(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 15711, 473]*) (*NotebookOutlinePosition[ 16609, 502]*) (* CellTagsIndexPosition[ 16565, 498]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "\[CenterDot]\[Del]", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], " = -", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "\[Times]\[Del]\[Times]", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "+\[Del]", Cell[BoxData[ \(TraditionalForm\`U\^2\/2\)]] }], "Title", TextAlignment->Center], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "As a boy, Richard Feynman wrote in his notebook:\nTHE MOST REMARKABLE\n\ FORMULA\nIN MATH\n", Cell[BoxData[ \(TraditionalForm\`e\^i\[Pi]\)]], "+1=0\nAt the risk of seeming somewhat less sophisticated,\nlet me express \ similar enthusiam for:\n", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "\[CenterDot]\[Del]", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], " = -", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "\[Times]\[Del]\[Times]", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "+\[Del]", Cell[BoxData[ \(TraditionalForm\`U\^2\/2\)]] }], "Text", TextAlignment->Center], Cell[TextData[{ "In finding the components of ", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "\[CenterDot]\[Del]", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], ", one might expect much tedium in differentiating unit vectors in \ non-Cartesian coordinate systems. But with the use of ", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "\[CenterDot]\[Del]", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], " = -", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "\[Times]\[Del]\[Times]", Cell[BoxData[ \(TraditionalForm\`U\&\[RightVector]\)]], "+\[Del]", Cell[BoxData[ \(TraditionalForm\`U\^2\/2\)]], ", we can avoid that tedium. This treatise ", StyleBox["The ARPS and COAMPS Coordinate System", FontSlant->"Italic"], " should be read before attempting to understand this notebook." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Vector Calculus Functions", "Section", InitializationCell->True], Cell[TextData[{ "Given a list ", StyleBox["a", FontColor->RGBColor[0, 0, 1]], " with the three compents of ", Cell[BoxData[ \(TraditionalForm\`\(a\& \[RightVector] \)\)]], ", a list ", StyleBox["q", FontColor->RGBColor[0, 0, 1]], " with the three coordinates, and a list ", StyleBox["h ", FontColor->RGBColor[0, 0, 1]], "with three scale factors find the curl. The formula has two flavors ", StyleBox["CurlExpl", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["Curl", FontColor->RGBColor[0, 0, 1]], ". ", StyleBox["CurlExpl", FontColor->RGBColor[0, 0, 1]], " uses the total derivative", StyleBox[" Dt", FontColor->RGBColor[0, 0, 1]], ", and explicity expresses what coordinates are held constant so the \ differentiation is a partial differentiation. ", StyleBox["Curl", FontColor->RGBColor[0, 0, 1]], " is useful if neither ", StyleBox["a", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["h", FontColor->RGBColor[0, 0, 1]], " are expressed explicitly in terms of the ", StyleBox["q", FontColor->RGBColor[0, 0, 1]], "; it's only benefit being that the output is slightly easier to read. " }], "Text", InitializationCell->True], Cell[BoxData[ \(CurlExpl[a_, q_, h_] := {\n\t \((Dt[h[\([3]\)] a[\([3]\)], q[\([2]\)], \n\t\t\t\t Constants -> {q[\([1]\)], q[\([3]\)]}\ ]\n\t\t - Dt[h[\([2]\)] a[\([2]\)], q[\([3]\)], \n\t\t\t\t\t Constants -> {q[\([1]\)], q[\([2]\)]}\ ])\)\n\t/ \((h[\([2]\)] h[\([3]\)])\), \n\t\t\t \((Dt[h[\([1]\)] a[\([1]\)], q[\([3]\)], \n\t\t\t\t Constants -> {q[\([1]\)], q[\([2]\)]}\ ]\n\t\t - Dt[h[\([3]\)] a[\([3]\)], q[\([1]\)], \n\t\t\t\t\t Constants -> {q[\([2]\)], q[\([3]\)]}\ ])\)\n\t/ \((h[\([1]\)] h[\([3]\)])\), \t\n\t\t \((Dt[h[\([2]\)] a[\([2]\)], q[\([1]\)], \n\t\t\t\t Constants -> {q[\([2]\)], q[\([3]\)]}\ ]\n\t\t - Dt[h[\([1]\)] a[\([1]\)], q[\([2]\)], \n\t\t\t\t\t Constants -> {q[\([1]\)], q[\([3]\)]}\ ])\)\n\t/ \((h[\([1]\)] h[\([2]\)])\)}\)], "Input", InitializationCell->True], Cell[BoxData[ \(Curl[a_, q_, h_] := {\n\t\t \((\((Dt[h[\([3]\)] a[\([3]\)], q[\([2]\)]\ ]\n\t\t - Dt[h[\([2]\)] a[\([2]\)], q[\([3]\)]\ ])\)\n\t/ \((h[\([2]\)] h[\([3]\)])\))\), \n\t\t \((\((Dt[h[\([1]\)] a[\([1]\)], q[\([3]\)]\ ]\n\t\t - Dt[h[\([3]\)] a[\([3]\)], q[\([1]\)]\ ])\)\n\t/ \((h[\([1]\)] h[\([3]\)])\))\), \n\t \((\((Dt[h[\([2]\)] a[\([2]\)], q[\([1]\)]\ ]\n\t\t - Dt[h[\([1]\)] a[\([1]\)], q[\([2]\)]\ ])\)\n\t/ \((h[\([1]\)] h[\([2]\)])\))\)}\)], "Input", InitializationCell->True], Cell[TextData[{ "Two flavors of divergence of ", Cell[BoxData[ \(TraditionalForm\`\(a\& \[RightVector] \)\)]], " , but they are not used here." }], "Text", InitializationCell->True], Cell[BoxData[ \(DivExpl[a_, q_, h_] := {\n\t Dt[h[\([2]\)] h[\([3]\)] a[\([1]\)], q[\([1]\)], \n\t\t\t Constants -> {q[\([2]\)], q[\([3]\)]}]\n\t\t\t\t/ \((h[\([1]\)] h[\([2]\)] h[\([3]\)])\), \n \(+Dt[h[\([1]\)] h[\([3]\)] a[\([2]\)], q[\([2]\)], \n\t\t\t\t Constants -> {q[\([1]\)], q[\([3]\)]}]\)\n\t\t\t\t/ \((h[\([1]\)] h[\([2]\)] h[\([3]\)])\), \n \(+Dt[h[\([1]\)] h[\([2]\)] a[\([3]\)], q[\([3]\)], \n\t\t\t Constants -> {q[\([1]\)], q[\([2]\)]}]\)\n\t\t\t\t/ \((h[\([1]\)] h[\([2]\)] h[\([3]\)])\)}\)], "Input", InitializationCell->True], Cell[BoxData[ \(Div[a_, q_, h_] := {\n\t Dt[h[\([2]\)] h[\([3]\)] a[\([1]\)], q[\([1]\)]]\n\t\t\t\t/ \((h[\([1]\)] h[\([2]\)] h[\([3]\)])\), \n \(+Dt[h[\([1]\)] h[\([3]\)] a[\([2]\)], q[\([2]\)]]\)\n\t\t\t\t/ \((h[\([1]\)] h[\([2]\)] h[\([3]\)])\), \n \(+Dt[h[\([1]\)] h[\([2]\)] a[\([3]\)], q[\([3]\)]]\)\n\t\t\t\t/ \((h[\([1]\)] h[\([2]\)] h[\([3]\)])\)}\)], "Input", InitializationCell->True], Cell[TextData[{ "Two flavors of the gradient of ", StyleBox["a", FontSlant->"Italic"], "." }], "Text", InitializationCell->True], Cell[BoxData[ \(GradExpl[a_, q_, h_] := {\n\t Dt[a, q[\([1]\)], \n\t\t\tConstants -> {q[\([2]\)], q[\([3]\)]}]/ h[\([1]\)], \n Dt[a, q[\([2]\)], \n\t\t\tConstants -> {q[\([1]\)], q[\([3]\)]}]/ h[\([2]\)], \n Dt[a, q[\([3]\)], \n\t\t\tConstants -> {q[\([1]\)], q[\([2]\)]}]/ h[\([3]\)]}\)], "Input", InitializationCell->True], Cell[BoxData[ \(Grad[a_, q_, h_] := {\nDt[a, q[\([1]\)]]/h[\([1]\)], \n Dt[a, q[\([2]\)]]/h[\([2]\)], \nDt[a, q[\([3]\)]]/h[\([3]\)]}\)], "Input", InitializationCell->True], Cell[TextData[{ "Two flavors of ", Cell[BoxData[ \(TraditionalForm\`\(U\& \[RightVector] \)\)]], "\[CenterDot]\[Del]", Cell[BoxData[ \(TraditionalForm\`\(U\& \[RightVector] \)\)]], ":" }], "Text", InitializationCell->True], Cell[BoxData[ \(UDelUExpl[u_, q_, h_] := \n\t \(-Cross[u, CurlExpl[u, q, h]]\)\n\t + GradExpl[Dot[u, u]/2, q, h] // Simplify\)], "Input", InitializationCell->True], Cell[BoxData[ \(UDelU[u_, q_, h_] := \n\t \(-Cross[u, Curl[u, q, h]]\)\n\t + Grad[Dot[u, u]/2, q, h] // Simplify \)], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Let's Try It!", "Section"], Cell[TextData[{ "We need to specify a list holding velocity components an a list holding \ coordinates. 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