Appendix A: Mathematica Functions

Transcription

1 Appendix A: Mathematica Functions In this appendix, we describe those Mathematica functions that we have used in this book. The use of each function is illustrated by examples. A more detailed description of these functions as well as many other functions may be found in the Mathematica 3.0 manual: S. Wolfram. The MATHEMATICA Book. Third Edition. Wolfram Media/Cambridge University Press, Cambridge (UK), New York, Melbourne, This version of Mathematica can be installed on various computer platforms, which have the following: 1) processor type Intel or higher; 2) operating system Windows 95 or Windows NT 3.51; 3) hard disk space 116 MB (needed if not running off CD-ROM); 116 MB recommended, 30 MB minimum; 4) system memory 16 MB recommended, 8 MB minimum. The Mathematica kernel uses PostScript language to represent all of the graphics it produces. The Mathematica for Windows front end includes a PostScript interpreter to translate PostScript code from the kernel into screen images. One of the features of Mathematica 3.0 is that the Mathematica documents are called Notebooks. Each Notebook is stored as a file name. nb. Notebooks can contain ordinary text and graphics as well as Mathematica input and output. The information in a Notebook is stored in cells, whose characteristics vary with the cell's function and the kind of information it holds. The square brackets on the right-hand side of the Notebook window indicate the extent of each cell. We have presented here the texts of our Notebooks together with the results of symbolic and numeric computations. Therefore, the corresponding Notebook cells include not only the input, but also the output of a computation made within a cell. It should be noted that a number of books have been published, in which the software system Mathematica is applied for the analytical or numerical solution of some continuum mechanics problems.

2 Appendix A: Mathematica Functions 527 The main feature of the use of M athematica 3.0 in this book is that we have applied this powerful software system for the computer implementation of analytical or approximate analytical methods for the solution of fluid mechanics problems or for the derivation and investigation of underlying mathematical models of various continua. In this respect, this book differs from Ganzha and Vorozhtsov's book}, where the emphasis was placed on the computer implementation of finite difference, finite volume, and finite element methods for the numerical solution of partial differential equations. As shown in Chapter I, many problems of tensor analysis arising at the derivation of the governing equations of fluid mechanics can be solved with the aid of Mathematica. Parker and Christensen 2 describe their MathTensor software, which provides a computer program that extends Mathematica capabilities to include tensor analysis. Kythe et al. and Vvedensky3,4 discuss the implementation of a number of analytical methods for the solution of partial differential equations: the method of characteristics, the separation of variables, etc. We have used the bisection method for finding the roots of transcendental equations in a number of our Mathematica notebooks. The interested reader can find in5 both the implementation of the bisection method and a discussion of its advantages and disadvantages. Baumann6 shows, in particular, how Mathematica can be used for the numerical solution of the Korteweg- de Vries equation (cf. Section 7.7 of the present book) with the aid of both the analytic method (the soliton solution) and the finite difference method by Zabusky and Kruskal. We may conclude from the above survey that there is at present a huge gap in the development of big industrial programs written in the language of Mathematica, which would solve such complex fluid mechanics problems as the three-dimensional transonic gas flow around the complete aircraft, the hypersonic three-dimensional gas flows around the reentry vehicles, three-dimensional gas flows between the compressor blades of an aircraft engine, two-dimensional flows of gas-particle mixtures in rocket nozzles, three-dimensional incompressible gas flows around the cars, etc. This book only partly bridges this gap. This is in line with the main purpose, which may be formulated as follows: the derivation, investigation, and comparison of the mathematical models of various continua. Before proceeding to the description of the built-in Mathematica functions we have used above, we present in Table A.l the names of these functions or the options used in functions in alphabetic order together with the numbers of the sections, where these functions are used.

3 528 Appendix A: Mathematica Functions Table A.I The Mathematica Functions Abs Flatten Plot VectorField3D AppendTo Floor PointSize ArcSin Graphics Polygon ArcTan G raphicsarray PowerExpand AspectRatio Hue Print Axes IdentityMatrix ReplacePart AxesLabel If RGBColor Circle 1m ScaleFactor ClearAll Interpolation Show Color Function Inverse Sign ComplexExpand Join Simplify ContourPlot Length Sin Contours Line Solve ContourS hading LinearSolve Sqrt Cos ListPlot Sum D Log Table Dashing MapThread Text DensityGraphics MatrixForm Thickness DisplayFunction N TraditionalForm DSolve Nlntegrate Transpose Eigenvalues Parametric Plot 3D TrigExpand Eigenvectors Pi TrigReduce Expand Plot 3D VectorHeads First PlotPoints While Abs[z] gives the absolute value of the real or complex number z. Ii Example 1 Abs[-3.5] 3.5 AppendTo[s, elem] appends elem to the value of s, and resets s to the result. Ii Example 2 xx = {Xl, X2, X3}; AppendTo[xx, a 2 ] {Xl, X2, X3, a 2 }

4 Appendix A: Mathematica Functions 529 ArcSin[z] gives the arc sine of the complex number z. Ii Example 3 ArcSin [1/2] II 6 II ArcTan[z] gives the arc tangent of the complex number z. ArcTan[x, y] gives the arc tangent of y/x, taking into account which quadrant the point (x, y) is in. Ii Example 4 ArcTan [1] II 4 II AspectRatio is an option for Show[..] and related functions that specifies the ratio of height to width for a plot. The default value AspectRatio -> l/goldenratio is used for two-dimensional plots. The setting AspectRatio -> Automatic forces Mathematica to take the actual ratio of height to width. Ii Example 5 al = ; a2 = ; a3 = ; a4 = ; a5 = ; tl = 0.2; body[z_]:= t1 (al Sqrt[z] + z (a2 + z (a3 + z (a4 + a5 z)))); gl = Plot[body[x], {x, 0, I}, DisplayFunction - > Identity]; g2 = Plot[-body[x], {x, 0, 1},DisplayFunction - > Identity]; Show[gl, g2, DisplayFunction - > $DisplayFunction] II The graphical image produced with the aid of this program is shown in Fig. A.l (a). Compare this picture with Fig. A.2 obtained in the case where the value AspectRatio -> Automatic is used (see the description of the option AxesLabel). Axes is an option for graphics functions that specifies whether axes should be drawn. Axes -> True draws all axes. Axes -> False removes the axes. [see also Fig. A.l (b)] II II

5 530 Appendix A: Mathematica Functions (a) (b) Figure A.l: Illustration to the description of the option (a) AspectRatio and (b) Axes -> False. Ii Example 6,I al = ; a2 = ; a3 = ; a4 = ; a5 = ; tl = 0.2; body[z_]:= tl (al Sqrt[z] + z (a2 + z (a3 + z (a4 + a5 z)))); gl = Plot[body[x]' {x, 0, I}, DisplayFunction - > Identity]; g2 = Plot[-body[x]' {x, 0, I}, DisplayFunction - > Identity]; Show[gl, g2, Axes - > False, DisplayFunction - > $DisplayFunction] II II AxesLabel is an option for graphics functions that specifies labels for axes (see also Fig. A.2). Ii Example 7 II al = ; a2 = ; a3 = ; a4 = ; a5 = ; tl = 0.2; body[z_]:= tl (al Sqrt[z] + z (a2 + z (a3 + z (a4 + a5 z)))); gl = Plot[body[x]' {x, 0, I}, DisplayFunction - > Identity]; g2 = Plot[-body[x]' {x, 0, 1},DisplayFunction - > Identity]; Show[gl,g2, AxesLabel - > {" x", "y"}, AspectRatio - > Automatic, DisplayFunction - > $DisplayFunction] II Circle[{x, y}, r] is a two-dimensional graphics primitive that represents a circle of radius r centered at point x, y. Circle[{x,y}, {rx, ry}] yields an ellipse with semi-axes rx and ry. Circle[{x,y}, r, {theta!, theta2}] represents a circular arc. Example: see the description of functions ContourPlot[... ], Text[... ].

6 Appendix A: Mathematica Functions 531 y ~ ~ 0.4 j:6:i x ~ Figure A.2: Illustration to the description of the option AxesLabel. Clear All [symbl, symb2,... ] clears all values, definitions, attributes, messages and defaults associated with symbols. Clear All["forml", "form2",... ] clears all symbols whose names textually match any of the forms. II xl = 2Cos[t] 2 Cos [t] ClearAll[x1] xl Example 8 Color Function is an option for various graphics functions that specifies a function to apply to z values to determine the color to use for a particular x, y region. ComplexExpand[expr] expands expr assuming that all variables are real. ComplexExpand[expr, {xl, x2,... }] expands expr assuming that variables matching any of the xi are complex. II Example 9 II ComplexExpand[Im[Cos [z2]j. Z - > x + I y]l/ /TraditionalForm - sin(x 2 - y2) sinh(2xy) II ContourPlot[f, {x, xmin, xmax}, {y, ymin, ymax}] generates a contour plot of f as a function of x and y. This function has the following default options: Options [ContourPlot] = {AspectRatio - > 1,

7 532 Appendix A: Mathematica Functions -4L- - = ~ = = = = ~ = _ Figure A.3: Illustration to the description of the function ContourPlot. Axes - > False, AxesLabel - > None, AxesOrigin - > Automatic, AxesStyle - > Automatic, Background - > Automatic, ColorFunction - > Automatic, ColorOutput - > Automatic, Compiled - > True, ContourLines - > True, Contours - > 10, ContourS hading - > True, ContourSmoothing - > True, ContourStyle - > Automatic, DefaultColor - > Automatic, Epilog - > {}, Frame - > True, FrameLabel - > None, FrameStyle - > Automatic, FrameTicks - > Automatic, Plot Label - > None, PlotPoints - > 15, PlotRange - > Automatic, PlotRegion - > Automatic, Prolog - > {}, RotateLabel - > True, Ticks - > Automatic, DefaultFont :> $DefaultFont, DisplayFunetion : > $DisplayFunction} (see also Fig. A.3). II Example 10 ); ~ [ x y-] _,:= y ( 1 - X ; ~ : 2 rc=2; streamlines = C o n t o u r P l o t [ ~ [ x, y ], PlotPoints { x, - 6, , }, { y, - 4, 4 }, Contours --+ {-4,-3,-2,-1,-0.5,0,0.5,1,2,3,4}, ContourS hading --+ False, ContourSmoothing --+ Automatic, DisplayFunction --+ Identity]; bound = Graphics[Circle[{O,O}, re]]; Show[streamlines, bound, AspectRatio --+ Automatic, DisplayFunction --+ $DisplayFunetion];

8 Appendix A: Mathematica Functions 533 Figure AA: Illustration to the descriptions of the functions Dashing, GraphicsArraY,Plot3D, Thickness, and PointSize. Contours is an option for ContourGraphics specifying the contours to use. The contour values are specified in the form of a list of numerical values. Example: see the description of the function ContourPlot[... ]. ContourS hading is an option for contour plots that specifies whether the regions between contour lines should be shaded. Example: see the description of the function Contour Plot [... ]. Cos[z] gives the cosine of z. The argument of Cos is assumed to be in radians. Ii Example 11 Tradi tionalform [ Cos [Pi /10 II D[f, x] gives the partial derivative of f with respect to x. D[f, {x, n}] gives the nth partial derivative of f with respect to x. D[f,xl,x2,... ] gives a mixed derivative. Ii Example 12 x = k[l] Sin[k[2JJ Sin[k[311; D[x, k[2jj cos[k[2jj k[l] sin[k[3jj Dashing[{rl, r2,... }] is a two-dimensional graphics directive that specifies that lines which follow are to be drawn dashed, with successive segments of lengths rl, r2,... (repeated cyclically). The ri is given as a fraction of the total width of the graph (see also Fig. AA).

9 534 Appendix A: Mathematica Functions Example 13 ========:::;'lll gi = Plot[Sin[xJ, {x,d, 27r}, PlotStyle - > {Dashing[{0.05,0.03}]}, DisplayFunction - > Identity]; xx = Table[N[(k-I) 7r/1OJ, {k, 21}]; yy = Table[N[Cos[xx[[klllL {k, 21}]; g2=listplot[mapthread[list, {xx, yy}], Plot Style - > {PointSize[0.02]}, DisplayFunction - > Identity]; g3=listplot[mapthread[list,{xx, 0.5yy}J, PlotJoined - > True, PlotStyle - > {Thickness[0.03]}, DisplayFunction - > Identity]; A2 = Plot3D[Sin[x] Cos[2yJ, {x, 0, 2Pi}, {y, 0, Pi}, DisplayFunction - > Identity]; Al = Show[g2,gI,g3, AspectRatio - > Automatic, DisplayFunction - > Identity]; Show[GraphicsArray[{AI, A2}]]; DensityGraphics[array] is a representation of a density plot. Options [DensityGraphics] = {AspectRatio - > 1, Axes - > False, AxesLabel - > None, AxesOrigin - > Automatic, AxesStyle - > Automatic, Background - > Automatic, ColorFunction - > Automatic, ColorOutput -i, Automatic, DefaultColor - > Automatic, Epilog - > {}, Frame - > True, FrameLabel - > None, FrameStyle - > Automatic, FrameTicks - > Automatic, ImageSize - > Automatic, Mesh - > True, MeshRange - > Automatic, MeshStyle - > Automatic, PlotLabel - > None, PlotRange - > Automatic, Plot Region - > Automatic, Prolog - > {}, RotateLabel - > True, Ticks - > Automatic, DefaultFont:>$DefaultFont, DisplayFunction: >$Display Function, FormatType: >$FormatType, TextStyle: >$TextStyle}. Display Function is an option for graphics and sound functions that specifies the function to apply to graphics and sound objects in order to display them. The default setting for DisplayFunction in graphics functions is $DisplayFunction. Setting DisplayFunction - > Identity will cause the objects to be returned, but no display to be generated. Examples: see the descriptions of options AspectRatio, Axes, AxesLabel. DSolve[eqn, y, x] solves a differential equation for the function y, with independent variable x. DSolve[{eqnl, eqn2,... }, {yl, y2,...

10 Appendix A: Mathematica Functions 535 }, xl solves a list of differential equations. DSolve[eqn, y, {xl, x2,... } 1 solves a partial differential equation. Example 14 DSolve[{x2'[x] == -x2[x] x, x2[6] == 6}, x2[x], x] x 2 ~ {{x2[x]- > E-T+ 2 6}} Eigenvalues[m] gives a list of the eigenvalues of the square matrix m. II Example 15 ========:::;'\1 A = {{-I, 0, 3}, {2,I,I}, {5,I,4}}; Eigenvalues[A] Eigenvectors[A] 1 1 { - 3, 2 (7 - v'33), 2 (7 + v'33)} { { 6 2 (9+ 2 v'33) I} { 6 2( -9+2v'33) I}} {-3, 1, 2}, - -9+$3' -9+$3', 9+$3' 9+$3, TraditionalForm [Inverse [ A]] Eigenvectors[m] gives a list of the eigenvectors of the square matrix m. Options[Eigenvectors]={ZeroTest - > Automatic}. Example: see the description of function Eigenvalues[... l. Expand[expr] expands out products and positive integer powers in expr. Expand[expr, pattlleaves unexpanded any parts of expr that are free of the pattern patt. II TraditionalForm [Expand [ (a - 3,8)5]] Example 16 ========:::;'\1 a 5 - I5,8a ,82a3-270,83a ~ 243,85 a

11 536 Appendix A: Mathematica Functions First[expr] gives the first element in expr. Ii lis = {ai, Sin[t], 7r/8}; First[lis] al Example 17 =========iid Flatten [list ] flattens out nested lists. Flatten [list, n] flattens to level n. Flatten [list, n, h] flattens sub expressions with head h. Ii Example 18 Flatten[{al,b2, {{2,3}, {a4, s5}}}] {al,b2,2,3,a4,s5} Floor[x] gives the greatest integer less than or equal to x. Ii Example 19 Floor[-2.4] -3 Floor[2.4] 2 Graphics[primitives, options] represents a two-dimensional graphical image. The default options are: Options[Graphics]={AspectRatio - > GoldenRatio< - 1), Axes - > False, AxesLabel - > None, AxesOrigin - > Automatic, AxesStyle - > Automatic, Background - > Automatic, Color Output - > Automatic, DefaultColor - > Automatic, Epilog - > {}, Frame - > False, FrameLabel - > None, FrameStyle - > Automatic, FrameTicks - > Automatic, GridLines - > None, ImageSize - > Automatic, PlotLabel - > None, Plot Range - > Automatic, PlotRegion - > Automatic, Prolog - >{}, RotateLabel - > True, Ticks - > Automatic, DefaultFont: >$DefaultFont, DisplayFunction: >$Display Function, FormatType: >$FormatType, TextStyle: >$TextStyle}. Example: see the description of the function Line[... ].

12 Appendix A: Mathematica Functions 537 GraphicsArray[{gl, g2,... }] represents a row of graphics objects. GraphicsArray [{{gll, g12,... },... }] produces a twodimensional array of graphics objects. Options[GraphicsArray] = {Aspect Ratio - > Automatic, Axes - > False, AxesLabel - > None, AxesOrigin - > Automatic, AxesStyle - > Automatic, Background - > Automatic, Color Output - > Automatic, DefaultColor - > Automatic, Epilog - > {}, Frame - > False, FrameLabel - > None, FrameStyle - > Automatic, FrameTicks - > None, GraphicsSpacing - > 0.1, GridLines - > None, ImageSize - > Automatic, PlotLabel - > None, PlotRange - > Automatic, PlotRegion - > Automatic, Prolog - > {}, RotateLabel - > True, Ticks - > None, DefaultFont:> $DefaultFont, Display Function: > $Display Function, FormatType: > $FormatType, TextStyle: > $TextStyle}. See Example 13. Hue[h] is a graphics directive that specifies that graphical objects which follow are to be displayed, if possible, in a color corresponding to hue h. Hue [h, s, b] specifies colors in terms of hue, saturation, and brightness. IdentityMatrix[n] gives the n by n identity matrix. Ii Example 20 TraditionalForm [Identity Matrix [3]] ( ~ ~ ~ ) 001 If[condition, t, f] gives t if condition evaluates to True, and f if it evaluates to False. If[condition, t, f, u] gives u if condition evaluates to neither True nor False. Ii Example 21 II b = 5i c = Sin[t]i d = t2i If[b < 4, c = COS[t]i d = 5t, c = Cosh[t]i d = 4t] i Print [" c =", c ". d =" d] " " c = Cosh[t]i d = 4 t

13 538 Appendix A: Mathematica Functions Im[z] gives the imaginary part of the complex number z. Example: see the description of the function ComplexExpand[... J. Interpolation[data] constructs an InterpolatingFunction object representing an approximate function that interpolates the data. The data can have the forms {{xl, fl}, {x2, f2},... } or {fl, f2,... }, where in the second case, the xi are taken to have values 1, 2,... Options[Interpolation] = InterpolationOrder - > 3 Inverse[m] gives the inverse of a square matrix m. Options [Inverse] = {Method - > Cofactor Expansion, Modulus - > 0, Zero Test - > (#1 == 0 & )}. Example: see the description of function Eigenvalues[...). Join[listl, list2,... ] concatenates lists together. Ii lsi = {Xl, X2}; Is2 = {Sqrt[l + b 2 ], Sin[t]}; TraditionalForm[Join[lsl, ls2]] {Xl, X2, Vb 2 + 1, sin(t)} Example 22 =========i111 Length[expr] gives the number of elements in expr. Ii Example 23 ========"11 Length[{a, x 3, 1995}] 3 Line [{ptl, pt2,... }] is a graphics primitive that represents a line joining a sequence of points [see Fig. A.5 (a)]. Ii Example 24 II Show[Graphics[Line[{{4,3}, {5,4}, {2,7}, {l,6}, {4,3}, {1,3}, {l,l}, {7,1}, {7,3}, {4,3}}]]' AspectRatio - > Automatic] LinearSolve[m, b] finds an x that solves the matrix equations m.x == b. Ii Example 25 II n = 3; b = Table[k, {k, n}]; A = Table[k/(i + k), {i, n}, {k, n}]; LinearSolve[A, b] {132, -300, l80}

14 Appendix A: Mathematica Functions 539 (a) (b) Figure A.5: Illustration to the description of the functions (a) Line and (b) ListPlot. ListPlot[{yl, y2,... }] plots a list ofvalues. The x coordinates for each point are taken to be 1, 2,... ListPlot[{{xl, yl}, {x2, y2},... }] plots a list of values with specified x and y coordinates. Options[ListPlot] = {AspectRatio - > GoldenRatio(-ll, Axes - > Automatic, AxesLabel - > None, AxesOrigin - > Automatic, AxesStyle - > Automatic, Background - > Automatic, Color Output - > Automatic, DefaultColor - > Automatic, Epilog - > {}, Frame - > False, FrameLabel - > None, FrameStyle - > Automatic, FrameTicks - > Automatic, GridLines - > None, ImageSize - > Automatic, PlotJoined - > False, Plot Label - > None, PlotRange - > Automatic, PlotRegion - > Automatic, PlotStyle - > Automatic, Prolog - > {}, RotateLabel - > True, Ticks - > Automatic, DefaultFont: > $DefaultFont, Display Function: >$Display Function, FormatType: >$FormatType, TextStyle: >$TextStyle} [see also Fig. A.5 (b)]. Ii Example 26 II ListPlot[{{O,1},{2,1}, {1,3}, {O,l}}, Axes - > False, PlotJoined - > True] Log[z] gives the natural logarithm of z (logarithm to base e). Log[b, z] gives the logarithm to base b.

15 540 Appendix A: Mathematica Functions Ii Example 27 ========\1.\ N[Log[lOlJ MapThread[f, {{ai, a2,... }, {bl, b2,... },... }] gives the list {Hal, bl,... ], f[a2, b2,... ],... }. MapThread[f, {exprl, expr2,... }, n]} applies f to the parts of expri at level n. Ii Example 28 II xx = {Xl, X2, X3}; YY = {YI, Y2, Y3}; MapThread[List, {xx, yy}j {{Xl, yd, {X2, Y2}, {X3, Y3}} MatrixForm[list] prints with the elements of list arranged in a regular array. Ii Example 29 =======::::::;JII Print["g = ", TraditionalForm[MatrixForm[IdentityMatrix[3JJJJ ( 100) g = o 0 1 N[expr] gives the numerical value of expr. N[expr, nj attempts to give a result with n-digit precision. Example 30 ========:::;'11 Ii N[Sqrt[2J, 45J Nlntegrate[f, x, xmin, xmax] gives a numerical approximation to the integral of f with respect to x from xmin to xmax. Options [Nlntegrate] = AccuracyGoal - > 00, Compiled - > True, GaussPoints - > Automatic, MaxPoints - > Automatic, MaxRecursion - > 6, Method - > Automatic, MinRecursion - > 0, PrecisionGoal - > Automatic, Singularity Depth - > 4, WorkingPrecision - > 16}.

16 Appendix A: Mathematica Functions 541 Ii Example 31 Nlntegrate[Sqrt[Sin [n/2]- Sin [x]], {x, 0, n/2}] We can check this result with the aid of the Mathematica function Integrate [J, which enables the user to obtain the expression for the definite integral in symbolic form: Integrate[Sqrt[Sin[n /2] - Sin[xll, {x, 0, n /2}] N[%] -2 +2J ParametricPlot3D[{fx, fy, fz}, {t, tmin, tmax}] produces a three-dimensional space curve parametrized by a variable t that runs from tmin to tmax. ParametricPlot3D[{fx, fy, fz}, {t, tmin, tmax}, {u, umin, umax} 1 produces a three-dimensional surface parametrized by t and u. ParametricPlot3D[{fx, fy, fz, s},... 1 shades the plot according to the color specification s. ParametricPlot3D[ {{fx, fy, fz}, {gx, gy, gz},... },... 1 plots several objects together. This function has the following default options. Options[ParametricPlot3D] = {AmbientLight - > GrayLevel[O.]' AspectRatio - > Automatic, Axes - > True, AxesEdge - > Automatic, AxesLabel - > None, AxesStyle - > Automatic, Background - > Automatic, Boxed - > True, BoxRatios - > Automatic, BoxStyle - > Automatic, ColorOutput - > Automatic, Compiled - > True, DefaultColor - > Automatic, Epilog - > n, FaceGrids - > None, ImageSize - > Automatic, Lighting - > True, LightSources - > {{ {L,O.,L}, RGBColor[l,O,O]}, {{L,L,L}, RGBColor[O,l,O]}, {{O.,L,L}, RGBColor[O,O,l]}}, Plot3Matrix - > Automatic, PlotLabel - > None, Plot Points - > Automatic, PlotRange - > Automatic, Plot Region - > Automatic, Polygonlntersections - > True, Prolog - > {}, RenderAll - > True, Shading - > True, SphericalRegion - > False, Ticks - > Automatic, ViewCenter - > Automatic, ViewPoint - > {L3, ,2.},

17 542 Appendix A: Mathematica Functions (a) (b) Figure A.6: Illustration to the descriptions of the function (a) ParametricPlot3D [... ] and (b) Polygon [... ]. ViewVertical - > {O.,O.,l.}, DefaultFont: >$DefaultFont, DisplayFunction: >$Display Function, FormatType: >$FormatType, Text Style: >$TextStyle} [see also Fig. A.6 (a)]. Ii Example 32 ParametricPlot3D[{2t, Sin[3t], Cos[3t]}, {t, 0, 5Pi}, Boxed - > False, Axes - > False] Pi (7r) is the constant pi, with numerical value approximately equal to Plot3D[f, {x, xmin, xmax}, {y, ymin, ymax}] generates a three-dimensional plot of j as a function of x and y. Plot3D [{ f, s}, {x, xmin, xmax}, {y, ymin, ymax} ] generates a three-dimensional plot in which the height of the surface is specified by j, and the shading is specified by s. Options[Plot3D] = AmbientLight - > GrayLevel[O], AspectRatio - > Automatic, Axes - > True, AxesEdge - > Automatic, AxesLabel - > None, AxesStyle - > Automatic, Background - > Automatic, Boxed - > True, BoxRatios - > {l,l,o.4}, BoxStyle - > Automatic, ClipFill - > Automatic, ColorFunction - > Automatic, Color Output - > Automatic, Compiled - > True, DefaultColor - > Automatic, Epilog - > 0, FaceGrids - > None,

18 Appendix A: Mathematica Functions 543 Figure A.7: Illustration to the descriptions of the function PlotVectorField3D[... J. HiddenSurface - > True, ImageSize - > Automatic, Lighting - > True, LightSources - > {{{l.,o.,l.}, RGBColor[l,O,Oj}, {{l.,l.,l.}, RGBColor[O,l,Oj}, {{O., l.,l.}, RGBColor[O,O,lj}}, Mesh - > True, MeshStyle - > Automatic, Plot3Matrix - > Automatic, PlotLabel - > None, PlotPoints - > 15, PlotRange - > Automatic, PlotRegion - > Automatic, Prolog - > {}, Shading - > True, SphericalRegion - > False, Ticks - > Automatic, ViewCenter - > Automatic, ViewPoint - > {1.3, ,2.}, ViewVertical - > {O.,O.,l.}, DefaultFont: >$DefaultFont, Display Function: > $Display Function, FormatType: >$FormatType, TextStyle: >$TextStyle}. Example: see the description of the M athematica function Dashing [J. Plot Points is an option for plotting functions that specifies how many sample points to use. Example: see the description of the function ContourPlot[... ), where the option PlotPoints -> 40 is used. The use of a larger number of plot points generally renders smoother curves. Plot VectorField3D[ {ix, fy, fz}, {x, xmin, xmax}, {y, ymin, ymax}, {z, zmin, zmax}] plots the vector field given by the vector function in the range specified. This function has the following default options: ScaleFactor - > Automatic, ScaleFunction - > None,

19 544 Appendix A: Mathematica Functions MaxArrowLength - > None, ColorFunction - > None, Plot Points - > 7, VectorHeads - > False. The specification VectorHeads - > True puts heads on arrows. The option PlotPoints specifies the number of evaluation points in each direction. The option ScaleFactor - > Automatic ensures that the largest vector fits in the mesh. If the vectors obtained with this default option are too short, one can rescale the vectors, for example, by setting ScaleFactor - > 1. 3 (see Fig. A.7). Example 33 < < Graphics'PlotField3D' PlotVectorField3D[{2x,y,z}, {x,0,2}, {y,o,l},{ z,o,l}, Plot Points - > 6, VectorHeads - > True] The function PlotVectorField3D [... ] is described in more detail in Martin's book7. PointSize[d] is a graphics directive that specifies that points which follow are to be shown if possible as circular regions with diameter d. The diameter d is given as a fraction of the total width of the graph. Example: see the description of the Mathematica function Dashing [J. Polygon[ {ptl, pt2,... }] is a graphics primitive that represents a filled polygon (see also Fig. A.6 (b)]. Ii Example 34 II Show[Graphics[{RGBColor[O.l, 1.0,1.0], P o l y g o n [ T a b l e 2[ S{ i2 n C [ o ~ s {n,80}]]}]] ;[ j~ 3 ; } j, 3, PowerExpand[expr] expands all powers of products and powers. The use of this function is efficient when one wishes to simplify expressions like J r 4 sin2 cp. Example 35 a = r 4 Sin[cpj2; TraditionalForm[Sqrt[a]] Jr 4 sin(cp)2 TraditionalForm[PowerExpand[a 1 / 2]] r2 sin( cp) II II

20 Appendix A : Mathematica Functions 545 Print [exprl, expr2,... ] prints the expri, followed by a newline (line feed). Example: see the description of function Traditional Form[... ]. ReplacePart[expr, new, n] yields an expression in which the nth part of expr is replaced by new. ReplacePart[expr, new, {i, j,... }] replaces the part at position {i, j,... }. ReplacePart[expr, new, {il, jl,.. }, {i2, j2,... },... }] replaces parts at several positions by new. Ii Example 36 ReplacePart[IdentityMatrix[3], Sin[4?], {3, l}li/traditionaiform o 0) 1 0 o 1 RGBColor[red, green, blue] is a graphics directive that specifies that graphical objects which follow are to be displayed, if possible, in the color given. Example: see the description of function Polygon[... ]. Show[graphics, options] displays two- and three-dimensional graphics, using the options specified. Show[gl, g2,.. ] shows several plots combined. Examples: see the descriptions of option DisplayFunction and functions Line[... ], ListPlot[... ]. Simplify[expr] performs a sequence of algebraic transformations on expr, and returns the simplest form it finds. This function has the following default options: > Automatic, TimeConst Options[Simplify]={ComplexityFunction - raint - > 300, Trig - > True}. Ii Example 37 = = = = = = = ~ Simplify[x 2 /3 + x y/(5 - x)] 1 3y 3 X (x- -5+x) Sign[x] gives -1, 0, or 1, depending on whether x is negative, zero, or positive.

21 546 Appendix A: Mathematica Functions Ii Sign[-1.35] ; -1 Sign[O.]; o Sign[0.005] 1 Example 38 Sin[z] gives the sine of z. The argument of Sin is assumed to be in radians. Ii Example 39 TraditionalForm[Sin[Pi/10]] 1 -(-1+J5) 4 Solve[eqns, vars] attempts to solve an equation or set of equations for the variables vars. Solve[eqns, vars, elims] attempts to solve the equations for vars, eliminating the variables elims. This function has the following default options: Options[Solve]= {InverseFunctions - > Automatic, MakeRules - > False, Method - > 3, Mode - > Generic, Sort - > True, Verify Solutions - > Automatic, WorkingPrecision - > 00 }. Ii ComplexExpand[Solve[x3-1/3 x-i == 0, xll Example 40 =======:::::;'11 {x (1 + iv3) ( 7 ~ 9 ) 1/3-135fZ9 - i(l- iv3) (H27 + 5V29))1/3}, {x (1- ~ iv3) 7 ~ 9 ( ) 1/3-135fZ9 - i (1 + iv3) ( ~ ( + 25V29)) 7 1/3 }}

22 Appendix A: Mathematica Functions 547 Sqrt [z] or, Vz gives the square root of z. Example: see the description of function N[...}. Sum[f, {i, imax}] evaluates the sum of the expressions f as evaluated for each i from 1 to imax. Sum[f, {i, imin, imax}] starts with i = imino Sum[f, {i, imin, imax, dill uses steps di. Sum[f, {i, imin, imax}, {j, jmin, jmax},... ] evaluates a sum over multiple indices. Ii Example 41 Simplify[Sum[( _l)j+l CosU x], {j, 4}]l/ /TraditionalForm cos(x) - cos(2x) + cos(3x) - cos(4x) Table [expr, {imax}] generates a list of imax copies of expr. Table[expr, {i, imax}] generates a list of the values of expr when i runs from 1 to imax. Table[expr, {i, imin, imax}l starts with i = imino Table[expr, {i, imin, imax, dill uses steps di. Table[expr, {i, imin, imax}, {j, jmin, jmax},... 1 gives a nested list. The list associated with i is outermost. Ii Example 42 TraditionaIForm[Table[ai IJ2, {i, 3}, {j, 3}]] Text [expr, coords] is a graphics primitive that represents text corresponding to the printed form of expr, centered at the point specified by coords (see also Fig. A.8). Ii Example 43 rc = 2; bound = Graphics[Circle[{O,O}, rc]]; arc = Graphics[Circle[{O, O}, 0.9, {O, Pi/6}]]; lin = Graphics[Line[{ {rc,o},{o,o}, {rc C o s [ ~ l, }JJ; gr = Graphics[Text["R", {0.7, 1.1 }JJ ; gt = Graphics[Text["t", {1.3, 0.37}]]; Show[bound, arc, lin, gr, gt, AspectRatio - > Automatic] rc S i n [ ~ ] } Thickness[r] is a graphics directive that specifies that lines which follow are to be drawn with a thickness r. The thickness r is given as a fraction of the total width of the graph. Example: see the description of the Mathematica function Dashing [J. II

23 548 Appendix A: Mathematica Functions R t Figure A.8: Illustration to the description of the function Text [... ]. TraditionaIForm[expr] prints as an approximation to the traditional mathematical notation for expr. Example: see the descriptions of the functions TrigExpand[... ], Sin[... ]. Transpose [list ] transposes the first two levels in list. Transpose [list, {nl, n2,... }] transposes the list so that the levels 1, 2,... in the list correspond to levels n1, n2,... in the result. Ii Example 44 A = {{a,b,c}, {l,2,3}}; TraditionaIForm[A] TraditionalForm [Transpose [A]] ( a b e ) TrigExpand [expr] expands out trigonometric functions in expr. Ii TraditionaIForm[TrigExpand[Cos[3x] Sin[4y]ll Example 45 =======:::::::;'11 4cos3(x) sin(y)cos 3 (y) - 12 cos(x )sin 2 (x) sin(y)cos 3 (y) -4cos3 (x)sin 3 (y) cos(y) + 12 cos(x)sin 2 (x)sin 3 (y) cos(y) TrigReduce[expr] rewrites products and powers of trigonometric functions in expr in terms of trigonometric functions with combined arguments.

24 Appendix A: Mathematica Functions 549 Ii Example 46 TraditionaIForm[TrigReduce[Cos[x]5 + Sin[x]5]] ~ ( cos(x) 1 O + 5 cos(3x) + cos(5x) + 10 sin(x) - 5 sin(3x) + sin(5x)) 16 VectorHeads is an option for plotting the three-dimensional vector field with the aid ofthe function PlotVectorField3D[... ]. The specification VectorHeads - > True puts heads on arrows (see Fig. A.7). While[test, body] evaluates the test, then the body, repetitively, until the test first fails to give the True. Ii Example 47 II ~ = yo=o; O ; y={yo}; W h i l e, [ ~ ~ = < ~ l+ yo=yo+0.1e; 0. 1 ; AppendTo[y, yo]]; y {O, 0.001, , 0.014, 0.03, 0.055, 0.091, 0.14, 0.204, , , } References 1. Ganzha, V.G. and Vorozhtsov, E.V., Numerical Solutions for Partial Differential Equations: Problem Solving Using Mathematica, CRC Press, Boca Raton, Parker, L. and Christensen, S.M., MathTensor: a System for Doing Tensor Analysis by Computer, Addison-Wesley, Reading, Kythe, P.K., Puri, P., and Schaferkotter, M.R., Partial Differential Equations and Mathematica, CRC Press, Boca Raton, Vvedensky, D., Partial Differential Equations with Mathematica, Addison-Wesley, Reading, Skeel, R.D. and Keiper, J.B., Elementary Numerical Computing with Mathematica, McGraw-Hill, Inc., New York, Baumann, G., Mathematica in Theoretical Physics: Selected Examples from Classical Mechanics to Fractals, TELOS/Springer-Verlag, New York, Martin, E. (Ed.), Mathematica 3.0. Standard Add-on Packages, Wolfram Media, Cambridge University Press, Champaign, IL, 1996.

25 Appendix B: Glossary of Programs As shown in Chapters 1-7, a feature of this book is the use of the advanced software system Mathematica 3.0 to illustrate various concepts or to implement some of the analytical or numerical methods, which are presented in this book. In this appendix, we briefly outline the purposes of all Mathematica notebooks, which we have made and tested on a personal computer using DOS. In order to use any specific program or the file of input data you can simply copy the needed files on your computer from the following Birkhiiuser uri address: One of the features of Mathematica is that the Mathematica documents are called Notebooks. Notebooks can contain ordinary text and graphics as well as Mathematica input and output. We describe 24 Mathematica Notebooks in this appendix. All our Notebooks have the unified names progi - j.nb, where i is the chapter number (i = 1,2,3,4,5,6,7) and j is the Notebook number within a given chapter, j = 1,2,... The information in a Notebook is stored in cells, whose characteristics vary with the cell's function and the kind of information it holds. The brackets on the right side of the Notebook window indicate the extent of each cell. The typical structure of the Notebooks for this book is as follows. Each Notebook begins with a title, for example, Metric Tensor Components. The contents of a Notebook are subdivided into the following sections: Impressum. This section contains the text as follows. This Mathematica Notebook is part of the book entitled S.P. Kiselev, E.V. Vorozhtsov, and V.M. Fomin. Foundations of Fluid Mechanics with Applications Problem Solving Using Mathematica. Birkhiiuser Boston, Basel, General Description. This section explains the purpose of the Notebook, the method implemented in a specific program, the limitations of the program, and refers the reader to a specific section of the book, where the implemented method is described in detail.

26 Appendix B: Glossary of Programs 551 User's Guide. This section represents an internal instruction of the Notebook for the practical use of a Mathematica program contained in the Notebook. This instruction can include several steps. Program Listing. This section contains the listing of a program written in the language of Mathematica 3.0 and usually includes the major part of the Mathematica program. Parameters Used in Program *.nb. This section explains in detail the physical or mathematical meaning of each input parameter and gives the limitations on the numerical values of parameters if necessary. It should be noted that this section is absent in a number of our Notebooks, which, in particular, solve certain specific tasks of symbolic computations. Examples of the Input Data. This section gives one or several examples of the specification of the input parameters. These examples enable the user to reproduce those analytical, numerical, or graphics examples, which were obtained with Mathematica in this book. At the end of this appendix, we present Tables B.13- B.14, which should help the interested reader to easily reproduce any of the figures whose numbers are indicated in the tables. If you want to introduce some change into a file of input data, please always save our original version of the file. This will help you to find an error in your file of input data if our program fails at your data. One of the possible errors in the process of the preparation of the file of input data is that one of the entries (divided by commas from each other) is missing, so that the total number of the entries is less than the number needed by our program. Then the Mathematica system shows your file of input data on the screen of the computer monitor, for example, as follows: PlateFlow[l, Pi/4, -4, 4, -3, 3, 40] and thus does not proceed to the computation. In this case we recommend you to compare your prepared file of input data with one of the examples of the input data in a given Notebook. The Structure of the Output. This section explains the meaning of each entry of the data that are printed out in the process of the work of the program. We give below the lists of our Mathematica Notebooks for each chapter. After the name of each Notebook file (given by bold face letters), we reproduce the meaning of each entry of the input data by using the corresponding cell of the Notebook. If the given program uses some functions, we also describe briefly the purpose of these functions (the names of the functions are printed in italics).

27 552 Appendix B: Glossary of Programs The limitations for the book size have not allowed us to include in this appendix the listings of our Mathematica Notebooks. Chapter 1 progl-1.nb This Notebook solves analytically Problem 1.1 (see Section of this book). It enables the user to find the analytic expressions for the components of the metric tensor gij in the cylindrical coordinates r, cp, z. This program also computes the following analytic expressions: 1) the squared length element ds 2 in cylindrical coordinates; 2) the expressions for the basis vectors el, e2, e3 of the Cartesian coordinate system in terms of the basis vectors El, E2, E3 of the cylindrical coordinate system. progl-2.nb This Notebook solves analytically Problem 1.2 (see Section of this book). It enables the user to find the analytic expressions for the components of the metric tensor gij in the spherical coordinates r, cp, B. This program also finds the analytic expression for the squared length element ds 2 in spherical coordinates. progl-3.nb enables the user to compute the analytic expressions for the metric tensor components gij, i, j = 1, 2, as well as the expressions for the covariant basis vectors e l and e 2 in terms of the contravariant basis vectors el and e2 in the oblique-angled coordinate system (see Problem 1.3 at the end of Section of this book). progl-4.nb enables the user to find: 1) the motion trajectories, 2) the components of velocity and acceleration, 3) the streamlines for a given velocity field where Xl(t), X2(t), and X3(t) are the coordinates of a material point moving with time t in the (Xl, X2, X3) space. The formulation of this problem can be found in Section 1.2 (see Problem 1.4). progl-5.nb enables the user to find the analytic expressions for the components of the Lagrangian ij and Eulerian Cij strain tensors for a displacement field U(Ul' U2, U3) given in the Cartesian basis (6,6,6); that is, Uj = uj(6, 6, 6), j = 1,2, 3. This Notebook solves Problem 1.5 (see Section 1.2 of this book).

28 Appendix B: Glossary of Programs 553 progl-6.nb enables the user to find the analytic expressions for the strain tensor c and the rotation tensor Wij for the case in which the displacement vector components UI, U2, U3 are given in a Cartesian basis XI, X2, X3. The Notebook also computes the analytic expressions for the principal strains Ci and the eigenvectors iii. This Notebook solves Problem 1.6 (see Section of this book). progl-7. nb enables the user to find the analytic expressions for the vortex lines governed by the equations from a velocity field given in a Cartesian basis (Xl, X2, X3). The coordinates (Xl, X2(XI), X3(X I)) of vortex lines are found by the solution of the ordinary differential equations (ODEs) where the quantities WI, W2, and W3 denote the components of angular velocity w = (1/2) rot V, v is the given velocity vector. This Notebook solves Problem 1.7 (see Section of this book). progl-8.nb is similar to the Notebook progl-7.nb and differs only by another form of the functions Vj(XI,X2,X3), j = 1,2,3, which specify the velocity field in a Cartesian basis (Xl, X2, X3). This Notebook solves Problem 1.8 (see Section of this book). Chapter 2 prog2-1.nb enables the user to compute the analytic expressions for the Christoffel symbols rfj in the spherical coordinates. The obtained expressions for the rf j are then used in Section 2.1 of this book. Chapter 4 prog4-1.nb implements with Mathematica the analytical solution (4.2.12) of the problem of a noncirculatory stationary irrotational flow of an incompressible fluid past the cylinder. The formulation of this problem and the discussion of its solution may be found in Section of this book. 'ljj ' [x_, y J is the function to compute the value of the stream function, (x, y) at (x, y) point. The main program of this Notebook is NoncircFlow [R_, xl, xr _, yb_, yt_, nppj.

29 554 Appendix B: Glossary of Programs The meaning of the input data is explained in Table B.1. Table B.1 Parameters Used in Program prog4-1.nb Parameter R xl xr yb yt npp Description the cylinder radius, R > 0; the abscissa of the left boundary of a rectangular region D in the (x, y) plane, in which the fluid flow is to be determined; Ixll 2: R; the abscissa of the right boundary of region D, xr > R 2: xl; the ordinate of the lower (bottom) boundary of region D, Iybl 2: R; the ordinate of the upper boundary of region D, Iytl > R; the number of points of each streamline for the graphics function ContourPlot, npp> 10. prog4-2.nb implements with Mathematica the analytical solution (4.2.14) of the problem of a circulatory stationary irrotational flow of an incompressible fluid past the cylinder for the case of one or two critical points. The formulation of this problem and the discussion of its solution may be found in Section of this book. 'ljj ~ [x_, y J is the function to compute the value of the stream function ~ ( y) x, at (x, y) point. The main program of this Notebook is CircFlow2 [R_, g_, xl, xr _, yb_, yt_, nppj The meaning of the input data is explained in Table B.2. prog4-3.nb implements with Mathematica the analytical solution (4.2.14) of the problem of a circulatory stationary irrotational flow of an incompressible fluid past the cylinder for the case of a single critical point lying on the y-axis above the cylinder. The formulation of this problem and the discussion of its solution may be found in Section of this book. 'ljj ~ [x_, y J is the function to compute the value of the stream function ~(, xy) at (x, y) point. The main program of this Notebook is CircFlowl [R_, g_, xl, xr _, yb_, yt_, nppj.

30 Appendix B: Glossary of Programs 555 The meaning of the input data is explained in Table B.2. prog4-4.nb implements with Mathematica the analytical solution (4.2.29), (4.2.30) of the problem of a circulatory planar stationary flow of an incompressible fluid past a plate. Parameter R g xl xr yb yt npp Table B.2 Parameters Used in Program prog4-2.nb Description the cylinder radius, R > 0; o < g = r / (27rvoo ) < 2R to ensure the existence of two critical points; 0 < g = 2R to ensure that the two critical points merge into one point; the abscissa of the left boundary of a rectangular region D in the (x, y) plane, in which the fluid flow is to be determined; Ixll 2: R; the abscissa of the right boundary of region D, xr > R 2: xl; the ordinate of the lower (bottom) boundary of region D, Iybl 2: R; the ordinate of the upper boundary of region D, Iytl > R ; the number of points of each streamline for the graphics function ContourPlot, npp> 10. Table B.3 Parameters Used in Program prog4-4.nb Parameter 2a v 0: xl xr yb yt npp Description 2a is the plate length; the plate lies in the interval -a :::; x :::; a, a > 0; the magnitude of the freestream velocity, v > 0; the angle between the vector of freestream velocity and the positive direction of the x-axis; the value of 0: is specified in radians; the abscissa of the left boundary of a rectangular region D in the (x, y) plane, in which the fluid flow is to be determined; Ixll > a; the abscissa of the right boundary of D, xr > a; the ordinate of the lower (bottom) boundary of region D, yb < 0; the ordinate of the upper boundary of region D, Iytl > 0; the number of points of each streamline for the graphics function ContourPlot, npp> 10.

31 556 Appendix B: Glossary of Programs The formulation of this problem and the discussion of its solution may be found in Section of this book. w w [zj is the function to compute the value of the complex potential w(z) for a given value of complex variable z in accordance with formula (4.2.29). The main program of this Notebook is PlateFlow[a_, v_, CL, xl, XL, yb_, yt_, nppj. The meaning of the input data is explained in Table B.3. prog4-5.nb solves the problem of axially symmetric incompressible fluid flow around a body of revolution by the method of sources and sinks in the case in which the freest ream velocity is directed along the body axis. The formulation of this problem and the discussion of its solution may be found in Section of this book. body body [zj is the function specifying the form of the body generatrix x = x(z) as the NACA OOt1 profile; the value t1 is specified by the program user (see below the section parameters used in program prog4-5.nb). 'ljjz_, xj is the approximation of the stream function with the aid of a quadrature formula. The main program of this Notebook is Axisym[vO_, tprl, n_, nppj. The meaning of the input data is explained in Table B.4. Table B.4 Parameters Used in Program prog4-5.nb Parameter vo tprf n npp Description the magnitude of the freestream velocity, vo>o; the relative thickness of the body profile, that is tprf is the ratio of the profile thickness and chord length; o <tprf< 1; the number of sources and sinks; n is a positive integer; the number of points of each streamline for the graphics function ContourPlot, npp> 10. prog4-6.nb solves the problem of three-dimensional incompressible fluid flow around a body of revolution by the method of sources

32 Appendix B: Glossary of Programs 557 and sinks in the case in which the freest ream velocity is directed along a normal to the symmetry axis of the body. The formulation of this problem and the discussion of its solution may be found in Section of this book. body body [zj is the function specifying the form of the body generatrix x = x(z) as the NACA OOtl profile; the value tl is specified by the program user (see below the section parameters used in program prog4-6.nb). uxyz uxyz [x5_, y5_, z5j is the function that computes the approximations by the method of sources and sinks of the velocity components in the Cartesian coordinate system in terms of the velocity components in the cylindrical coordinate system. grlin grlin[jj is the function that computes the (x,y,z) coordinates of the individual (jth) streamline and generates a graphics picture of this streamline, which is plotted subsequently on a single graphics picture along with all other streamlines. The streamline coordinates (x, y, z) are determined by numerical integration of the system of equations (4.3.19). The main program of this Notebook is ThreeDimFlow[vO_, tprl, nj. The meaning of the input data is explained in Table B.5. Parameter vo tprf n vpts Table B.5 Parameters Used in Program prog4-6.nb Description the magnitude of the freestream velocity, vo>o; the relative thickness of the body profile; that is, tprf is the ratio of the profile thickness and chord length; 0 <tprf< 1; the number of sources and sinks; n is a positive integer; the number of evaluation points in each of the spatial directions x, y, z for the plot of the velocity vectors in a region around the body; vpts is a positive integer, vpts> 1. Chapter 5 prog5-1.nh enables the user to compute the analytic expressions for the components of the velocity Laplacian llij in the spherical coordinates T, cp, e. The discussion of underlying formulas may be found in Section 5.1 of this book.

33 558 Appendix B: Glossary of Programs prog5-2.nb enables the user to solve numerically the problem of a boundary layer of a flat plate. The problem under consideration is the initial-value problem (5.3.18), (5.3.19) (see Section 5.3 of this book). It is solved numerically with the aid of the standard fourth-order Runge Kutta method. fyzfyz[l, y_, zj is the function to compute the right-hand sides of equations (5.3.20). The main program of this Notebook is B l a y e r hminj [ ~ _, The meaning of the input data is explained in Table B.6. Parameter ~ m a x hmin Table B.6 Parameters Used in Program prog5-2.nb Description the abscissa of the right end of the integration interval 0 ::; ~ ::; ~ m a ~ x, m > a 0; x the start value of the integration stepsize along the ( axis, 0 < hmin < 1. Chapter 6 prog6-1.nb enables the user to solve numerically the problem of a quasi-one-dimensional stationary isentropic gas flow in a variable section duct. The derivation of the underlying solution formulas may be found in Section of this book. fw fw [xj computes the ordinate y of the nozzle wall in accordance with Fig fm fm [MJ is a function constructed to solve numerically the transcendental equation (6.1.10) by the bisection method. The main program of this Notebook is nozzle [he, cyo_. cro_. cr_. cxo_, cx4_, thetl, thet2_. 1_' Ml_, Mmax_, eps_, ix_. regj. The meaning of the input data is explained in Table B.7. prog6-2.nb enables the user to determine the curve V = V(x) in the shock transition in a viscous, non-heat-conducting compressible gas. The formulation of this problem and the discussion of its solution may be found in Section 6.2 of this book. The main program of this Notebook is