightVector] \)[t]\/\[PartialD]t\^2}\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["[", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector])\), \((\[Phi], A\&\[RightVector])\)\), StyleBox["]", FontWeight->"Bold"]}]}], StyleBox["]", FontWeight->"Bold"]}], "\n", "\t", "+", " ", RowBox[{ StyleBox["{", FontWeight->"Bold"], RowBox[{ \((\[PartialD]\/\[PartialD]t, \[EmptyDownTriangle]\&\[RightVector]) \), ",", RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((\[PartialD]\/\[PartialD]t, \(-\[EmptyDownTriangle]\&\[RightVector]\))\), \((\[Phi], \(-A\&\[RightVector]\))\)\), StyleBox["}", FontWeight->"Bold"]}]}], StyleBox["}", FontWeight->"Bold"]}]}], "\n", "=", \(\((\[PartialD]\^2 \[Phi]\/\[PartialD]t\^2 + \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]\ \[EmptyDownTriangle]\&\[RightVector]\ \[Phi]\ , \(-\(\[PartialD]\^2 A\&\[RightVector]\/\[PartialD]t\^2\)\) + \ \[EmptyDownTriangle]\&\[RightVector]\ X\ \ \((\[EmptyDownTriangle]\&\[RightVector]\ X\ A\&\[RightVector]) \) - \ \[EmptyDownTriangle]\&\[RightVector]\ \[EmptyDownTriangle]\&\[RightVector]\[CenterDot]A \&\[RightVector])\)\n = \(\((\ \[PartialD]\^2 \[Phi]\/\[PartialD]t\^2 + \[EmptyDownTriangle]\&\[RightVector]\^2\ \[Phi]\ , \(-\(\[PartialD]\^2 A\&\[RightVector]\/\[PartialD]t\^2\)\) - \ \[EmptyDownTriangle]\&\[RightVector]\^2\ A\&\[RightVector]\ ) \) = 4\ \[Pi]\ \((\[Rho], J\&\[RightVector])\)\)\)}]], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ Notice how the scalar and vector parts have neatly partitioned themselves. \ This is a wave equation, except that a sign is flipped. Here is the equation \ for a longitudinal wave like sound.\ \>", "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[ \(0\ == \ \[EmptyDownTriangle]\&\[RightVector]\^2\ w\&\[RightVector][t]\ - \[PartialD]\^2 w\&\[RightVector][t]\/\[PartialD]t\^2\)], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\[PartialD]\^2 w\&\[RightVector]\/\[PartialD]t\^2 - \ \[EmptyDownTriangle]\&\[RightVector]\^2\ w\&\[RightVector]\ = 0\)], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "The second time derivative of w must be the same as Del^2 w. This has a \ solution which depends on sines and cosines (for simplicity, the details of \ initial and boundary conditions are skipped, and the infinite sum has been \ made finite)."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(w\& \[RightVector] \)[t]\ = \ \[Sum]\+\(n = 1\)\%10 Cos[n\ \[Pi]\ t] Sin[n\ \[Pi]\ R]; \n \[PartialD]\_t\ \(\[PartialD]\_t\ \(w\& \[RightVector] \)[t]\)\ - \[PartialD]\_R\ \(\[PartialD]\_R\ \(w\& \[RightVector] \)[t]\)\), \(0\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(w\&\[RightVector]\ = \ \[Sum]\+\(n = 1\)\%\[Infinity] Cos[n\ \[Pi]\ t] Sin[n\ \[Pi]\ R]\), \(\[PartialD]\_t\ \(\[PartialD]\_t\ w\&\[RightVector]\)\ - \[PartialD]\_R\ \(\[PartialD]\_R\ w\&\[RightVector]\)\ = 0\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "Hit w with two time derivatives, and out comes -n^2 pi^2 w. Take Del^2, and \ that creates the same results. Thus every value of n will satisfy the \ longitudinal wave equation."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "Now to find the solution for the sum of the second time derivative and \ Del^2. One of the signs must be switched by doing some operation twice. \ Sounds like a job for i! With quaternions, the square of a normalized \ 3-vector equals (-1, 0), and it is i if y = z = 0 . The solution to \ Maxwell's equations in the light gauge is"]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(w\& \[RightVector] \)[t]\ = \ \[Sum]\+\(n = 1\)\%10 Cos[n\ \[Pi]\ t] Sin[n\ \[Pi]\ R\ \(V\& \[RightVector] \)]; \n \[PartialD]\_t\ \(\[PartialD]\_t\ \(w\& \[RightVector] \)[t]\) + \[PartialD]\_R\ \(\[PartialD]\_R\ \(w\& \[RightVector] \)[t]\)\ /. \ \(V\& \[RightVector] \) -> \ I\), \(0\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(\(w\& \[RightVector] \)\ = \ \[Sum]\+\(n = 1\)\%\[Infinity] Cos[n\ \[Pi]\ t] Sin[n\ \[Pi]\ R\ \(V\& \[RightVector] \)]\), \(if\ \ \ \(V\& \[RightVector] \)\^2 = \ \(-1\), \ then\ \ \ \[PartialD]\_t\ \(\[PartialD]\_t\ \(w\& \[RightVector] \)\) + \[PartialD]\_R\ \(\[PartialD]\_R\ \(w\& \[RightVector] \)\)\ = 0\)}], "Input", CellMargins->{{18, 23}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox[ "Hit this two time derivatives yields -n^2 pi^2 w. Del^2 w has all of this \ and the normalized phase factor V^2 = (-1,0). V acts like an imaginary phase \ factor that rotates the spatial component. The sum for any n is zero (the \ details of the solution depend on the initial and boundary conditions)."]], "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Implications"]], "Subsection", CellMargins->{{0, 23}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"post"], Cell["\<\ The solution to the Maxwell equations in the light gauge is a superposition \ of waves--each with a separate value of n--where the spatial part gets \ rotated by the 3D analogue of i. That is a quantized, transverse wave. \ That's fortunate, because light is a quantized transverse wave. The \ equations were generated by taking the classical Maxwell equations, and \ making them simpler.\ \>", "Text", CellMargins->{{Inherited, 23}, {Inherited, Inherited}}] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 451}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{580, 270}, WindowMargins->{{14, Automatic}, {Automatic, 18}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{False, Inherited}, PrintingOptions->{"PrintingMargins"->{{72, 57.5625}, {57.5625, 72}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False, "FirstPageHeader"->False, "FirstPageFooter"->False}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Style Definitions"]], "Subtitle"], Cell[TextData[StyleBox[ "Modify the definitions below to change the default appearance of all cells \ in a given style. Make modifications to any definition using commands in the \ Format menu."]], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Style Environment Names"]], "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, ScriptMinSize->12, FontSize->16], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellBracketOptions->{"Margins"->{1, 1}, "Widths"->{0, 5}}, ScriptMinSize->8, FontSize->11], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ScriptMinSize->5, FontSize->10, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Notebook Options"]], "Section", FontFamily->"New York"], Cell[TextData[StyleBox[ "The options defined for the style below will be used at the Notebook \ level."]], "Text", FontFamily->"New York"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, CellFrameLabelMargins->6, StyleMenuListing->None, FontFamily->"New York"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Headings"]], "Section", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellMargins->{{7, Inherited}, {8, 40}}, Evaluatable->False, CellGroupingRules->{"TitleGrouping", 0}, CellHorizontalScrolling->False, PageBreakBelow->False, TextAlignment->Center, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subtitle", 0}, {"Subsubtitle", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->24, FontWeight->"Bold"], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, 10}, {20, 40}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->44], Cell[StyleData["Title", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, FontFamily->"Times New Roman", FontSize->20], Cell[StyleData["Title", "Printout"], CellMargins->{{2, 10}, {12, 30}}, FontFamily->"Times New Roman", FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{7, Inherited}, {6, 15}}, Evaluatable->False, CellGroupingRules->{"TitleGrouping", 10}, CellHorizontalScrolling->False, PageBreakBelow->False, TextAlignment->Center, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subsubtitle", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->18], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->36], Cell[StyleData["Subtitle", "Condensed"], CellMargins->{{8, 10}, {4, 4}}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontFamily->"Times New Roman", FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellMargins->{{7, Inherited}, {6, 15}}, Evaluatable->False, CellGroupingRules->{"TitleGrouping", 20}, CellHorizontalScrolling->False, PageBreakBelow->False, TextAlignment->Center, CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->14, FontSlant->"Italic"], Cell[StyleData["Subsubtitle", "Presentation"], CellMargins->{{24, 10}, {20, 20}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->24], Cell[StyleData["Subsubtitle", "Condensed"], CellMargins->{{8, 10}, {8, 8}}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontFamily->"Times New Roman", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[GraySquare]", CellMargins->{{22, Inherited}, {8, 20}}, Evaluatable->False, CellGroupingRules->{"SectionGrouping", 30}, CellHorizontalScrolling->False, PageBreakBelow->False, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->18, FontWeight->"Bold", FontVariations->{"Underline"->True}], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{13, 0}, {7, 22}}, FontFamily->"Times New Roman", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{19, Inherited}, {8, 15}}, Evaluatable->False, CellGroupingRules->{"SectionGrouping", 40}, CellHorizontalScrolling->False, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{36, 10}, {11, 32}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->22], Cell[StyleData["Subsection", "Condensed"], CellMargins->{{16, Inherited}, {6, 12}}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["Subsection", "Printout"], CellMargins->{{9, 0}, {7, 22}}, FontFamily->"Times New Roman", FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[EmptySquare]", CellMargins->{{18, Inherited}, {8, 12}}, Evaluatable->False, CellGroupingRules->{"SectionGrouping", 50}, CellHorizontalScrolling->False, PageBreakBelow->False, CounterIncrements->"Subsubsection", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{34, 10}, {11, 26}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{17, Inherited}, {6, 12}}, FontFamily->"Times New Roman", FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{9, 0}, {7, 14}}, FontFamily->"Times New Roman", FontSize->11] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Body Text"]], "Section", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{7, 10}, {7, 7}}, Evaluatable->False, CellHorizontalScrolling->False, PageBreakWithin->Automatic, LineSpacing->{1, 3}, CounterIncrements->"Text", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontFamily->"Times New Roman"], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontFamily->"Times New Roman"], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}, FontFamily->"Times New Roman"] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{7, 10}, {6, 6}}, Evaluatable->False, CellHorizontalScrolling->False, PageBreakWithin->Automatic, LineSpacing->{1, 3}, CounterIncrements->"SmallText", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->10], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontFamily->"Times New Roman", FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, FontFamily->"Times New Roman", FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Input/Output"]], "Section", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain"], Cell[TextData[StyleBox[ "The cells in this section define styles used for input and output to the \ kernel. Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names."]], "Text", FontFamily->"Times New Roman", FontSize->14], Cell[CellGroupData[{ Cell[StyleData["Input"], PageWidth->Infinity, CellMargins->{{42, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain"], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain"], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain"] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", StyleMenuListing->None, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[StyleData["Output"], PageWidth->Infinity, CellMargins->{{42, 10}, {7, 5}}, CellEditDuplicate->True, Evaluatable->False, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontFamily->"Times New Roman", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], PageWidth->Infinity, CellMargins->{{42, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0.500008, 0, 0], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontFamily->"Times New Roman", FontSize->14, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], PageWidth->Infinity, CellMargins->{{42, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontFamily->"Times New Roman", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], PageWidth->Infinity, CellMargins->{{7, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", AspectRatioFixed->True, ImageSize->{387, 393}, ImageMargins->{{34, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Graphics", "Condensed"], ImageSize->{175, 175}, ImageMargins->{{38, Inherited}, {Inherited, 0}}, FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["Graphics", "Printout"], ImageSize->{250, 250}, ImageMargins->{{30, Inherited}, {Inherited, 0}}, FontFamily->"Times New Roman", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["CellLabel", "Presentation"], FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["CellLabel", "Condensed"], FontFamily->"Times New Roman", FontSize->14], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Times New Roman", FontSize->14, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Formulas and Programming"]], "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}], Cell[StyleData["DisplayFormula", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Headers and Footers"]], "Section"], Cell[StyleData["Header"], CellMargins->{{7, 0}, {4, 1}}, Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{7, 0}, {0, 4}}, Evaluatable->False, PageBreakWithin->Automatic, TextAlignment->Center, AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Palette Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles that define standard ButtonFunctions, for \ use in palette buttons."]], "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Hyperlink Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles useful for making hypertext ButtonBoxes. \ The \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks."]], "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[TextData[StyleBox[ "The following styles are for linking automatically to the on-line help \ system."]], "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Placeholder Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles useful for making placeholder objects in \ palette templates."]], "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->Primary, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["FormatType Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles that are mixed in with the styles of most \ cells. If a cell's FormatType matches the name of one of the styles defined \ below, then that style is applied between the cell's style and its own \ options."]], "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell[TextData[StyleBox[ "The style defined below is mixed in to any cell that is in an inline cell \ within another."]], "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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