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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[ 305652, 8283]*) (*NotebookOutlinePosition[ 313613, 8461]*) (* CellTagsIndexPosition[ 311990, 8426]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Generalizations of the Fundamental Theorem of Calculus (Part \ II)\ \>", "Title", CellTags->"mmtag:16:divergence_theorem"], Cell["\<\ Using Divergence theorem to find London attraction potential \ between point particle and a finite cylinder.\ \>", "Subtitle"], Cell[CellGroupData[{ 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"] }, Open ]], Cell[CellGroupData[{ 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[CellGroupData[{ 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 ]], Cell[CellGroupData[{ Cell["\<\ Here we will use a trickof using Evaluate[] in a function \ definition to save time--a small diversion here will demonstrate why this is \ efficient\ \>", "Text", 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[ \(Timing[ f[c_]\ := \ Evaluate[Integrate[Exp[Tan[x]], {x, 0, c}]]]\)], "Input"], Cell[BoxData[ \({7.776986999999998`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[g[c_]\ := \ Integrate[Exp[Tan[x]], {x, 0, c}]]\)], "Input"], Cell[BoxData[ \({0.000032999999998395424`\ Second, Null}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?f\)\)], "Input"], Cell["Global`f", "Print", CellTags->"Info3339740806-1604724"], 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->"Info3339740806-1604724"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?g\)\)], "Input"], Cell["Global`g", "Print", CellTags->"Info3339740806-3383149"], 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->"Info3339740806-3383149"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[f[0.5]]\)], "Input"], Cell[BoxData[ \({0.0009340000000008786`\ Second, \(\(0.6572099322091018`\)\(\ \[InvisibleSpace]\)\) + 5.551115123125783`*^-17\ \[ImaginaryI]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[g[0.5]]\)], "Input"], Cell[BoxData[ \({0.12612800000000135`\ Second, \(\(0.6572099313456736`\)\(\ \[InvisibleSpace]\)\) + 5.551115123125783`*^-17\ \[ImaginaryI]}\)], "Output"] }, Open ]] }, Closed]], 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[ \(\(-0.011617409378892947`\)\)], "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[ \(\(-0.01161740937889297`\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ 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...)\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[TopIntegrand\[Zeta]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[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(TopIntegrand\[Zeta]r\\)\\\" is similar to \ existing symbol \\\"\\!\\(TopIntegrand\[Theta]r\\)\\\". \\!\\(\\*ButtonBox[\\\ \"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \ ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], 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], Cell[BoxData[ \(2\ \[Pi]\ TopIntegrand\[Zeta]r\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ 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[ \(0.019746507925389237`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TopContribution[2, 1/3, 1, 2]\)], "Input"], Cell[BoxData[ \(0.0197465079253876`\)], "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[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(BotIntegrand\[Theta]\\)\\\" is similar to \ existing symbol \\\"\\!\\(BotIntegrand\[Theta]r\\)\\\". \\!\\(\\*ButtonBox[\\\ \"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \ ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell[BoxData[ \(BotContribution[xpos_, \ ypos_, \ zpos_, \ CylRad_\ , \ CylLen_]\ := \ NIntegrate[ Evaluate[ BotIntegrand\[Theta][xpos, ypos, zpos, \ CylRad, \ CylLen], {t, 0, 2\ \[Pi]}]]\)], "Input"], Cell[BoxData[ \(BotContribution[dist_, \ zpos_, \ CylRad_\ , \ CylLen_]\ := \ NIntegrate[ Evaluate[ BotIntegrand\[Theta][dist, 0, zpos, \ CylRad, \ CylLen], {t, 0, 2\ \[Pi]}]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(BotContribution[Sqrt[2], Sqrt[2], 1/3, 1, 2]\)], "Input"], Cell[BoxData[ \(0.011074868917683132`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(BotContribution[2, 1/3, 1, 2]\)], "Input"], Cell[BoxData[ \(0.011074868917684923`\)], "Output"] }, Open ]], Cell["Adding up all the surface integral contributions", "Text"], Cell[BoxData[ \(LondonCylinderPotential[dist_, \ zpos_, \ CylRad_\ , \ CylLen_]\ \ := CylinderContribution[dist, zpos, CylRad, CylLen]\ + \ TopContribution[dist, zpos, CylRad, CylLen]\ + \ BotContribution[dist, zpos, CylRad, CylLen]\)], "Input", CellTags-> "mmtag:16:dipole_interaction_between_cylinder_and_pont__function_for"], Cell[CellGroupData[{ Cell[BoxData[ \(LondonCylinderPotential[2, .5, 1, 3]\)], "Input"], Cell[BoxData[ \(\(-0.252331039462739`\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LondonCylinderPotential[ .25, .5, 1, 3]\)], "Input"], Cell[BoxData[ \(3.4393174758259777`\)], "Output"] }, Open ]], Cell[BoxData[ \( (*Plot[ LondonCylinderPotential[dist, zpos, 1, 2], {dist, 1.1, 3}, {zpos, 0, 3}]*) \)], "Input"], Cell[BoxData[ \(<< Graphics`Graphics`\)], "Input"], Cell["\<\ Visualize result as a function of radial distance at different \ 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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"] }, Open ]] }, Open ]] }, Open ]], 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 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