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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[ 518882, 12265]*) (*NotebookOutlinePosition[ 527296, 12454]*) (* CellTagsIndexPosition[ 525606, 12418]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Generalizations of the Fundamental Theorem of Calculus (Part II)\n", StyleBox["(note: some of the symbolic integrations in this notebook will \ take a long time to evaluate)", FontSize->18, FontColor->GrayLevel[0.666667]] }], "Title", CellTags->"mmtag:16:divergence_theorem"], Cell[CellGroupData[{ Cell[TextData[{ "Using Divergence theorem to find London Dispersion Potential (here modeled \ as point-wise", Cell[BoxData[ \(TraditionalForm\`1\/r\^6\)]], ") for a Finite Body" }], "Subtitle"], Cell[BoxData[ \(The\ \(dipole--\) dipole\ interaction\ force\ goes\ like\ \(1\/r\^6\) . \ \ The\ London\ \ interaction\ that\ gives\ the\ \ force\ between\ two\ bodies\ due\ to\ the\ \ superposition\ of\ all\ pointwise\ dipole\ interactions . \ \ \[IndentingNewLine]\[IndentingNewLine]To\ find\ such\ forces, \ it\ is\ necessary\ to\ integrate\ over\ the\ volumes\ of\ both\ \(bodies\ --\) an\ integral\ over\ six\ dimensions . \ \ There\ are\ a\ few\ \ \(cases--\) such\ as\ two\ spheres, \ a\ sphere\ and\ a\ semi - infinite\ solid, \ etc . \(--\(-where\)\)\ closed\ form\ solutions\ can\ be\ obtained . \ \ \ \ However, \ in\ general, \ it\ is\ necessary\ to\ perform\ a\ numerical\ integration . \ \ If\ the\ \ integral\ over\ both\ volumes\ could\ be\ converted\ to\ integrals\ over\ \ both\ surfaces, then\ the\ numerical\ integration\ is\ considerably\ more\ efficient . \ \ \ Such\ conversions, \ can\ be\ obtained\ though\ application\ of\ the\ divergence\ theorem . \ \[IndentingNewLine]\[IndentingNewLine]\ Here\ is\ an\ example\ for\ the\ interaction\ between\ a\ single\ \ \((point)\)\ and\ a\ finite\ volume\ \((a\ cylinder)\) . \ \ To\ proceed, \ \ \ a\ vector\ \ integrand\ \ must\ be\ found\ for\ which\ the\ divergence\ is\ \ the\ desired\ \ pointwise\ \(\(interaction\)\(.\)\)\)], "Subsection", CellTags->{ "mmtag:16:pointwise_interaction_between_two_volumes", "mmtag:16:numerical_integration__reducing_the_amount_of_brute_force", "mmtag:16:brute_force__reducing_in_numerical_integration"}], Cell[TextData[{ "Find ", Cell[BoxData[ \(TraditionalForm\`\(F\& \[Rule] \)\)]], " such that div ", Cell[BoxData[ \(TraditionalForm\`\(F\& \[Rule] \)\)]], " is ", Cell[BoxData[ \(TraditionalForm\`\(-1\)\/\((\(r\& \[Rule] \) - \(x\& \[Rule] \))\)\^6\ \)]], " where ", Cell[BoxData[ \(TraditionalForm\`\(r\& \[Rule] \)\)]], " = (\[Xi],\[Eta],\[Zeta]) is a position in the cylinder and ", Cell[BoxData[ \(TraditionalForm\`\(x\& \[Rule] \)\)]], "=(x,y,z) is a general position in space (A fairly general method to do \ this and similar problems can be found in ", StyleBox["Argento C; Jagota A; Carter WC", FontSize->12], StyleBox[" ``", FontSize->12, FontWeight->"Bold"], StyleBox["Surface formulation for molecular interactions of macroscopic \ bodies'' J.Mech. Physics Solids 1997, pp 1161-1183 .", FontSize->12] }], "Text", CellTags->{ "mmtag:16:divergence_theorem__example_usage_to_convert_volume_integration_\ to_surface_integrals", "mmtag:16:dipole_interaction_between_cylinder_and_pont"}], Cell["\<\ The following is a ``guess'' at the vector potential; it will be \ verified as the correct one by checking its divergence.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FVecLondon\ = \ \(1\/\(3 \((\((\[Xi] - x)\)\^2\ + \((\[Eta] - y)\)\^2\ \ + \ \((\[Zeta] - z)\)\^2)\)\^3\)\) {\[Xi] - x, \ \[Eta] - y, \ \[Zeta] - z}\)], "Input", CellTags->"mmtag:16:London_interaction__vector_potential"], Cell[BoxData[ \({\(\(-x\) + \[Xi]\)\/\(3\ \((\((\(-z\) + \[Zeta])\)\^2 + \((\(-y\) + \ \[Eta])\)\^2 + \((\(-x\) + \[Xi])\)\^2)\)\^3\), \(\(-y\) + \[Eta]\)\/\(3\ \((\ \((\(-z\) + \[Zeta])\)\^2 + \((\(-y\) + \[Eta])\)\^2 + \((\(-x\) + \ \[Xi])\)\^2)\)\^3\), \(\(-z\) + \[Zeta]\)\/\(3\ \((\((\(-z\) + \[Zeta])\)\^2 \ + \((\(-y\) + \[Eta])\)\^2 + \((\(-x\) + \[Xi])\)\^2)\)\^3\)}\)], "Output", CellTags->"mmtag:16:London_interaction__vector_potential"] }, Open ]], Cell[BoxData[ \(<< Calculus`VectorAnalysis`\)], "Input"], Cell["\<\ The following verifies that the correct vector potential is \ obtained\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ Div[FVecLondon, Cartesian[\[Xi], \[Eta], \[Zeta]]]]\)], "Input", CellTags->"mmtag:16:Div[]"], Cell[BoxData[ \(\(-\(1\/\((\((z - \[Zeta])\)\^2 + \((y - \[Eta])\)\^2 + \((x - \[Xi])\)\ \^2)\)\^3\)\)\)], "Output", CellTags->"mmtag:16:Div[]"] }, Open ]], Cell[TextData[{ "Therefore, ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(Cylinder\[IndentingNewLine] Volume\)\(1\/\((\(r\& \[Rule] \) - \(x\& \[Rule] \))\)\^6\) \ \[DifferentialD]V\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(Cylinder\[IndentingNewLine] Surfaces\)\(FVecLondon\&\[RightVector]\) \[Bullet] \[DifferentialD]\ \(A\& \[Rule] \)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(Cylinder\[IndentingNewLine] Surface\)\(FVecLondon\&\[RightVector]\) \[Bullet] \ \[DifferentialD]\(A\& \[Rule] \)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(Cylinder\[IndentingNewLine] Ends\)\(FVecLondon\&\[RightVector]\) \[Bullet] \ \[DifferentialD]\(A\& \[Rule] \)\)]], " is the total interaction between a point an a cylinder." }], "Text"], Cell[TextData[{ "Cylindrical coordinates: (\[Xi],\[Eta],\[Zeta]) = (r Cos[t], r Sin[t], \ \[Zeta]):\nOne cylindrical surface r= R = const. t \[Element] (0, 2\[Pi]), \ \[Zeta] \[Element] (-", Cell[BoxData[ \(TraditionalForm\`L\/2\)]], " , ", Cell[BoxData[ \(TraditionalForm\`L\/2\)]], ")\nTwo caps r \[Element] (0,R) t \[Element] (0, 2\[Pi]), \[Zeta]=\ \[PlusMinus]", Cell[BoxData[ \(TraditionalForm\`L\/2\)]] }], "Text", CellTags->"mmtag:16:integral__over_cylinder"], Cell["Cylinder Surface normals and differential quantities", "Section"], Cell["\<\ The following is a parametric representation of a cylinder surface \ that is coaxial with the z-axis (the cylinder ends will be included \ later)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CylSurf\ = \ {R\ Cos[t], \ R\ \ Sin[t], \[Zeta]}\)], "Input"], Cell[BoxData[ \({R\ Cos[t], R\ Sin[t], \[Zeta]}\)], "Output"] }, Open ]], Cell[TextData[{ "The infinitessimal surface vectors ", Cell[BoxData[ \(TraditionalForm\`\(R\_u\)\&\[RightVector]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(R\_v\)\&\[RightVector]\)]], " for the cylinder surface are obtained by differentiation; they will be \ used to find the surface patch d", Cell[BoxData[ \(TraditionalForm\`\(A\& \[Rule] \)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(CylSurfRt\ = \ D[CylSurf, t]\), "\[IndentingNewLine]", \(CylSurfRz\ \ \ = \ D[CylSurf, \[Zeta]]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \({\(-R\)\ Sin[t], R\ Cos[t], 0}\)], "Output"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(CylSurfRz\\)\\\" is similar to existing \ symbol \\\"\\!\\(CylSurfRt\\)\\\". \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ \({0, 0, 1}\)], "Output"] }, Open ]], Cell[TextData[{ "The surface normal given by ", Cell[BoxData[ \(TraditionalForm\`\(R\_u\)\&\[RightVector]\)]], " \[Times] ", Cell[BoxData[ \(TraditionalForm\`\(R\_v\)\&\[RightVector]\)]], " for the cylinder surface, there for the following (multiplied by \ d\[Theta] dz) is the infinitessimal oriented surface patch d", Cell[BoxData[ \(TraditionalForm\`\(A\& \[Rule] \)\)]], "." }], "Text", CellTags->"mmtag:16:surface_normal__example"], Cell[CellGroupData[{ Cell[BoxData[ \(NormalVecCylSurf\ = \ Cross[CylSurfRt, CylSurfRz]\)], "Input", CellTags->"mmtag:16:Cross[]"], Cell[BoxData[ \({R\ Cos[t], R\ Sin[t], 0}\)], "Output", CellTags->"mmtag:16:Cross[]"] }, Open ]], Cell["\<\ The integrand to be evaluated over the cylinder surface is the \ vector potential, dotted into \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CylinderIntegrand\[Theta]\[Zeta]\ = \ FullSimplify[\((FVecLondon /. {\[Xi] \[Rule] R\ Cos[t], \ \[Eta] \[Rule] \ R\ Sin[t]})\) . NormalVecCylSurf]\)], "Input"], Cell[BoxData[ \(\(R\ \((R - x\ Cos[t] - y\ Sin[t])\)\)\/\(3\ \((R\^2 + x\^2 + y\^2 + \ \((z - \[Zeta])\)\^2 - 2\ R\ \((x\ Cos[t] + y\ Sin[t])\))\)\^3\)\)], "Output"] }, Open ]], Cell[TextData[{ "Closed form for integral is un-findable, use a numeric integration--this \ will be written in two forms\n1) A general function of x,y,z of the point \ particle and the cylinder radius and length.\n2) Because of the polar \ symmetry, the integral should be a function of only r = ", Cell[BoxData[ \(TraditionalForm\`\@\(x\^2\ + \ y\^2\)\)]], "\n\nWe can use the same function name twice, but with a different number \ of arguments. But, first we will try to do at least one of the two \ integrations symbolically. If we can do this, it will reduce the original \ numerical integration from three dimensions down to 1---which is a nice \ bargain if we can find it." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Try and see if we can do one of the two integrals. First, let's \ try theta:\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[CylinderIntegrand\[Theta]\[Zeta], t, \ Assumptions -> R\ > \ 0\ && \ \ \[Zeta]\ \[Element] \ Reals\ && \ x\ \[Element] \ Reals && y \[Element] \ Reals && \ z\ \[Element] \ Reals]\)], "Input"], Cell[BoxData[ \(1\/3\ R\ Integrate[\(R - x\ Cos[t] - y\ Sin[t]\)\/\((R\^2 + x\^2 + y\^2 \ + \((z - \[Zeta])\)\^2 - 2\ R\ \((x\ Cos[t] + y\ Sin[t])\))\)\^3, t, Assumptions \[Rule] R > 0 && \[Zeta] \[Element] Reals && x \[Element] Reals && y \[Element] Reals && z \[Element] Reals]\)], "Output"] }, Open ]] }, Open ]], Cell["\<\ The integral over theta gives no symbolic form...Try and see if we \ can do one of the two integrals. So, let's try \[Zeta]:\ \>", "Subsubsection",\ CellTags->{ "mmtag:16:numerical_integration__reducing_the_amount_of_brute_force_even_\ more_with_finesse", "mmtag:16:brute_force__finesse_and_reducing_in_numerical_integration"}], Cell[CellGroupData[{ Cell[BoxData[ \(CylinderIntegrand\[Theta]Ind\[Zeta] = Integrate[CylinderIntegrand\[Theta]\[Zeta], \[Zeta]\ , \ Assumptions -> R\ > \ 0\ && \ \ L > 0\ && \ \ x\ \[Element] \ Reals && y \[Element] \ Reals && \ z\ \[Element] \ Reals && \ t \[Element] \ Reals]\)], "Input"], Cell[BoxData[ \(1\/3\ R\ \((R - x\ Cos[t] - y\ Sin[t])\)\ \((\(-\(\(3\ ArcTan[\(z - \[Zeta]\)\/\@\(R\^2 + x\^2 \ + y\^2 - 2\ R\ x\ Cos[t] - 2\ R\ y\ Sin[t]\)]\)\/\(8\ \((R\^2 + x\^2 + y\^2 - \ 2\ R\ x\ Cos[t] - 2\ R\ y\ Sin[t])\)\^\(5/2\)\)\)\) - \(3\ \((z - \ \[Zeta])\)\)\/\(8\ \((R\^2 + x\^2 + y\^2 - 2\ R\ x\ Cos[t] - 2\ R\ y\ Sin[t])\ \)\^2\ \((R\^2 + x\^2 + y\^2 + \((z - \[Zeta])\)\^2 - 2\ R\ x\ Cos[t] - 2\ R\ \ y\ Sin[t])\)\) + \(z - \[Zeta]\)\/\(4\ \((R\^2 + x\^2 + y\^2 + \((z - \ \[Zeta])\)\^2 - 2\ R\ x\ Cos[t] - 2\ R\ y\ Sin[t])\)\^2\ \((\(-R\^2\) - x\^2 \ - y\^2 + 2\ R\ x\ Cos[t] + 2\ R\ y\ Sin[t])\)\))\)\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Speed and Efficiency: When to Evaluate the Right-Hand-Side during a \ Function Definition\ \>", "Subtitle", CellTags->{ "mmtag:16:Evaluate[]__in_function_definitions", "mmtag:16:defining_functions_from_previously_evaluated_functions", "mmtag:16:functions__definiting_from_previously_defined_expressions_\ without_using_replacement_rules"}], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[Integrate[Exp[Tan[x]], {x, 0, c}]]\)], "Input"], Cell[BoxData[ \({1.2728500000000622`\ Second, c\ If[\(-\(\[Pi]\/2\)\) \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0, \(\(1\/\(2\ c\)\)\((\[ImaginaryI]\ \[ExponentialE]\^\(-\ \[ImaginaryI]\)\ \((\(-ExpIntegralEi[\[ImaginaryI]]\) + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\)\ \((ExpIntegralEi[\(-\[ImaginaryI]\)] - ExpIntegralEi[\(-\[ImaginaryI]\) + Tan[c]])\) + ExpIntegralEi[\[ImaginaryI] + Tan[c]])\))\)\), Integrate[\[ExponentialE]\^Tan[c\ x], {x, 0, 1}, Assumptions \[Rule] \(! \((\(-\(\[Pi]\/2\)\) \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0)\)\)]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[Integrate[Exp[Tan[x]], {x, 0, c}]]\)], "Input"], Cell[BoxData[ \({1.225153999999975`\ Second, c\ If[\(-\(\[Pi]\/2\)\) \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0, \(\(1\/\(2\ c\)\)\((\[ImaginaryI]\ \[ExponentialE]\^\(-\ \[ImaginaryI]\)\ \((\(-ExpIntegralEi[\[ImaginaryI]]\) + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\)\ \((ExpIntegralEi[\(-\[ImaginaryI]\)] - ExpIntegralEi[\(-\[ImaginaryI]\) + Tan[c]])\) + ExpIntegralEi[\[ImaginaryI] + Tan[c]])\))\)\), Integrate[\[ExponentialE]\^Tan[c\ x], {x, 0, 1}, Assumptions \[Rule] \(! \((\(-\(\[Pi]\/2\)\) \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0)\)\)]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ f[c_]\ := \ Evaluate[Integrate[Exp[Tan[x]], {x, 0, c}]]]\)], "Input"], Cell[BoxData[ \({1.2282090000001062`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[h[c_]\ = Integrate[Exp[Tan[x]], {x, 0, c}]]\)], "Input"], Cell[BoxData[ \({1.3342329999999976`\ Second, c\ If[\(-\(\[Pi]\/2\)\) \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0, \(\[ImaginaryI]\ \[ExponentialE]\^\(-\[ImaginaryI]\)\ \ \((\(-ExpIntegralEi[\[ImaginaryI]]\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\)\ \ \((ExpIntegralEi[\(-\[ImaginaryI]\)] - ExpIntegralEi[\(-\[ImaginaryI]\) + \ Tan[c]])\) + ExpIntegralEi[\[ImaginaryI] + Tan[c]])\)\)\/\(2\ c\), Integrate[\[ExponentialE]\^Tan[c\ x], {x, 0, 1}, Assumptions \[Rule] \(! \((\(-\(\[Pi]\/2\)\) \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0)\)\)]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[g[c_]\ := \ Integrate[Exp[Tan[x]], {x, 0, c}]]\)], "Input"], Cell[BoxData[ \({0.000053999999998666226`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?f\)\)], "Input"], Cell["Global`f", "Print", CellTags->"Info3371201135-1412415"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {\(f[c_] := c\ If[\(-\(\[Pi]\/2\)\) \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0, \(\[ImaginaryI]\ \[ExponentialE]\^\(-\ \[ImaginaryI]\)\ \((\(-ExpIntegralEi[\[ImaginaryI]]\) + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\)\ \((ExpIntegralEi[\(-\[ImaginaryI]\)] - ExpIntegralEi[\(-\ \[ImaginaryI]\) + Tan[c]])\) + ExpIntegralEi[\[ImaginaryI] + \ Tan[c]])\)\)\/\(2\ c\), Integrate[\[ExponentialE]\^Tan[c\ x], {x, 0, 1}, Assumptions \[Rule] \(! \((\(-\(\[Pi]\/2\)\) \ \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0)\)\)]]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ "f"], Editable->False]], "Print", CellTags->"Info3371201135-1412415"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?h\)\)], "Input"], Cell["Global`h", "Print", CellTags->"Info3371201135-2068537"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {\(h[c_] = c\ If[\(-\(\[Pi]\/2\)\) \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0, \(\[ImaginaryI]\ \[ExponentialE]\^\(-\ \[ImaginaryI]\)\ \((\(-ExpIntegralEi[\[ImaginaryI]]\) + \[ExponentialE]\^\(2\ \ \[ImaginaryI]\)\ \((ExpIntegralEi[\(-\[ImaginaryI]\)] - ExpIntegralEi[\(-\ \[ImaginaryI]\) + Tan[c]])\) + ExpIntegralEi[\[ImaginaryI] + \ Tan[c]])\)\)\/\(2\ c\), Integrate[\[ExponentialE]\^Tan[c\ x], {x, 0, 1}, Assumptions \[Rule] \(! \((\(-\(\[Pi]\/2\)\) \ \[LessEqual] Re[c] \[LessEqual] \[Pi]\/2 || Im[c] \[NotEqual] 0)\)\)]]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ "h"], Editable->False]], "Print", CellTags->"Info3371201135-2068537"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?g\)\)], "Input"], Cell["Global`g", "Print", CellTags->"Info3371201136-8490394"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {\(g[ c_] := \[Integral]\_0\%c\( \[ExponentialE]\^Tan[ x]\) \[DifferentialD]x\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnWidths->0.999, ColumnAlignments->{Left}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], Definition[ "g"], Editable->False]], "Print", CellTags->"Info3371201136-8490394"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[f[0.5]]\)], "Input"], Cell[BoxData[ \({0.004230999999997209`\ Second, \(\(0.6572099322091018`\)\(\ \[InvisibleSpace]\)\) + 5.551115123125783`*^-17\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[h[0.5]]\)], "Input"], Cell[BoxData[ \({0.003494000000003439`\ Second, \(\(0.6572099322091018`\)\(\ \[InvisibleSpace]\)\) + 5.551115123125783`*^-17\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[g[0.5]]\)], "Input"], Cell[BoxData[ \({0.3632350000000102`\ Second, \(\(0.6572099313456736`\)\(\ \[InvisibleSpace]\)\) + 5.551115123125783`*^-17\ \[ImaginaryI]}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" London Dispersion Potential for a Finite Cylinder", "Subtitle"], Cell[BoxData[ \(CylinderIntegrand\[Theta][x_, y_, z_, CylRad_, CylLen_]\ := \ Evaluate[Simplify[\((CylinderIntegrand\[Theta]Ind\[Zeta] /. {\[Zeta] -> CylLen/2, R -> CylRad})\) - \((CylinderIntegrand\[Theta]Ind\[Zeta] /. \ {\[Zeta] -> \(-CylLen\)/2, R -> CylRad})\), Assumptions -> CylRad\ > \ 0\ && \ \ CylLen > 0\ && \ \ x\ \[Element] \ Reals && y \[Element] \ Reals && \ z\ \[Element] \ Reals && \ t \[Element] \ Reals]]\)], "Input"], Cell[BoxData[ \(CylinderContribution[x_, \ y_, \ z_, \ CylRad_\ , \ CylLen_]\ := \ NIntegrate[ Evaluate[ CylinderIntegrand\[Theta][x, y, z, CylRad, CylLen], {t, 0, \ 2 \[Pi]}]]\)], "Input", CellTags->"mmtag:16:NIntegrate[]"], Cell[BoxData[ \(Because\ of\ the\ polar\ symmetry, \ this\ contribution\ should\ only\ be\ a\ function\ of\ the\ distance\ \((\ \@\(x\^2 + y\^2\))\), \ so\ we\ can\ write\ a\ new\ function\ \((with\ the\ same\ name)\)\ with\ \ fewer\ arguments\)], "Text", CellTags->"mmtag:16:functions_overloading"], Cell[BoxData[ \(CylinderContribution[dist_, \ z_, \ CylRad_\ , \ CylLen_]\ := NIntegrate[ Evaluate[ CylinderIntegrand\[Theta][dist, 0, z, CylRad, CylLen], {t, 0, \ 2 \[Pi]}]]\)], "Input"], Cell["\<\ Checking the numerical integration for the first form of the \ function\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CylinderContribution[2, 2, 1/2, 1, 1]\)], "Input"], Cell[BoxData[ \(\(-0.011617409378892943`\)\)], "Output"] }, Open ]], Cell["\<\ Checking the numerical integration for the second form of the \ function\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CylinderContribution[2\ Sqrt[2], 1/2, 1, 1]\)], "Input"], Cell[BoxData[ \(\(-0.011617409378892964`\)\)], "Output"] }, Open ]], Cell["\<\ Top Cap Surface normals and differential quantities, parallel \ method to cylinder surface\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TopSurf\ = \ {r\ Cos[t], r\ Sin[t], L\/2}\)], "Input"], Cell[BoxData[ \({r\ Cos[t], r\ Sin[t], L\/2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(TopSurfRt\ = \ D[TopSurf, t]\), "\[IndentingNewLine]", \(TopSurfRr\ \ \ = \ D[TopSurf, r]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \({\(-r\)\ Sin[t], r\ Cos[t], 0}\)], "Output"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(TopSurfRr\\)\\\" is similar to existing \ symbol \\\"\\!\\(TopSurfRt\\)\\\". \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ \({Cos[t], Sin[t], 0}\)], "Output"] }, Open ]], Cell[BoxData[ \(The\ oriented\ surface\ patch\ d \( A\& \[Rule] \)\ for\ the\ top\ surface\ is\ the\ following\ \ \((multiplied\ by\ dr\ d\[Theta])\)\)], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NormalVecTopSurf\ = \ FullSimplify[Cross[TopSurfRr, TopSurfRt]]\)], "Input"], Cell[BoxData[ \({0, 0, r}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TopIntegrand\[Theta]r\ \ = \ FullSimplify[\((FVecLondon /. {\[Xi] \[Rule] r\ Cos[t], \ \[Eta] \[Rule] \ r\ Sin[t], \[Zeta] \[Rule] L/2})\) . NormalVecTopSurf]\)], "Input"], Cell[BoxData[ \(\(r\ \((L - 2\ z)\)\)\/\(6\ \((1\/4\ \((L - 2\ z)\)\^2 + \((x - r\ \ Cos[t])\)\^2 + \((y - r\ Sin[t])\)\^2)\)\^3\)\)], "Output"] }, Open ]], Cell["\<\ Because we will need to numerically integrate over r and over t, we \ might try the same trick as above and see if one of these integrals can be \ expressed in closed form.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Integrating over \[Theta] doesn't provide a closed form (this next \ step takes a while to evaluate... probably better to skip the \ evaluation.)\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[TopIntegrand\[Theta]r, \ {t, 0, 2 \[Pi]}, Assumptions\ \[Rule] \ r\ \[GreaterEqual] \ 0\ && \ L\ > \ 0\ && \ \ x \[Element] \ Reals\ && \ y\ \[Element] \ Reals\ && \ z\ \[Element] \ Reals]\)], "Input", Editable->False], Cell[BoxData[ \(Integrate[\(r\ \((L - 2\ z)\)\)\/\(6\ \((1\/4\ \((L - 2\ z)\)\^2 + \((x \ - r\ Cos[t])\)\^2 + \((y - r\ Sin[t])\)\^2)\)\^3\), {t, 0, 2\ \[Pi]}, Assumptions \[Rule] {r > 0, L > 0, x \[Element] Reals, y \[Element] Reals, z \[Element] Reals}]\)], "Output", Editable->False] }, Open ]] }, Open ]], Cell["Integrating over r does work", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(TopIntegrand\[Theta]Indr = \ Integrate[TopIntegrand\[Theta]r, \ r, Assumptions\ \[Rule] \ t\ \[GreaterEqual] \ 0\ && \ L\ > \ 0\ && \ \ \ x \[Element] \ Reals\ && \ y\ \[Element] \ Reals\ && \ z\ \[Element] \ Reals]\)], "Input"], Cell[BoxData[ \(1\/6\ \((L - 2\ z)\)\ \((\(-\(\(12\ ArcTan[\(2\ \((\(-r\) + x\ Cos[t] + y\ \ Sin[t])\)\)\/\@\(L\^2 + 2\ x\^2 + 2\ y\^2 - 4\ L\ z + 4\ z\^2 - 2\ x\^2\ \ Cos[2\ t] + 2\ y\^2\ Cos[2\ t] - 4\ x\ y\ Sin[2\ t]\)]\ \((x\ Cos[t] + y\ Sin[t])\)\)\/\((L\^2 + 2\ x\^2 + 2\ y\^2 - 4\ L\ z + \ 4\ z\^2 - 2\ x\^2\ Cos[2\ t] + 2\ y\^2\ Cos[2\ t] - 4\ x\ y\ Sin[2\ \ t])\)\^\(5/2\)\)\) + \(12\ \((\(-x\^2\) - y\^2 + 2\ r\ x\ Cos[t] - x\^2\ \ Cos[2\ t] + y\^2\ Cos[2\ t] + 2\ r\ y\ Sin[t] - 2\ x\ y\ Sin[2\ \ t])\)\)\/\(\((L\^2 + 4\ r\^2 + 4\ x\^2 + 4\ y\^2 - 4\ L\ z + 4\ z\^2 - 8\ r\ \ x\ Cos[t] - 8\ r\ y\ Sin[t])\)\ \((L\^2 + 2\ x\^2 + 2\ y\^2 - 4\ L\ z + 4\ \ z\^2 - 2\ x\^2\ Cos[2\ t] + 2\ y\^2\ Cos[2\ t] - 4\ x\ y\ Sin[2\ t])\)\^2\) - \ \(4\ \((\(-L\^2\) - 4\ x\^2 - 4\ y\^2 + 4\ L\ z - 4\ z\^2 + 4\ r\ x\ Cos[t] + \ 4\ r\ y\ Sin[t])\)\)\/\(\((L\^2 + 4\ r\^2 + 4\ x\^2 + 4\ y\^2 - 4\ L\ z + 4\ \ z\^2 - 8\ r\ x\ Cos[t] - 8\ r\ y\ Sin[t])\)\^2\ \((\(-L\^2\) - 2\ x\^2 - 2\ y\ \^2 + 4\ L\ z - 4\ z\^2 + 2\ x\^2\ Cos[2\ t] - 2\ y\^2\ Cos[2\ t] + 4\ x\ y\ \ Sin[2\ t])\)\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TopIntegrand\[Theta][x_, y_, z_, CylRad_, CylLen_]\ := Evaluate[Simplify[\((TopIntegrand\[Theta]Indr /. {r -> CylRad, L -> CylLen})\) - \((TopIntegrand\[Theta]Indr /. {r -> 0, L -> CylLen})\)]]\)], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(TopIntegrand\[Theta]\\)\\\" is similar to \ existing symbol \\\"\\!\\(TopIntegrand\[Theta]r\\)\\\". \\!\\(\\*ButtonBox[\\\ \"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \ ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell[BoxData[ \(TopContribution[xpos_, \ ypos_, \ zpos_, \ CylRad_\ , \ CylLen_]\ := \ NIntegrate[ Evaluate[ TopIntegrand\[Theta][xpos, ypos, zpos, \ CylRad, \ CylLen], {t, 0, 2\ \[Pi]}]]\)], "Input"], Cell[BoxData[ \(TopContribution[dist_, \ zpos_, \ CylRad_\ , \ CylLen_]\ := \ NIntegrate[ Evaluate[ TopIntegrand\[Theta][dist, 0, zpos, \ CylRad, \ CylLen], {t, 0, 2\ \[Pi]}]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(TopContribution[Sqrt[2], Sqrt[2], 1/3, 1, 2]\)], "Input"], Cell[BoxData[ \(0.019746507925389352`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TopContribution[2, 1/3, 1, 2]\)], "Input"], Cell[BoxData[ \(0.019746507925387565`\)], "Output"] }, Open ]], Cell["Bottom Cap Surface normals and differential quantities", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(BotSurf\ = \ {r\ Cos[t], r\ Sin[t], \(-L\)\/2}\)], "Input"], Cell[BoxData[ \({r\ Cos[t], r\ Sin[t], \(-\(L\/2\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(BotSurfRt\ = \ D[BotSurf, t]\), "\[IndentingNewLine]", \(BotSurfRr\ \ \ = \ D[BotSurf, r]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \({\(-r\)\ Sin[t], r\ Cos[t], 0}\)], "Output"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(BotSurfRr\\)\\\" is similar to existing \ symbol \\\"\\!\\(BotSurfRt\\)\\\". \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ \({Cos[t], Sin[t], 0}\)], "Output"] }, Open ]], Cell[BoxData[ \(The\ oriented\ surface\ patch\ d \( A\& \[Rule] \)\ for\ the\ bottom\ surface\ is\ the\ following\ \ \((multiplied\ by\ dr\ d\[Theta])\)\)], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NormalVecBotSurf\ = \ FullSimplify[Cross[BotSurfRt, BotSurfRr]]\)], "Input"], Cell[BoxData[ \({0, 0, \(-r\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(BotIntegrand\[Theta]r\ = \ FullSimplify[\((FVecLondon /. {\[Xi] \[Rule] r\ Cos[t], \ \[Eta] \[Rule] \ r\ Sin[t], \[Zeta] \[Rule] \(-L\)/2})\) . NormalVecBotSurf]\)], "Input"], Cell[BoxData[ \(\(32\ r\ \((L + 2\ z)\)\)\/\(3\ \((4\ \((r\^2 + x\^2 + y\^2)\) + \((L + \ 2\ z)\)\^2 - 8\ r\ \((x\ Cos[t] + y\ Sin[t])\))\)\^3\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(BotIntegrand\[Theta]Indr = \ Integrate[BotIntegrand\[Theta]r, \ r, Assumptions\ \[Rule] \ t\ \[GreaterEqual] \ 0\ && \ L\ > \ 0\ && \ \ \ x \[Element] \ Reals\ && \ y\ \[Element] \ Reals\ && \ z\ \[Element] \ Reals]\)], "Input"], Cell[BoxData[ \(32\/3\ \((L + 2\ z)\)\ \((\(-\(\((3\ ArcTan[\(2\ \((\(-r\) + x\ Cos[t] + y\ \ Sin[t])\)\)\/\@\(L\^2 + 2\ x\^2 + 2\ y\^2 + 4\ L\ z + 4\ z\^2 - 2\ x\^2\ \ Cos[2\ t] + 2\ y\^2\ Cos[2\ t] - 4\ x\ y\ Sin[2\ t]\)]\ \((x\ Cos[t] + y\ Sin[t])\))\)/\((16\ \@\(L\^2 + 2\ x\^2 + 2\ y\^2 + 4\ \ L\ z + 4\ z\^2 - 2\ x\^2\ Cos[2\ t] + 2\ y\^2\ Cos[2\ t] - 4\ x\ y\ Sin[2\ \ t]\)\ \((\(-L\^2\) - 2\ x\^2 - 2\ y\^2 - 4\ L\ z - 4\ z\^2 + 2\ x\^2\ Cos[2\ \ t] - 2\ y\^2\ Cos[2\ t] + 4\ x\ y\ Sin[2\ t])\)\^2)\)\)\) + \(3\ \((\(-x\^2\) \ - y\^2 + 2\ r\ x\ Cos[t] - x\^2\ Cos[2\ t] + y\^2\ Cos[2\ t] + 2\ r\ y\ \ Sin[t] - 2\ x\ y\ Sin[2\ t])\)\)\/\(16\ \((L\^2 + 4\ r\^2 + 4\ x\^2 + 4\ y\^2 \ + 4\ L\ z + 4\ z\^2 - 8\ r\ x\ Cos[t] - 8\ r\ y\ Sin[t])\)\ \((\(-L\^2\) - 2\ \ x\^2 - 2\ y\^2 - 4\ L\ z - 4\ z\^2 + 2\ x\^2\ Cos[2\ t] - 2\ y\^2\ Cos[2\ t] \ + 4\ x\ y\ Sin[2\ t])\)\^2\) + \(L\^2 + 4\ x\^2 + 4\ y\^2 + 4\ L\ z + 4\ z\^2 \ - 4\ r\ x\ Cos[t] - 4\ r\ y\ Sin[t]\)\/\(16\ \((L\^2 + 4\ r\^2 + 4\ x\^2 + 4\ \ y\^2 + 4\ L\ z + 4\ z\^2 - 8\ r\ x\ Cos[t] - 8\ r\ y\ Sin[t])\)\^2\ \ \((\(-L\^2\) - 2\ x\^2 - 2\ y\^2 - 4\ L\ z - 4\ z\^2 + 2\ x\^2\ Cos[2\ t] - 2\ \ y\^2\ Cos[2\ t] + 4\ x\ y\ Sin[2\ t])\)\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(BotIntegrand\[Theta][x_, y_, z_, CylRad_, CylLen_]\ := Evaluate[Simplify[\((BotIntegrand\[Theta]Indr /. {r -> CylRad, L -> CylLen})\) - \((BotIntegrand\[Theta]Indr /. {r -> 0, L -> CylLen})\)]]\)], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), 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.917 r .34275 .75469 .35688 .76785 .37983 .76193 .3658 .74882 Metetra .641 .738 .919 r .3658 .74882 .37983 .76193 .40298 .75594 .38906 .74288 Metetra .641 .74 .921 r .38906 .74288 .40298 .75594 .42633 .7499 .41252 .73687 Metetra .641 .742 .922 r .41252 .73687 .42633 .7499 .44989 .74378 .4362 .73078 Metetra .641 .744 .924 r .4362 .73078 .44989 .74378 .47366 .7376 .4601 .72462 Metetra .642 .746 .925 r .4601 .72462 .47366 .7376 .49765 .73134 .48421 .71836 Metetra .642 .748 .926 r .48421 .71836 .49765 .73134 .52185 .72502 .50854 .71203 Metetra .642 .75 .927 r .50854 .71203 .52185 .72502 .54627 .71861 .5331 .7056 Metetra .642 .751 .928 r .5331 .7056 .54627 .71861 .5709 .71213 .55788 .69909 Metetra .643 .753 .929 r .55788 .69909 .5709 .71213 .59577 .70558 .58289 .6925 Metetra .643 .754 .929 r .58289 .6925 .59577 .70558 .62086 .69895 .60812 .68581 Metetra .643 .755 .93 r .60812 .68581 .62086 .69895 .64617 .69224 .6336 .67904 Metetra .643 .756 .931 r .6336 .67904 .64617 .69224 .67172 .68545 .6593 .67219 Metetra .644 .757 .931 r .6593 .67219 .67172 .68545 .69751 .67859 .68525 .66525 Metetra .644 .757 .931 r .68525 .66525 .69751 .67859 .72353 .67165 .71144 .65823 Metetra .644 .758 .932 r .71144 .65823 .72353 .67165 .74979 .66464 .73787 .65112 Metetra .644 .759 .932 r .73787 .65112 .74979 .66464 .7763 .65755 .76455 .64393 Metetra .644 .759 .932 r .76455 .64393 .7763 .65755 .80306 .65038 .79148 .63666 Metetra .645 .76 .932 r .79148 .63666 .80306 .65038 .83006 .64314 .81866 .62931 Metetra .645 .76 .933 r .81866 .62931 .83006 .64314 .85732 .63581 .84611 .62187 Metetra .645 .76 .933 r .84611 .62187 .85732 .63581 .88484 .62842 .87381 .61436 Metetra .645 .76 .933 r .87381 .61436 .88484 .62842 .91261 .62094 .90178 .60677 Metetra .645 .761 .933 r .90178 .60677 .91261 .62094 .94066 .61339 .93002 .59909 Metetra .645 .761 .933 r .93002 .59909 .94066 .61339 .96897 .60576 .95854 .59133 Metetra .638 .724 .91 r .32845 .74058 .34275 .75469 .3658 .74882 .35159 .73485 Metetra .638 .727 .913 r .35159 .73485 .3658 .74882 .38906 .74288 .37494 .72905 Metetra .638 .731 .915 r .37494 .72905 .38906 .74288 .41252 .73687 .39852 .72315 Metetra .638 .734 .918 r .39852 .72315 .41252 .73687 .4362 .73078 .42231 .71716 Metetra .639 .737 .92 r .42231 .71716 .4362 .73078 .4601 .72462 .44632 .71108 Metetra .639 .74 .922 r .44632 .71108 .4601 .72462 .48421 .71836 .47056 .70488 Metetra .64 .743 .924 r .47056 .70488 .48421 .71836 .50854 .71203 .49502 .69859 Metetra .64 .745 .925 r .49502 .69859 .50854 .71203 .5331 .7056 .51971 .69218 Metetra .641 .748 .926 r .51971 .69218 .5331 .7056 .55788 .69909 .54464 .68567 Metetra .641 .75 .927 r .54464 .68567 .55788 .69909 .58289 .6925 .56979 .67906 Metetra .642 .751 .928 r .56979 .67906 .58289 .6925 .60812 .68581 .59518 .67235 Metetra .642 .753 .929 r .59518 .67235 .60812 .68581 .6336 .67904 .6208 .66554 Metetra .643 .754 .93 r .6208 .66554 .6336 .67904 .6593 .67219 .64667 .65863 Metetra .643 .755 .93 r .64667 .65863 .6593 .67219 .68525 .66525 .67278 .65162 Metetra .643 .756 .931 r .67278 .65162 .68525 .66525 .71144 .65823 .69913 .64452 Metetra .644 .757 .931 r .69913 .64452 .71144 .65823 .73787 .65112 .72573 .63733 Metetra .644 .758 .932 r .72573 .63733 .73787 .65112 .76455 .64393 .75259 .63005 Metetra .644 .758 .932 r .75259 .63005 .76455 .64393 .79148 .63666 .7797 .62267 Metetra .644 .759 .932 r .7797 .62267 .79148 .63666 .81866 .62931 .80707 .61522 Metetra .644 .759 .932 r .80707 .61522 .81866 .62931 .84611 .62187 .8347 .60767 Metetra .645 .76 .933 r .8347 .60767 .84611 .62187 .87381 .61436 .86259 .60004 Metetra .645 .76 .933 r .86259 .60004 .87381 .61436 .90178 .60677 .89076 .59232 Metetra .645 .76 .933 r .89076 .59232 .90178 .60677 .93002 .59909 .9192 .58452 Metetra .645 .761 .933 r .9192 .58452 .93002 .59909 .95854 .59133 .94792 .57664 Metetra .632 .705 .899 r .314 .72507 .32845 .74058 .35159 .73485 .33722 .71965 Metetra .632 .711 .903 r .33722 .71965 .35159 .73485 .37494 .72905 .36066 .71413 Metetra .633 .717 .908 r .36066 .71413 .37494 .72905 .39852 .72315 .38432 .70848 Metetra .634 .722 .911 r .38432 .70848 .39852 .72315 .42231 .71716 .40822 .70271 Metetra .635 .727 .915 r .40822 .70271 .42231 .71716 .44632 .71108 .43234 .69681 Metetra .636 .732 .917 r .43234 .69681 .44632 .71108 .47056 .70488 .4567 .69076 Metetra .637 .736 .92 r .4567 .69076 .47056 .70488 .49502 .69859 .48129 .68458 Metetra .637 .739 .922 r .48129 .68458 .49502 .69859 .51971 .69218 .50611 .67826 Metetra .638 .743 .924 r .50611 .67826 .51971 .69218 .54464 .68567 .53117 .67181 Metetra .639 .746 .926 r .53117 .67181 .54464 .68567 .56979 .67906 .55647 .66522 Metetra .64 .748 .927 r .55647 .66522 .56979 .67906 .59518 .67235 .58201 .65851 Metetra .641 .75 .928 r .58201 .65851 .59518 .67235 .6208 .66554 .60779 .65168 Metetra .641 .752 .929 r .60779 .65168 .6208 .66554 .64667 .65863 .63382 .64474 Metetra .642 .753 .93 r .63382 .64474 .64667 .65863 .67278 .65162 .66009 .63768 Metetra .642 .755 .93 r .66009 .63768 .67278 .65162 .69913 .64452 .68661 .63051 Metetra .643 .756 .931 r .68661 .63051 .69913 .64452 .72573 .63733 .71339 .62325 Metetra .643 .757 .931 r .71339 .62325 .72573 .63733 .75259 .63005 .74042 .61588 Metetra .644 .758 .932 r .74042 .61588 .75259 .63005 .7797 .62267 .76771 .60841 Metetra .644 .758 .932 r .76771 .60841 .7797 .62267 .80707 .61522 .79526 .60085 Metetra .644 .759 .932 r .79526 .60085 .80707 .61522 .8347 .60767 .82308 .5932 Metetra .644 .759 .933 r .82308 .5932 .8347 .60767 .86259 .60004 .85117 .58545 Metetra .645 .76 .933 r .85117 .58545 .86259 .60004 .89076 .59232 .87954 .57761 Metetra .645 .76 .933 r .87954 .57761 .89076 .59232 .9192 .58452 .90818 .56968 Metetra .645 .76 .933 r .90818 .56968 .9192 .58452 .94792 .57664 .9371 .56166 Metetra .622 .677 .882 r .29947 .70737 .314 .72507 .33722 .71965 .32273 .70255 Metetra .623 .686 .889 r .32273 .70255 .33722 .71965 .36066 .71413 .34623 .69758 Metetra .624 .695 .895 r .34623 .69758 .36066 .71413 .38432 .70848 .36997 .69243 Metetra .626 .704 .901 r .36997 .69243 .38432 .70848 .40822 .70271 .39395 .68709 Metetra .628 .712 .907 r .39395 .68709 .40822 .70271 .43234 .69681 .41817 .68154 Metetra .63 .719 .911 r .41817 .68154 .43234 .69681 .4567 .69076 .44264 .6758 Metetra .631 .725 .915 r .44264 .6758 .4567 .69076 .48129 .68458 .46734 .66986 Metetra .633 .731 .918 r .46734 .66986 .48129 .68458 .50611 .67826 .4923 .66372 Metetra .635 .736 .921 r .4923 .66372 .50611 .67826 .53117 .67181 .51749 .6574 Metetra .636 .74 .923 r .51749 .6574 .53117 .67181 .55647 .66522 .54293 .65091 Metetra .637 .743 .925 r .54293 .65091 .55647 .66522 .58201 .65851 .56862 .64425 Metetra .639 .747 .927 r .56862 .64425 .58201 .65851 .60779 .65168 .59456 .63744 Metetra .64 .749 .928 r .59456 .63744 .60779 .65168 .63382 .64474 .62074 .63049 Metetra .64 .751 .929 r .62074 .63049 .63382 .64474 .66009 .63768 .64718 .62341 Metetra .641 .753 .93 r .64718 .62341 .66009 .63768 .68661 .63051 .67387 .61619 Metetra .642 .754 .93 r .67387 .61619 .68661 .63051 .71339 .62325 .70082 .60886 Metetra .642 .756 .931 r .70082 .60886 .71339 .62325 .74042 .61588 .72803 .60142 Metetra .643 .757 .931 r .72803 .60142 .74042 .61588 .76771 .60841 .75551 .59386 Metetra .643 .758 .932 r .75551 .59386 .76771 .60841 .79526 .60085 .78325 .5862 Metetra .644 .758 .932 r .78325 .5862 .79526 .60085 .82308 .5932 .81126 .57844 Metetra .644 .759 .932 r .81126 .57844 .82308 .5932 .85117 .58545 .83955 .57058 Metetra .644 .759 .933 r .83955 .57058 .85117 .58545 .87954 .57761 .86811 .56262 Metetra .644 .76 .933 r .86811 .56262 .87954 .57761 .90818 .56968 .89696 .55456 Metetra .645 .76 .933 r .89696 .55456 .90818 .56968 .9371 .56166 .92609 .54641 Metetra .604 .633 .854 r .28498 .68603 .29947 .70737 .32273 .70255 .30822 .68237 Metetra .606 .648 .865 r .30822 .68237 .32273 .70255 .34623 .69758 .33173 .67846 Metetra .609 .662 .876 r .33173 .67846 .34623 .69758 .36997 .69243 .35549 .67426 Metetra .613 .675 .886 r .35549 .67426 .36997 .69243 .39395 .68709 .37953 .66973 Metetra .616 .688 .894 r .37953 .66973 .39395 .68709 .41817 .68154 .40382 .66487 Metetra .62 .7 .902 r .40382 .66487 .41817 .68154 .44264 .6758 .42838 .65968 Metetra .623 .71 .908 r .42838 .65968 .44264 .6758 .46734 .66986 .45319 .65419 Metetra .626 .719 .913 r .45319 .65419 .46734 .66986 .4923 .66372 .47826 .6484 Metetra .629 .726 .917 r .47826 .6484 .4923 .66372 .51749 .6574 .50359 .64234 Metetra .632 .732 .92 r .50359 .64234 .51749 .6574 .54293 .65091 .52917 .63603 Metetra .634 .738 .923 r .52917 .63603 .54293 .65091 .56862 .64425 .555 .6295 Metetra .636 .742 .925 r .555 .6295 .56862 .64425 .59456 .63744 .58109 .62277 Metetra .637 .745 .927 r .58109 .62277 .59456 .63744 .62074 .63049 .60744 .61586 Metetra .638 .748 .928 r .60744 .61586 .62074 .63049 .64718 .62341 .63404 .60877 Metetra .64 .751 .929 r .63404 .60877 .64718 .62341 .67387 .61619 .66091 .60154 Metetra .641 .753 .93 r .66091 .60154 .67387 .61619 .70082 .60886 .68803 .59416 Metetra .641 .754 .931 r .68803 .59416 .70082 .60886 .72803 .60142 .71542 .58665 Metetra .642 .756 .931 r .71542 .58665 .72803 .60142 .75551 .59386 .74308 .57901 Metetra .643 .757 .932 r .74308 .57901 .75551 .59386 .78325 .5862 .77101 .57126 Metetra .643 .758 .932 r .77101 .57126 .78325 .5862 .81126 .57844 .79922 .56339 Metetra .643 .758 .932 r .79922 .56339 .81126 .57844 .83955 .57058 .82771 .55541 Metetra .644 .759 .932 r .82771 .55541 .83955 .57058 .86811 .56262 .85647 .54733 Metetra .644 .759 .933 r .85647 .54733 .86811 .56262 .89696 .55456 .88553 .53915 Metetra .644 .76 .933 r .88553 .53915 .89696 .55456 .92609 .54641 .91488 .53086 Metetra .572 .567 .809 r .27078 .65821 .28498 .68603 .30822 .68237 .29387 .65689 Metetra .577 .588 .827 r .29387 .65689 .30822 .68237 .33173 .67846 .31727 .65508 Metetra .583 .61 .845 r .31727 .65508 .33173 .67846 .35549 .67426 .34099 .6527 Metetra .59 .632 .861 r .34099 .6527 .35549 .67426 .37953 .66973 .36501 .64971 Metetra .597 .652 .876 r .36501 .64971 .37953 .66973 .40382 .66487 .38933 .64611 Metetra .603 .671 .888 r .38933 .64611 .40382 .66487 .42838 .65968 .41394 .64194 Metetra .61 .687 .897 r .41394 .64194 .42838 .65968 .45319 .65419 .43884 .63724 Metetra .616 .701 .905 r .43884 .63724 .45319 .65419 .47826 .6484 .46402 .63206 Metetra .62 .712 .912 r .46402 .63206 .47826 .6484 .50359 .64234 .48946 .62645 Metetra .625 .722 .916 r .48946 .62645 .50359 .64234 .52917 .63603 .51518 .62048 Metetra .628 .73 .92 r .51518 .62048 .52917 .63603 .555 .6295 .54115 .61419 Metetra .631 .736 .923 r .54115 .61419 .555 .6295 .58109 .62277 .5674 .60761 Metetra .634 .741 .925 r .5674 .60761 .58109 .62277 .60744 .61586 .5939 .60079 Metetra .636 .745 .927 r .5939 .60079 .60744 .61586 .63404 .60877 .62067 .59375 Metetra .638 .748 .928 r .62067 .59375 .63404 .60877 .66091 .60154 .64771 .58652 Metetra .639 .751 .929 r .64771 .58652 .66091 .60154 .68803 .59416 .67501 .57912 Metetra .64 .753 .93 r .67501 .57912 .68803 .59416 .71542 .58665 .70259 .57155 Metetra .641 .754 .931 r .70259 .57155 .71542 .58665 .74308 .57901 .73043 .56385 Metetra .642 .756 .931 r .73043 .56385 .74308 .57901 .77101 .57126 .75856 .55601 Metetra .642 .757 .932 r .75856 .55601 .77101 .57126 .79922 .56339 .78696 .54804 Metetra .643 .758 .932 r .78696 .54804 .79922 .56339 .82771 .55541 .81565 .53995 Metetra .643 .758 .932 r .81565 .53995 .82771 .55541 .85647 .54733 .84462 .53175 Metetra .644 .759 .933 r .84462 .53175 .85647 .54733 .88553 .53915 .87389 .52344 Metetra .644 .76 .933 r .87389 .52344 .88553 .53915 .91488 .53086 .90345 .51502 Metetra .522 .474 .744 r .25742 .61801 .27078 .65821 .29387 .65689 .28007 .62161 Metetra .528 .501 .77 r .28007 .62161 .29387 .65689 .31727 .65508 .30314 .62413 Metetra .538 .532 .797 r .30314 .62413 .31727 .65508 .34099 .6527 .32662 .62541 Metetra .55 .565 .823 r .32662 .62541 .34099 .6527 .36501 .64971 .35051 .62541 Metetra .563 .598 .846 r .35051 .62541 .36501 .64971 .38933 .64611 .37477 .62418 Metetra .576 .628 .866 r .37477 .62418 .38933 .64611 .41394 .64194 .39938 .62184 Metetra .588 .654 .882 r .39938 .62184 .41394 .64194 .43884 .63724 .42432 .61853 Metetra .598 .676 .895 r .42432 .61853 .43884 .63724 .46402 .63206 .44957 .61438 Metetra .607 .694 .904 r .44957 .61438 .46402 .63206 .48946 .62645 .47512 .60954 Metetra .615 .708 .911 r .47512 .60954 .48946 .62645 .51518 .62048 .50096 .60412 Metetra .621 .719 .917 r .50096 .60412 .51518 .62048 .54115 .61419 .52707 .59821 Metetra .625 .728 .921 r .52707 .59821 .54115 .61419 .5674 .60761 .55347 .5919 Metetra .629 .735 .924 r .55347 .5919 .5674 .60761 .5939 .60079 .58013 .58525 Metetra .632 .74 .926 r .58013 .58525 .5939 .60079 .62067 .59375 .60707 .57831 Metetra .635 .745 .928 r .60707 .57831 .62067 .59375 .64771 .58652 .63428 .57112 Metetra .637 .748 .929 r .63428 .57112 .64771 .58652 .67501 .57912 .66176 .56372 Metetra .638 .751 .93 r .66176 .56372 .67501 .57912 .70259 .57155 .68952 .55613 Metetra .64 .753 .931 r .68952 .55613 .70259 .57155 .73043 .56385 .71755 .54836 Metetra .641 .755 .931 r .71755 .54836 .73043 .56385 .75856 .55601 .74587 .54044 Metetra .642 .756 .932 r .74587 .54044 .75856 .55601 .78696 .54804 .77447 .53238 Metetra .642 .757 .932 r .77447 .53238 .78696 .54804 .81565 .53995 .80336 .52419 Metetra .643 .758 .932 r .80336 .52419 .81565 .53995 .84462 .53175 .83255 .51587 Metetra .643 .759 .933 r .83255 .51587 .84462 .53175 .87389 .52344 .86203 .50742 Metetra .644 .759 .933 r .86203 .50742 .87389 .52344 .90345 .51502 .89181 .49887 Metetra .45 .358 .66 r .24617 .55205 .25742 .61801 .28007 .62161 .26768 .56686 Metetra .453 .385 .69 r .26768 .56686 .28007 .62161 .30314 .62413 .28988 .57895 Metetra .465 .422 .726 r .28988 .57895 .30314 .62413 .32662 .62541 .31275 .58798 Metetra .484 .467 .764 r .31275 .58798 .32662 .62541 .35051 .62541 .33623 .59399 Metetra .507 .516 .801 r .33623 .59399 .35051 .62541 .37477 .62418 .36025 .59729 Metetra .53 .563 .833 r .36025 .59729 .37477 .62418 .39938 .62184 .38474 .59828 Metetra .552 .605 .859 r .38474 .59828 .39938 .62184 .42432 .61853 .40966 .59738 Metetra .571 .64 .88 r .40966 .59738 .42432 .61853 .44957 .61438 .43494 .59496 Metetra .587 .667 .894 r .43494 .59496 .44957 .61438 .47512 .60954 .46057 .59135 Metetra .599 .689 .905 r .46057 .59135 .47512 .60954 .50096 .60412 .48651 .58679 Metetra .609 .706 .913 r .48651 .58679 .50096 .60412 .52707 .59821 .51276 .58148 Metetra .617 .718 .918 r .51276 .58148 .52707 .59821 .55347 .5919 .5393 .57558 Metetra .623 .728 .922 r .5393 .57558 .55347 .5919 .58013 .58525 .56612 .5692 Metetra .628 .735 .925 r .56612 .5692 .58013 .58525 .60707 .57831 .59322 .56242 Metetra .631 .741 .927 r .59322 .56242 .60707 .57831 .63428 .57112 .6206 .55533 Metetra .634 .745 .928 r .6206 .55533 .63428 .57112 .66176 .56372 .64827 .54796 Metetra .636 .749 .93 r .64827 .54796 .66176 .56372 .68952 .55613 .67621 .54035 Metetra .638 .751 .93 r .67621 .54035 .68952 .55613 .71755 .54836 .70444 .53255 Metetra .639 .753 .931 r .70444 .53255 .71755 .54836 .74587 .54044 .73295 .52456 Metetra .641 .755 .932 r .73295 .52456 .74587 .54044 .77447 .53238 .76175 .51641 Metetra .641 .756 .932 r .76175 .51641 .77447 .53238 .80336 .52419 .79085 .50811 Metetra .642 .757 .932 r .79085 .50811 .80336 .52419 .83255 .51587 .82025 .49967 Metetra .643 .758 .933 r .82025 .49967 .83255 .51587 .86203 .50742 .84994 .4911 Metetra .643 .759 .933 r .84994 .4911 .86203 .50742 .89181 .49887 .87995 .4824 Metetra .429 .296 .597 r .23889 .44355 .24617 .55205 .26768 .56686 .25882 .47006 Metetra .353 .25 .596 r .25882 .47006 .26768 .56686 .28988 .57895 .27877 .50525 Metetra .36 .283 .633 r .27877 .50525 .28988 .57895 .31275 .58798 .30009 .5318 Metetra .382 .335 .681 r .30009 .5318 .31275 .58798 .33623 .59399 .32257 .55046 Metetra .416 .4 .734 r .32257 .55046 .33623 .59399 .36025 .59729 .346 .5626 Metetra .456 .47 .784 r .346 .5626 .36025 .59729 .38474 .59828 .37016 .56966 Metetra .495 .534 .826 r .37016 .56966 .38474 .59828 .40966 .59738 .39492 .57289 Metetra .529 .589 .859 r .39492 .57289 .40966 .59738 .43494 .59496 .42017 .57329 Metetra .556 .632 .882 r .42017 .57329 .43494 .59496 .46057 .59135 .44583 .57159 Metetra .577 .664 .897 r .44583 .57159 .46057 .59135 .48651 .58679 .47185 .56833 Metetra .594 .689 .908 r .47185 .56833 .48651 .58679 .51276 .58148 .49821 .5639 Metetra .606 .706 .915 r .49821 .5639 .51276 .58148 .5393 .57558 .52489 .55859 Metetra .615 .72 .92 r .52489 .55859 .5393 .57558 .56612 .5692 .55186 .5526 Metetra .622 .729 .924 r .55186 .5526 .56612 .5692 .59322 .56242 .57913 .54607 Metetra .627 .737 .926 r .57913 .54607 .59322 .56242 .6206 .55533 .60668 .53912 Metetra .631 .742 .928 r .60668 .53912 .6206 .55533 .64827 .54796 .63452 .53181 Metetra .634 .746 .929 r .63452 .53181 .64827 .54796 .67621 .54035 .66266 .52422 Metetra .636 .75 .93 r .66266 .52422 .67621 .54035 .70444 .53255 .69108 .51639 Metetra .638 .752 .931 r .69108 .51639 .70444 .53255 .73295 .52456 .71979 .50834 Metetra .639 .754 .932 r .71979 .50834 .73295 .52456 .76175 .51641 .7488 .50011 Metetra .641 .756 .932 r .7488 .50011 .76175 .51641 .79085 .50811 .7781 .4917 Metetra .642 .757 .932 r .7781 .4917 .79085 .50811 .82025 .49967 .80771 .48315 Metetra .642 .758 .933 r .80771 .48315 .82025 .49967 .84994 .4911 .83763 .47445 Metetra .643 .759 .933 r .83763 .47445 .84994 .4911 .87995 .4824 .86786 .46561 Metetra .664 .741 .907 r .22357 .42924 .23889 .44355 .25882 .47006 .24638 .42139 Metetra .613 .5 .701 r .24638 .42139 .25882 .47006 .27877 .50525 .26939 .41348 Metetra .397 .286 .609 r .26939 .41348 .27877 .50525 .30009 .5318 .2901 .44036 Metetra .24 .174 .578 r .2901 .44036 .30009 .5318 .32257 .55046 .31025 .48647 Metetra .283 .249 .643 r .31025 .48647 .32257 .55046 .346 .5626 .33235 .51598 Metetra .342 .342 .715 r .33235 .51598 .346 .5626 .37016 .56966 .35579 .53395 Metetra .406 .437 .78 r .35579 .53395 .37016 .56966 .39492 .57289 .38019 .54409 Metetra .464 .52 .83 r .38019 .54409 .39492 .57289 .42017 .57329 .40528 .54889 Metetra .511 .585 .866 r .40528 .54889 .42017 .57329 .44583 .57159 .43091 .55003 Metetra .546 .633 .889 r .43091 .55003 .44583 .57159 .47185 .56833 .45698 .54863 Metetra .572 .668 .903 r .45698 .54863 .47185 .56833 .49821 .5639 .48344 .54542 Metetra .591 .693 .913 r .48344 .54542 .49821 .5639 .52489 .55859 .51024 .54092 Metetra .604 .71 .919 r .51024 .54092 .52489 .55859 .55186 .5526 .53736 .53544 Metetra .614 .723 .923 r .53736 .53544 .55186 .5526 .57913 .54607 .56478 .52925 Metetra .621 .732 .926 r .56478 .52925 .57913 .54607 .60668 .53912 .59251 .52249 Metetra .627 .739 .928 r .59251 .52249 .60668 .53912 .63452 .53181 .62053 .51529 Metetra .631 .744 .929 r .62053 .51529 .63452 .53181 .66266 .52422 .64885 .50774 Metetra .634 .748 .93 r .64885 .50774 .66266 .52422 .69108 .51639 .67747 .49989 Metetra .636 .751 .931 r .67747 .49989 .69108 .51639 .71979 .50834 .70638 .49179 Metetra .638 .753 .932 r .70638 .49179 .71979 .50834 .7488 .50011 .73559 .48348 Metetra .64 .755 .932 r .73559 .48348 .7488 .50011 .7781 .4917 .76511 .47498 Metetra .641 .756 .932 r .76511 .47498 .7781 .4917 .80771 .48315 .79494 .4663 Metetra .642 .757 .933 r .79494 .4663 .80771 .48315 .83763 .47445 .82508 .45747 Metetra .642 .758 .933 r .82508 .45747 .83763 .47445 .86786 .46561 .85553 .44849 Metetra .664 .741 .907 r .20801 .4147 .22357 .42924 .24638 .42139 .23093 .40672 Metetra .664 .741 .907 r .23093 .40672 .24638 .42139 .26939 .41348 .25406 .39868 Metetra .664 .741 .907 r .25406 .39868 .26939 .41348 .2901 .44036 .2774 .39056 Metetra .55 .459 .705 r .2774 .39056 .2901 .44036 .31025 .48647 .30037 .39058 Metetra .098 .065 .529 r .30037 .39058 .31025 .48647 .33235 .51598 .31977 .45276 Metetra .177 .178 .621 r .31977 .45276 .33235 .51598 .35579 .53395 .34184 .48928 Metetra .275 .308 .715 r .34184 .48928 .35579 .53395 .38019 .54409 .36555 .51021 Metetra .369 .429 .793 r .36555 .51021 .38019 .54409 .40528 .54889 .39033 .52147 Metetra .446 .526 .846 r .39033 .52147 .40528 .54889 .43091 .55003 .41584 .52658 Metetra .503 .596 .879 r .41584 .52658 .43091 .55003 .45698 .54863 .44191 .52767 Metetra .543 .644 .899 r .44191 .52767 .45698 .54863 .48344 .54542 .46843 .52605 Metetra .571 .677 .911 r .46843 .52605 .48344 .54542 .51024 .54092 .49534 .52257 Metetra .591 .7 .919 r .49534 .52257 .51024 .54092 .53736 .53544 .5226 .51775 Metetra .605 .716 .923 r .5226 .51775 .53736 .53544 .56478 .52925 .55018 .51197 Metetra .615 .727 .926 r .55018 .51197 .56478 .52925 .59251 .52249 .57808 .50546 Metetra .622 .736 .928 r .57808 .50546 .59251 .52249 .62053 .51529 .60629 .4984 Metetra .627 .742 .93 r .60629 .4984 .62053 .51529 .64885 .50774 .63479 .4909 Metetra .631 .746 .931 r .63479 .4909 .64885 .50774 .67747 .49989 .6636 .48305 Metetra .634 .75 .931 r .6636 .48305 .67747 .49989 .70638 .49179 .69272 .47491 Metetra .637 .752 .932 r .69272 .47491 .70638 .49179 .73559 .48348 .72214 .46652 Metetra .638 .754 .932 r .72214 .46652 .73559 .48348 .76511 .47498 .75187 .45792 Metetra .64 .756 .932 r .75187 .45792 .76511 .47498 .79494 .4663 .78192 .44912 Metetra .641 .757 .933 r .78192 .44912 .79494 .4663 .82508 .45747 .81228 .44016 Metetra .642 .758 .933 r .81228 .44016 .82508 .45747 .85553 .44849 .84297 .43103 Metetra .664 .741 .907 r .19218 .39992 .20801 .4147 .23093 .40672 .21523 .39181 Metetra .664 .741 .907 r .21523 .39181 .23093 .40672 .25406 .39868 .23848 .38363 Metetra .664 .741 .907 r .23848 .38363 .25406 .39868 .2774 .39056 .26194 .37538 Metetra .664 .741 .907 r .26194 .37538 .2774 .39056 .30037 .39058 .28561 .36705 Metetra .437 .546 .869 r .28561 .36705 .30037 .39058 .31977 .45276 .30865 .37113 Metetra 0 0 .496 r .30865 .37113 .31977 .45276 .34184 .48928 .32846 .43546 Metetra .084 .143 .627 r .32846 .43546 .34184 .48928 .36555 .51021 .35107 .47148 Metetra .232 .314 .743 r .35107 .47148 .36555 .51021 .39033 .52147 .37533 .4913 Metetra .356 .453 .823 r .37533 .4913 .39033 .52147 .41584 .52658 .40063 .50146 Metetra .446 .552 .87 r .40063 .50146 .41584 .52658 .44191 .52767 .42664 .50561 Metetra .507 .618 .896 r .42664 .50561 .44191 .52767 .46843 .52605 .4532 .5059 Metetra .548 .661 .911 r .4532 .5059 .46843 .52605 .49534 .52257 .48021 .50362 Metetra .575 .69 .919 r .48021 .50362 .49534 .52257 .5226 .51775 .50759 .49957 Metetra .594 .709 .924 r .50759 .49957 .5226 .51775 .55018 .51197 .53533 .49426 Metetra .607 .723 .927 r .53533 .49426 .55018 .51197 .57808 .50546 .56339 .48805 Metetra .617 .733 .929 r .56339 .48805 .57808 .50546 .60629 .4984 .59178 .48115 Metetra .624 .74 .93 r .59178 .48115 .60629 .4984 .63479 .4909 .62047 .47372 Metetra .629 .745 .931 r .62047 .47372 .63479 .4909 .6636 .48305 .64948 .46587 Metetra .632 .749 .932 r .64948 .46587 .6636 .48305 .69272 .47491 .6788 .45769 Metetra .635 .751 .932 r .6788 .45769 .69272 .47491 .72214 .46652 .70843 .44923 Metetra .637 .754 .932 r .70843 .44923 .72214 .46652 .75187 .45792 .73838 .44052 Metetra .639 .755 .933 r .73838 .44052 .75187 .45792 .78192 .44912 .76865 .43161 Metetra .64 .757 .933 r .76865 .43161 .78192 .44912 .81228 .44016 .79924 .42251 Metetra .641 .758 .933 r .79924 .42251 .81228 .44016 .84297 .43103 .83016 .41323 Metetra .039 .168 .672 r .33671 .42968 .35107 .47148 .37533 .4913 .36027 .45937 Metetra .63 .748 .932 r .63509 .44837 .64948 .46587 .6788 .45769 .66461 .44014 Metetra .634 .751 .932 r .66461 .44014 .6788 .45769 .70843 .44923 .69446 .4316 Metetra .636 .753 .933 r .69446 .4316 .70843 .44923 .73838 .44052 .72462 .42279 Metetra 0 0 .561 r .32232 .38788 .33671 .42968 .36027 .45937 .34509 .42725 Metetra .628 .747 .933 r .62042 .43054 .63509 .44837 .66461 .44014 .65015 .42227 Metetra .632 .75 .933 r .65015 .42227 .66461 .44014 .69446 .4316 .68021 .41364 Metetra .635 .753 .933 r .68021 .41364 .69446 .4316 .72462 .42279 .71059 .40473 Metetra .664 .741 .907 r .14311 .35409 .15974 .36962 .18302 .36124 .16652 .34557 Metetra .664 .741 .907 r .16652 .34557 .18302 .36124 .20652 .35279 .19013 .33697 Metetra .664 .741 .907 r .19013 .33697 .20652 .35279 .23023 .34425 .21397 .32829 Metetra .664 .741 .907 r .21397 .32829 .23023 .34425 .25416 .33564 .23803 .31953 Metetra .664 .741 .907 r .23803 .31953 .25416 .33564 .27831 .32695 .26231 .31069 Metetra .664 .741 .907 r .26231 .31069 .27831 .32695 .30268 .31818 .28682 .30176 Metetra .864 .425 .024 r .28682 .30176 .30268 .31818 .32232 .38788 .3077 .34903 Metetra 0 0 .397 r .3077 .34903 .32232 .38788 .34509 .42725 .32969 .39647 Metetra 0 .141 .684 r .32969 .39647 .34509 .42725 .36971 .44875 .35392 .42286 Metetra .171 .393 .837 r .35392 .42286 .36971 .44875 .39548 .45971 .37954 .43683 Metetra .351 .538 .898 r .37954 .43683 .39548 .45971 .42203 .46419 .40607 .4432 Metetra .456 .619 .92 r .40607 .4432 .42203 .46419 .44918 .46452 .43326 .44472 Metetra .519 .666 .929 r .43326 .44472 .44918 .46452 .47679 .46212 .46098 .44307 Metetra .557 .695 .932 r .46098 .44307 .47679 .46212 .50481 .45783 .48914 .43925 Metetra .583 .714 .933 r .48914 .43925 .50481 .45783 .5332 .45223 .51769 .43391 Metetra .6 .727 .934 r .51769 .43391 .5332 .45223 .56194 .44566 .54661 .42749 Metetra .612 .736 .934 r .54661 .42749 .56194 .44566 .59102 .43838 .57587 .42026 Metetra .62 .742 .934 r .57587 .42026 .59102 .43838 .62042 .43054 .60547 .41241 Metetra .626 .747 .934 r .60547 .41241 .62042 .43054 .65015 .42227 .63541 .40408 Metetra .63 .75 .934 r .63541 .40408 .65015 .42227 .68021 .41364 .66568 .39536 Metetra .634 .753 .934 r .66568 .39536 .68021 .41364 .71059 .40473 .69629 .38632 Metetra .636 .755 .934 r .69629 .38632 .71059 .40473 .74131 .39555 .72724 .37701 Metetra .638 .756 .934 r .72724 .37701 .74131 .39555 .77237 .38616 .75853 .36746 Metetra .64 .757 .934 r .75853 .36746 .77237 .38616 .80376 .37657 .79017 .3577 Metetra .664 .741 .907 r .1262 .3383 .14311 .35409 .16652 .34557 .14973 .32963 Metetra .664 .741 .907 r .14973 .32963 .16652 .34557 .19013 .33697 .17347 .32088 Metetra .664 .741 .907 r .17347 .32088 .19013 .33697 .21397 .32829 .19743 .31205 Metetra .664 .741 .907 r .19743 .31205 .21397 .32829 .23803 .31953 .22162 .30314 Metetra .664 .741 .907 r .22162 .30314 .23803 .31953 .26231 .31069 .24603 .29415 Metetra .664 .741 .907 r .24603 .29415 .26231 .31069 .28682 .30176 .27068 .28507 Metetra .672 .18 0 r .27068 .28507 .28682 .30176 .3077 .34903 .29269 .31464 Metetra .707 .35 0 r .29269 .31464 .3077 .34903 .32969 .39647 .31398 .36798 Metetra 0 .017 .588 r .31398 .36798 .32969 .39647 .35392 .42286 .33785 .39814 Metetra .052 .333 .813 r .33785 .39814 .35392 .42286 .37954 .43683 .36331 .41453 Metetra .29 .514 .899 r .36331 .41453 .37954 .43683 .40607 .4432 .38982 .42244 Metetra .423 .609 .926 r .38982 .42244 .40607 .4432 .43326 .44472 .41707 .42496 Metetra .499 .662 .934 r .41707 .42496 .43326 .44472 .46098 .44307 .44488 .42393 Metetra .546 .693 .936 r .44488 .42393 .46098 .44307 .48914 .43925 .47318 .4205 Metetra .575 .713 .936 r .47318 .4205 .48914 .43925 .51769 .43391 .50189 .41538 Metetra .595 .726 .936 r .50189 .41538 .51769 .43391 .54661 .42749 .53098 .40906 Metetra .608 .735 .936 r .53098 .40906 .54661 .42749 .57587 .42026 .56043 .40186 Metetra .617 .742 .935 r .56043 .40186 .57587 .42026 .60547 .41241 .59023 .39397 Metetra .624 .747 .935 r .59023 .39397 .60547 .41241 .63541 .40408 .62038 .38557 Metetra .629 .75 .934 r .62038 .38557 .63541 .40408 .66568 .39536 .65087 .37674 Metetra .633 .753 .934 r .65087 .37674 .66568 .39536 .69629 .38632 .6817 .36757 Metetra .636 .755 .934 r .6817 .36757 .69629 .38632 .72724 .37701 .71288 .3581 Metetra .638 .756 .934 r .71288 .3581 .72724 .37701 .75853 .36746 .74441 .34839 Metetra .639 .757 .934 r .74441 .34839 .75853 .36746 .79017 .3577 .77629 .33845 Metetra .664 .741 .907 r .109 .32223 .1262 .3383 .14973 .32963 .13265 .31342 Metetra .664 .741 .907 r .13265 .31342 .14973 .32963 .17347 .32088 .15652 .30452 Metetra .664 .741 .907 r .15652 .30452 .17347 .32088 .19743 .31205 .18061 .29554 Metetra .664 .741 .907 r .18061 .29554 .19743 .31205 .22162 .30314 .20493 .28647 Metetra .664 .741 .907 r .20493 .28647 .22162 .30314 .24603 .29415 .22948 .27732 Metetra .664 .741 .907 r .22948 .27732 .24603 .29415 .27068 .28507 .25426 .26808 Metetra 0 .208 .705 r .25426 .26808 .27068 .28507 .29269 .31464 .27721 .28475 Metetra .821 .432 0 r .27721 .28475 .29269 .31464 .31398 .36798 .29789 .34199 Metetra 0 0 .479 r .29789 .34199 .31398 .36798 .33785 .39814 .32146 .37481 Metetra 0 .275 .779 r .32146 .37481 .33785 .39814 .36331 .41453 .34678 .39298 Metetra .229 .493 .897 r .34678 .39298 .36331 .41453 .38982 .42244 .37327 .40206 Metetra .391 .602 .931 r .37327 .40206 .38982 .42244 .41707 .42496 .40056 .40534 Metetra .481 .66 .94 r .40056 .40534 .41707 .42496 .44488 .42393 .42849 .40479 Metetra .535 .693 .941 r .42849 .40479 .44488 .42393 .47318 .4205 .45691 .40164 Metetra .568 .713 .94 r .45691 .40164 .47318 .4205 .50189 .41538 .48578 .39667 Metetra .59 .727 .939 r .48578 .39667 .50189 .41538 .53098 .40906 .51505 .39041 Metetra .605 .736 .937 r .51505 .39041 .53098 .40906 .56043 .40186 .54469 .38319 Metetra .615 .742 .936 r .54469 .38319 .56043 .40186 .59023 .39397 .57469 .37525 Metetra .622 .747 .936 r .57469 .37525 .59023 .39397 .62038 .38557 .60505 .36674 Metetra .628 .75 .935 r .60505 .36674 .62038 .38557 .65087 .37674 .63576 .35779 Metetra .632 .753 .935 r .63576 .35779 .65087 .37674 .6817 .36757 .66682 .34847 Metetra .635 .755 .935 r .66682 .34847 .6817 .36757 .71288 .3581 .69823 .33884 Metetra .637 .756 .934 r .69823 .33884 .71288 .3581 .74441 .34839 .73 .32895 Metetra .639 .757 .934 r .73 .32895 .74441 .34839 .77629 .33845 .76214 .31882 Metetra .664 .741 .907 r .09151 .30589 .109 .32223 .13265 .31342 .11528 .29692 Metetra .664 .741 .907 r .11528 .29692 .13265 .31342 .15652 .30452 .13927 .28787 Metetra .664 .741 .907 r .13927 .28787 .15652 .30452 .18061 .29554 .16349 .27873 Metetra .664 .741 .907 r .16349 .27873 .18061 .29554 .20493 .28647 .18794 .2695 Metetra .664 .741 .907 r .18794 .2695 .20493 .28647 .22948 .27732 .21263 .26019 Metetra .664 .741 .907 r .21263 .26019 .22948 .27732 .25426 .26808 .23755 .25079 Metetra .161 .501 .909 r .23755 .25079 .25426 .26808 .27721 .28475 .26123 .25861 Metetra .871 .471 .011 r .26123 .25861 .27721 .28475 .29789 .34199 .2814 .31828 Metetra .602 .164 0 r .2814 .31828 .29789 .34199 .32146 .37481 .3047 .35283 Metetra 0 .223 .739 r .3047 .35283 .32146 .37481 .34678 .39298 .32991 .3722 Metetra .174 .474 .893 r .32991 .3722 .34678 .39298 .37327 .40206 .35639 .38208 Metetra .363 .597 .936 r .35639 .38208 .37327 .40206 .40056 .40534 .38375 .38588 Metetra .466 .659 .945 r .38375 .38588 .40056 .40534 .42849 .40479 .41177 .38565 Metetra .525 .693 .945 r .41177 .38565 .42849 .40479 .45691 .40164 .44033 .38268 Metetra .562 .714 .943 r .44033 .38268 .45691 .40164 .48578 .39667 .46936 .3778 Metetra .586 .728 .941 r .46936 .3778 .48578 .39667 .51505 .39041 .4988 .37154 Metetra .602 .737 .939 r .4988 .37154 .51505 .39041 .54469 .38319 .52864 .36427 Metetra .613 .743 .938 r .52864 .36427 .54469 .38319 .57469 .37525 .55884 .35623 Metetra .621 .747 .937 r .55884 .35623 .57469 .37525 .60505 .36674 .58941 .3476 Metetra .627 .751 .936 r .58941 .3476 .60505 .36674 .63576 .35779 .62034 .33851 Metetra .631 .753 .935 r .62034 .33851 .63576 .35779 .66682 .34847 .65163 .32902 Metetra .634 .755 .935 r .65163 .32902 .66682 .34847 .69823 .33884 .68328 .31922 Metetra .637 .756 .935 r .68328 .31922 .69823 .33884 .73 .32895 .7153 .30913 Metetra .639 .758 .934 r .7153 .30913 .73 .32895 .76214 .31882 .74768 .2988 Metetra .664 .741 .907 r .07371 .28927 .09151 .30589 .11528 .29692 .0976 .28014 Metetra .664 .741 .907 r .0976 .28014 .11528 .29692 .13927 .28787 .12172 .27093 Metetra .664 .741 .907 r .12172 .27093 .13927 .28787 .16349 .27873 .14607 .26163 Metetra .664 .741 .907 r .14607 .26163 .16349 .27873 .18794 .2695 .17065 .25224 Metetra .664 .741 .907 r .17065 .25224 .18794 .2695 .21263 .26019 .19547 .24276 Metetra .664 .741 .907 r .19547 .24276 .21263 .26019 .23755 .25079 .22054 .23318 Metetra .382 .626 .949 r .22054 .23318 .23755 .25079 .26123 .25861 .24476 .23533 Metetra .893 .489 .067 r .24476 .23533 .26123 .25861 .2814 .31828 .26449 .29642 Metetra .67 .212 0 r .26449 .29642 .2814 .31828 .3047 .35283 .28756 .33203 Metetra 0 .181 .699 r .28756 .33203 .3047 .35283 .32991 .3722 .31269 .35213 Metetra .127 .459 .887 r .31269 .35213 .32991 .3722 .35639 .38208 .33916 .3625 Metetra .339 .594 .94 r .33916 .3625 .35639 .38208 .38375 .38588 .36659 .36661 Metetra .453 .66 .95 r .36659 .36661 .38375 .38588 .41177 .38565 .39472 .36655 Metetra .518 .695 .949 r .39472 .36655 .41177 .38565 .44033 .38268 .42343 .36365 Metetra .557 .716 .946 r .42343 .36365 .44033 .38268 .46936 .3778 .45261 .35877 Metetra .583 .729 .943 r .45261 .35877 .46936 .3778 .4988 .37154 .48224 .35245 Metetra .6 .738 .941 r .48224 .35245 .4988 .37154 .52864 .36427 .51226 .34508 Metetra .612 .744 .939 r .51226 .34508 .52864 .36427 .55884 .35623 .54267 .33692 Metetra .62 .748 .938 r .54267 .33692 .55884 .35623 .58941 .3476 .57346 .32815 Metetra .626 .751 .937 r .57346 .32815 .58941 .3476 .62034 .33851 .60461 .31888 Metetra .63 .754 .936 r .60461 .31888 .62034 .33851 .65163 .32902 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for ", StyleBox["z", FontSlant->"Italic"], " values equal to and larger than the cylinder height. The cylinder height \ is 1 and its radius is 1;.the potential is plotted for ", StyleBox["z", FontSlant->"Italic"], " = 1.1, ", StyleBox["z", FontSlant->"Italic"], " = 1.2, ", StyleBox["z", FontSlant->"Italic"], " = 1.3, and ", StyleBox["z", FontSlant->"Italic"], "= 1.4. All these ", StyleBox["z", FontSlant->"Italic"], " values are beyond the end of the cylinder, and the potentials do not \ diverge at ", StyleBox["dist", FontSlant->"Italic"], " = 1. Note that unlike the previous plots, this one extends from ", StyleBox["dist", FontSlant->"Italic"], " = 0 to 3." }], "Subsubsection"], Cell["\<\ The contour plot below would take an enormously long time to \ compute if we had not employed all of the ``integral tricks''\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ContourPlot[ LondonCylinderPotential[dist, height, 1, 0.25], {dist, 0.001, 2}, {height, 0.001, 2}, Contours -> 25, ColorFunction -> \((Hue[0.6\ #] &)\)]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% ContourGraphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0187498 0.48101 0.0187498 0.48101 [ [.01875 -0.0125 -7 -20 ] [.01875 -0.0125 7 0 ] [.25925 -0.0125 -14.5 -20 ] [.25925 -0.0125 14.5 0 ] [.49976 -0.0125 -7 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.30391 L .29968 .30181 L .33239 .29968 L .33974 .29877 L .37981 .29293 L .41987 .2853 L .45994 .27564 L .5 .26307 L .50895 .25962 L .54006 .24265 L .57731 .21955 L .58013 .2171 L .6182 .17949 L .62019 .17716 L .6435 .13942 L .66026 .10501 L .66195 .09936 L .67019 .05929 L .67279 .01923 L .01923 .01923 L F 0 g .01923 .30836 m .05929 .30826 L .09936 .30796 L .13942 .30744 L .17949 .30664 L .21955 .30549 L .25962 .30391 L .29968 .30181 L .33239 .29968 L .33974 .29877 L .37981 .29293 L .41987 .2853 L .45994 .27564 L .5 .26307 L .50895 .25962 L .54006 .24265 L .57731 .21955 L .58013 .2171 L .6182 .17949 L .62019 .17716 L .6435 .13942 L .66026 .10501 L .66195 .09936 L .67019 .05929 L .67279 .01923 L s 1 .432 0 r .01923 .30346 m .05929 .30338 L .09936 .30313 L .13942 .30269 L .17949 .30202 L .21955 .30106 L .25962 .29976 L .26181 .29968 L .29968 .29696 L .33974 .29301 L .37981 .28764 L .41987 .28055 L .45994 .27133 L .49802 .25962 L .5 .25843 L .54006 .23813 L .57066 .21955 L .58013 .21138 L .61269 .17949 L .62019 .17073 L .63941 .13942 L .65666 .09936 L .66026 .09548 L .66698 .05929 L .66962 .01923 L .01923 .01923 L F 0 g .01923 .30346 m .05929 .30338 L .09936 .30313 L .13942 .30269 L .17949 .30202 L .21955 .30106 L .25962 .29976 L .26181 .29968 L .29968 .29696 L .33974 .29301 L .37981 .28764 L .41987 .28055 L .45994 .27133 L .49802 .25962 L .5 .25843 L .54006 .23813 L .57066 .21955 L .58013 .21138 L .61269 .17949 L .62019 .17073 L .63941 .13942 L .65666 .09936 L .66026 .09548 L .66698 .05929 L .66962 .01923 L s 1 .288 0 r .01923 .29936 m .05929 .29926 L .09936 .29894 L .13942 .29837 L .17949 .29749 L .21955 .2962 L .25962 .29435 L .29968 .29174 L .33974 .2881 L .37981 .28312 L .41987 .27643 L .45994 .2675 L .48711 .25962 L .5 .25284 L .54006 .23419 L .56453 .21955 L .58013 .20621 L .60768 .17949 L .62019 .16487 L .6358 .13942 L .65152 .09936 L .66026 .08567 L .66409 .05929 L .66679 .01923 L .01923 .01923 L F 0 g .01923 .29936 m .05929 .29926 L .09936 .29894 L .13942 .29837 L .17949 .29749 L .21955 .2962 L .25962 .29435 L .29968 .29174 L .33974 .2881 L .37981 .28312 L .41987 .27643 L .45994 .2675 L .48711 .25962 L .5 .25284 L .54006 .23419 L .56453 .21955 L .58013 .20621 L .60768 .17949 L .62019 .16487 L .6358 .13942 L .65152 .09936 L .66026 .08567 L .66409 .05929 L .66679 .01923 L s 1 .144 0 r .01923 .29419 m .05929 .2941 L .09936 .29381 L .13942 .29329 L .17949 .29249 L .21955 .29131 L .25962 .28962 L .29968 .28722 L .33974 .28384 L .37981 .27916 L .41987 .27277 L .45994 .26406 L .47614 .25962 L .5 .2484 L .54006 .23069 L .55888 .21955 L .58013 .20157 L .60314 .17949 L .62019 .15954 L .63257 .13942 L .64767 .09936 L .66026 .06995 L .66145 .05929 L .66422 .01923 L .01923 .01923 L F 0 g .01923 .29419 m .05929 .2941 L .09936 .29381 L .13942 .29329 L .17949 .29249 L .21955 .29131 L .25962 .28962 L .29968 .28722 L .33974 .28384 L .37981 .27916 L .41987 .27277 L .45994 .26406 L .47614 .25962 L .5 .2484 L .54006 .23069 L .55888 .21955 L .58013 .20157 L .60314 .17949 L .62019 .15954 L .63257 .13942 L .64767 .09936 L .66026 .06995 L .66145 .05929 L .66422 .01923 L s 1 0 0 r .01923 .28966 m .05929 .28958 L .09936 .28932 L .13942 .28885 L .17949 .28812 L .21955 .28703 L .25962 .28547 L .29968 .28324 L .33974 .28007 L .37981 .27563 L .41987 .26948 L .45994 .26091 L .46495 .25962 L .5 .24472 L .54006 .22752 L .55365 .21955 L .58013 .19741 L .59905 .17949 L .62019 .15468 L .62963 .13942 L .64454 .09936 L .65501 .05929 L .66026 .04501 L .66186 .01923 L .01923 .01923 L F 0 g .01923 .28966 m .05929 .28958 L .09936 .28932 L .13942 .28885 L .17949 .28812 L .21955 .28703 L .25962 .28547 L .29968 .28324 L .33974 .28007 L .37981 .27563 L .41987 .26948 L .45994 .26091 L .46495 .25962 L .5 .24472 L .54006 .22752 L .55365 .21955 L .58013 .19741 L .59905 .17949 L .62019 .15468 L .62963 .13942 L .64454 .09936 L .65501 .05929 L .66026 .04501 L .66186 .01923 L s 1 .144 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08076 L .29968 .08076 L .33974 .08077 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08076 L .29968 .08076 L .33974 .08077 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s 1 .288 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08076 L .29968 .08076 L .33974 .08077 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08076 L .29968 .08076 L .33974 .08077 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s 1 .432 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08077 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08077 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s 1 .576 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08076 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08076 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s 1 .72 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08076 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08076 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s 1 .864 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08076 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08076 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s .992 1 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08076 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08076 L .33974 .08076 L .37981 .08079 L .41987 .08089 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s .848 1 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08079 L .41987 .08088 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08079 L .41987 .08088 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s .704 1 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s .56 1 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08184 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08184 L .45994 .05929 L .45994 .01923 L s .416 1 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s .272 1 0 r .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08075 m .05929 .08075 L .09936 .08075 L .13942 .08075 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s .128 1 0 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08075 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s 0 1 .016 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08075 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s 0 1 .16 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s 0 1 .304 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08075 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s 0 1 .448 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s 0 1 .592 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08076 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s 0 1 .736 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08075 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08075 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s 0 1 .88 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08075 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08075 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L s 0 .976 1 r .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08075 L .37981 .08078 L .41987 .08088 L .45994 .08183 L .45994 .05929 L .45994 .01923 L .01923 .01923 L F 0 g .01923 .08074 m .05929 .08074 L .09936 .08074 L .13942 .08074 L .17949 .08074 L .21955 .08074 L .25962 .08074 L .29968 .08075 L .33974 .08075 L .37981 .08078 L 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