(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 5855, 189]*) (*NotebookOutlinePosition[ 6552, 213]*) (* CellTagsIndexPosition[ 6508, 209]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Mass Flux Integrals", "Title"], Cell[CellGroupData[{ Cell["Problem Statement", "Section"], Cell["Suppose that the velocity of a fluid is", "Text"], Cell[BoxData[{ \(Clear[v, x, y, z]\), "\n", \(\(v[x_, y_, z_]\ = {y\^2, x\^2 + z, x - z};\)\)}], "Input"], Cell["and that its density is", "Text"], Cell[BoxData[{ \(Clear[\[Rho], x, y, z]\), "\n", \(\(\[Rho][x_, y_, z_]\ = \ 1\/\(x\^2 + y\^2 + z\^2 + 1\);\)\)}], "Input"], Cell["\<\ We wish to find the outward mass flux across the sphere of radius 2 \ centered at the origin.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Solution", "Section"], Cell[CellGroupData[{ Cell["Parametric representation of the surface", "Subsection"], Cell["\<\ The sphere of radius 2 centered at the origin can be represented by\ \ \>", "Text"], Cell[BoxData[{ \(Clear[r, rx, ry, rz, \[Theta], z]\), "\n", \(\(rx[\[Theta]_, \[Phi]_]\ = \ 2\ Cos[\[Theta]]\ Sin[\[Phi]];\)\), "\n", \(\(ry[\[Theta]_, \[Phi]_]\ = \ 2\ Sin[\[Theta]]\ Sin[\[Phi]];\)\), "\n", \(\(rz[\[Theta]_, \[Phi]_]\ = \ 2\ Cos[\[Phi]];\)\), "\n", \(r[\[Theta]_, \[Phi]_]\ = \ {rx[\[Theta], \[Phi]], ry[\[Theta], \[Phi]], rz[\[Theta], \[Phi]]}\)}], "Input"], Cell[TextData[ "for 0 \[LessEqual] \[Theta] \[LessEqual] 2\[Pi] and 0 \[LessEqual] \[Phi] \ \[LessEqual] \[Pi]."], "Text"], Cell["Just to verify our result, here is a plot of the surface.", "Text"], Cell[BoxData[ \(\(ParametricPlot3D[ r[\[Theta], \[Phi]], \ {\[Theta], 0, 2 \[Pi]}, \ {\[Phi], 0, \[Pi]}];\)\)], "Input"], Cell[TextData[ "In terms of these parameters, \[Theta] and \[Phi], the velocity is"], "Text"], Cell[BoxData[{ \(Clear[vel, \[Theta], \[Phi]]\), "\[IndentingNewLine]", \(vel[\[Theta]_, \[Phi]_] = Simplify[v[x, y, z] /. {x \[Rule] rx[\[Theta], \[Phi]], y \[Rule] ry[\[Theta], \[Phi]], z \[Rule] rz[\[Theta], \[Phi]]}]\)}], "Input"], Cell["and the density is", "Text"], Cell[BoxData[{ \(Clear[density, \[Theta], \[Phi]]\), "\[IndentingNewLine]", \(density[\[Theta]_, \[Phi]_] = Simplify[\[Rho][x, y, z] /. {x \[Rule] rx[\[Theta], \[Phi]], y \[Rule] ry[\[Theta], \[Phi]], z \[Rule] rz[\[Theta], \[Phi]]}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Calculation of the normal vector", "Subsection"], Cell["The normal vector to the sphere is", "Text"], Cell[BoxData[{ \(Clear[n, \[Theta], \[Phi]]\), "\[IndentingNewLine]", \(n[\[Theta]_, \[Phi]_] = Simplify[Cross[\[PartialD]\_\[Theta]\ r[\[Theta], \[Phi]], \[PartialD]\_\[Phi]\ r[\[Theta], \[Phi]]]]\)}], "Input"], Cell["\<\ Oops, it looks like this is the inward normal, because the \ coefficients are all negative. We need to switch the terms in the cross \ product. The outward normal is\ \>", "Text"], Cell[BoxData[{ \(Clear[n, \[Theta], \[Phi]]\), "\[IndentingNewLine]", \(n[\[Theta]_, \[Phi]_] = Simplify[Cross[\[PartialD]\_\[Phi]\ r[\[Theta], \[Phi]], \[PartialD]\_\[Theta]\ r[\[Theta], \[Phi]]]]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Calculation of the mass flux", "Subsection"], Cell[TextData[{ "So, the outward mass flux across the sphere is ", Cell[BoxData[ \(\[Integral]\_0 \%\[Pi]\(\[Integral]\_0 \%\(2 \[Pi]\)\(( density[\[Theta], \[Phi]]\ vel[\[Theta], \[Phi]].n[\[Theta], \[Phi]])\) \[DifferentialD]\[Theta] \[DifferentialD]\[Phi]\)\)]], ". 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