Exploring Analytic Geometry With Mathematica
Exploring Analytic Geometry with Mathematica by Donald L. VosslerPaperback, 865 pagesAcademic Press, 1999Book Size: 2.13" x 9.19" x 7.48"ISBN: 0-12-728255-6PDF edition availableThis PDF file contains the complete published text of the book entitled Exploring AnalyticGeometry with Mathematica by author Donald L. Vossler published in 1999 by Academic Press.The book is out of print and no longer available as a paperback from the original publisher.Additional materials from the book’s accompanying CD, including the Descarta2D software, areavailable at the author’s web site http://www.descarta2d.com .AbstractThe study of two-dimensional analytic geometry has gone in and out of fashion several timesover the past century. However this classic field of mathematics has once again become populardue to the growing power of personal computers and the availability of powerful mathematicalsoftware systems, such as Mathematica, that can provide an interactive environment for studyingthe field. By combining the power of Mathematica with an analytic geometry software systemcalled Descarta2D, the author has succeeded in meshing an ancient field of study with moderncomputational tools, the result being a simple, yet powerful, approach to studying analyticgeometry. Students, engineers and mathematicians alike who are interested in analytic geometrycan use this book and software for the study, research or just plain enjoyment of analyticgeometry.Mathematica is a registered trademark of Wolfram Research.Descarta2D is a trademark of the author, Donald L. Vossler.Copyright 1999-2007 Donald L. Vossler
Exploring Analytic Geometrywith Mathematica Donald L. VosslerBME, Kettering University, 1978MM, Aquinas College, 1981Anaheim, California USA, 1999
Copyright 1999-2007 Donald L. Vossler
PrefaceThe study of two-dimensional analytic geometry has gone in and out of fashion several timesover the past century, however this classic field of mathematics has once again become populardue to the growing power of personal computers and the availability of powerful mathematicalsoftware systems, such as Mathematica, that can provide an interactive environment for studying the field. By combining the power of Mathematica with an analytic geometry softwaresystem called Descarta2D, the author has succeeded in meshing an ancient field of study withmodern computational tools, the result being a simple, yet powerful, approach to studyinganalytic geometry. Students, engineers and mathematicians alike who are interested in analytic geometry can use this book and software for the study, research or just plain enjoymentof analytic geometry.Mathematica provides an attractive environment for studying analytic geometry. Mathematica supports both numeric and symbolic computations, meaning that geometry problemscan be solved numerically, producing approximate or exact answers, as well as producing general formulas with variables. Mathematica also has good facilities for producing graphicalplots which are useful for visualizing the graphs of two-dimensional geometry.FeaturesExploring Analytic Geometry with Mathematica, Mathematica and Descarta2D provide thefollowing outstanding features: The book can serve as classical analytic geometry textbook with in-line Mathematicadialogs to illustrate key concepts. A large number of examples with solutions and graphics is keyed to the textual development of each topic. Hints are provided for improving the reader’s use and understanding of Mathematicaand Descarta2D. More advanced topics are covered in explorations provided with each chapter, and fullsolutions are illustrated using Mathematica.v
viPreface A detailed reference manual provides complete documentation for Descarta2D, with complete syntax for over 100 new commands. Complete source code for Descarta2D is provided in 30 well-documented Mathematicanotebooks. The complete book is integrated into the Mathematica Help Browser for easy access andreading. A CD-ROM is included for convenient, permanent storage of the Descarta2D software. A complete software system and mathematical reference is packaged as an affordablebook.Classical Analytic GeometryExploring Analytic Geometry with Mathematica begins with a traditional development of analytic geometry that has been modernized with in-line chapter dialogs using Descarta2D andMathematica to illustrate the underlying concepts. The following topics are covered in 21chapters:Coordinates Points Equations Graphs Lines Line Segments Circles Arcs Triangles Parabolas Ellipses Hyperbolas General Conics Conic Arcs Medial Curves Transformations Arc Length Area Tangent Lines Tangent Circles Tangent Conics Biarcs.Each chapter begins with definitions of underlying mathematical terminology and developsthe topic with more detailed derivations and proofs of important concepts.ExplorationsEach chapter in Exploring Analytic Geometry with Mathematica concludes with more advancedtopics in the form of exploration problems to more fully develop the topics presented in eachchapter. There are more than 100 of these more challenging explorations, and the full solutionsare provided on the CD-ROM as Mathematica notebooks as well as printed in Part VIII of thebook. Sample explorations include some of the more famous theorems from analytic geometry:Carlyle’s Circle Castillon’s Problem Euler’s Triangle Formula Eyeball Theorem Gergonne’s Point Heron’s Formula Inversion Monge’s Theorem Reciprocal Polars Reflection in a Point Stewart’s Theorem plus many more.
PrefaceviiDescarta2DDescarta2D provides a full-scale Mathematica implementation of the concepts developed inExploring Analytic Geometry with Mathematica. A reference manual section explains in detailthe usage of over 100 new commands that are provided by Descarta2D for creating, manipulating and querying geometric objects in Mathematica. To support the study and enhancementof the Descarta2D algorithms, the complete source code for Descarta2D is provided, both inprinted form in the book and as Mathematica notebook files on the CD-ROM.CD-ROMThe CD-ROM provides the complete text of the book in Abode Portable Document Format(PDF) for interactive reading. In addition, the CD-ROM provides the following Mathematicanotebooks: Chapters with Mathematica dialogs, 24 interactive notebooks Reference material for Descarta2D, three notebooks Complete Descarta2D source code, 30 notebooks Descarta2D packages, 30 loadable files Exploration solutions, 125 notebooks.These notebooks have been thoroughly tested and are compatible with Mathematica Version3.0.1 and Version 4.0. Maximum benefit of the book and software is gained by using it inconjunction with Mathematica, but a passive reading and viewing of the book and notebookfiles can be accomplished without using Mathematica itself.Organization of the BookExploring Analytic Geometry with Mathematica is a 900-page volume divided into nine parts: Introduction (Getting Started and Descarta2D Tour) Elementary Geometry (Points, Lines, Circles, Arcs, Triangles) Conics (Parabolas, Ellipses, Hyperbolas, Conics, Medial Curves) Geometric Functions (Transformations, Arc Length, Area) Tangent Curves (Lines, Circles, Conics, Biarcs) Descarta2D Reference (philosophy and command descriptions) Descarta2D Packages (complete source code)
viiiPreface Explorations (solution notebooks) Epilogue (Installation Instructions, Bibliography and a detailed index).About the AuthorDonald L. Vossler is a mechanical engineer and computer software designer with more than20 years experience in computer aided design and geometric modeling. He has been involvedin solid modeling since its inception in the early 1980’s and has contributed to the theoreticalfoundation of the subject through several published papers. He has managed the developmentof a number of commercial computer aided design systems and holds a US Patent involvingthe underlying data representations of geometric models.
ContentsIIntroduction11 Getting Started1.1 Introduction . . . . . . .1.2 Historical Background .1.3 What’s on the CD-ROM1.4 Mathematica . . . . . .1.5 Starting Descarta2D . .1.6 Outline of the Book . .33345672 Descarta2D Tour2.1 Points . . . . . . . .2.2 Equations . . . . . .2.3 Lines . . . . . . . . .2.4 Line Segments . . .2.5 Circles . . . . . . . .2.6 Arcs . . . . . . . . .2.7 Triangles . . . . . .2.8 Parabolas . . . . . .2.9 Ellipses . . . . . . .2.10 Hyperbolas . . . . .2.11 Transformations . .2.12 Area and Arc Length2.13 Tangent Curves . . .2.14 Symbolic Proofs . .2.15 Next Steps . . . . .991012131415161718192020212223II.Elementary Geometry253 Coordinates and Points273.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Rectangular Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28ix
xContents3.33.43.53.63.7Line Segments and Distance . .Midpoint between Two Points .Point of Division of Two PointsCollinear Points . . . . . . . . .Explorations . . . . . . . . . .39. 39. 39. 41. 42. 46. 47. 485 Lines and Line Segments5.1 General Equation . . . . . . . . . . .5.2 Parallel and Perpendicular Lines . .5.3 Angle between Lines . . . . . . . . .5.4 Two–Point Form . . . . . . . . . . .5.5 Point–Slope Form . . . . . . . . . . .5.6 Slope–Intercept Form . . . . . . . .5.7 Intercept Form . . . . . . . . . . . .5.8 Normal Form . . . . . . . . . . . . .5.9 Intersection Point of Two Lines . . .5.10 Point Projected Onto a Line . . . . .5.11 Line Perpendicular to Line Segment5.12 Angle Bisector Lines . . . . . . . . .5.13 Concurrent Lines . . . . . . . . . . .5.14 Pencils of Lines . . . . . . . . . . . .5.15 Parametric Equations . . . . . . . .5.16 Explorations . . . . . . . . . . . . .51. 51. 54. 55. 56. 58. 62. 64. 65. 69. 70. 72. 73. 74. 75. 78. 816 Circles6.1 Definitions and Standard Equation6.2 General Equation of a Circle . . .6.3 Circle from Diameter . . . . . . . .6.4 Circle Through Three Points . . .6.5 Intersection of a Line and a Circle6.6 Intersection of Two Circles . . . .6.7 Distance from a Point to a Circle .6.8 Coaxial Circles . . . . . . . . . . .6.9 Radical Axis . . . . . . . . . . . .6.10 Parametric Equations . . . . . . .4 Equations and Graphs4.1 Variables and Functions4.2 Polynomials . . . . . . .4.3 Equations . . . . . . . .4.4 Solving Equations . . .4.5 Graphs . . . . . . . . . .4.6 Parametric Equations .4.7 Explorations . . . . . .30333336378585888990919295969799
Contentsxi6.11 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017 Arcs7.1 Definitions . . . . . . . . . . . . .7.2 Bulge Factor Arc . . . . . . . . .7.3 Three–Point Arc . . . . . . . . .7.4 Parametric Equations . . . . . .7.5 Points and Angles at Parameters7.6 Arcs from Ray Points . . . . . .7.7 Explorations . . . . . . . . . . .105. 105. 107. 110. 111. 112. 113. 1148 Triangles8.1 Definitions . . . . . . .8.2 Centroid of a Triangle8.3 Circumscribed Circle .8.4 Inscribed Circle . . . .8.5 Solving Triangles . . .8.6 Cevian Lengths . . . .8.7 Explorations . . . . .III.Conics9 Parabolas9.1 Definitions . . . . . . . . . . . . . .9.2 General Equation of a Parabola . .9.3 Standard Forms of a Parabola . . .9.4 Reduction to Standard Form . . .9.5 Parabola from Focus and Directrix9.6 Parametric Equations . . . . . . .9.7 Explorations . . . . . . . . . . . 14210 Ellipses10.1 Definitions . . . . . . . . . . . . . . . .10.2 General Equation of an Ellipse . . . .10.3 Standard Forms of an Ellipse . . . . .10.4 Reduction to Standard Form . . . . .10.5 Ellipse from Vertices and Eccentricity10.6 Ellipse from Foci and Eccentricity . .10.7 Ellipse from Focus and Directrix . . .10.8 Parametric Equations . . . . . . . . .10.9 Explorations . . . . . . . . . . . . . .145145147147150151153153155156.
xiiContents11 Hyperbolas11.1 Definitions . . . . . . . . . . . . . . . . . .11.2 General Equation of a Hyperbola . . . . .11.3 Standard Forms of a Hyperbola . . . . . .11.4 Reduction to Standard Form . . . . . . .11.5 Hyperbola from Vertices and Eccentricity11.6 Hyperbola from Foci and Eccentricity . .11.7 Hyperbola from Focus and Directrix . . .11.8 Parametric Equations . . . . . . . . . . .11.9 Explorations . . . . . . . . . . . . . . . .15915916116116616716816917017312 General Conics12.1 Conic from Quadratic Equation . . . . .12.2 Classification of Conics . . . . . . . . . .12.3 Center Point of a Conic . . . . . . . . .12.4 Conic from Point, Line and Eccentricity12.5 Common Vertex Equation . . . . . . . .12.6 Conic Intersections . . . . . . . . . . . .12.7 Explorations . . . . . . . . . . . . . . .17517518418418518618919013 Conic Arcs13.1 Definition of a Conic Arc13.2 Equation of a Conic Arc .13.3 Projective Discriminant .13.4 Conic Characteristics . . .13.5 Parametric Equations . .13.6 Explorations . . . . . . .19319319419619619819914 Medial Curves14.1 Point–Point .14.2 Point–Line . .14.3 Point–Circle .14.4 Line–Line . .14.5 Line–Circle .14.6 Circle–Circle14.7 Explorations.201201202204206207210212IV.Geometric Functions21515 Transformations21715.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21715.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21915.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Contentsxiii15.4 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22415.5 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22616 Arc16.116.216.316.416.516.616.716.816.9LengthLines and Line Segments . . . . . .Perimeter of a Triangle . . . . . . .Polygons Approximating Curves .Circles and Arcs . . . . . . . . . .Ellipses and Hyperbolas . . . . . .Parabolas . . . . . . . . . . . . . .Chord Parameters . . . . . . . . .Summary of Arc Length FunctionsExplorations . . . . . . . . . . . .17 Area17.1 Areas of Geometric Figures17.2 Curved Areas . . . . . . . .17.3 Circular Areas . . . . . . .17.4 Elliptic Areas . . . . . . . .17.5 Hyperbolic Areas . . . . . .17.6 Parabolic Areas . . . . . . .17.7 Conic Arc Area . . . . . . .17.8 Summary of Area Functions17.9 Explorations . . . . . . . .V.237. 237. 240. 240. 242. 245. 246. 248. 249. 249Tangent Curves18 Tangent Lines18.1 Lines Tangent18.2 Lines Tangent18.3 Lines Tangent18.4 25526627328019 Tangent Circles19.1 Tangent Object, Center Point . . . . . . . .19.2 Tangent Object, Center on Object, Radius .19.3 Two Tangent Objects, Center on Object . .19.4 Two Tangent Objects, Radius . . . . . . . .19.5 Three Tangent Objects . . . . . . . . . . . .19.6 Explorations . . . . . . . . . . . . . . . . .283283285286287288289to a Circle . . . . .to Conics . . . . . .to Standard Conics. . . . . . . . . . . .
xivContents20 Tangent Conics20.1 Constraint Equations . . . . . . .20.2 Systems of Quadratics . . . . . .20.3 Validity Conditions . . . . . . . .20.4 Five Points . . . . . . . . . . . .20.5 Four Points, One Tangent Line .20.6 Three Points, Two Tangent Lines20.7 Conics by Reciprocal Polars . . .20.8 Explorations . . . . . . . . . . .21 Biarcs21.1 Biarc Carrier Circles . . . . .21.2 Knot Point . . . . . . . . . .21.3 Knot Circles . . . . . . . . . .21.4 Biarc Programming Examples21.5 Explorations . . . . . . . . .VI.293293294296296298301306310.311. 311. 314. 316. 317. 322Reference32322 Technical Notes22.1 Computation Levels . . . .22.2 Names . . . . . . . . . . . .22.3 Descarta2D Objects . . . .22.4 Descarta2D Packages . . . .22.5 Descarta2D Functions . . .22.6 Descarta2D Documentation.32532532632633733833923 Command Browser34124 Error Messages367VIIPackagesD2DArc2D . . . . .D2DArcLength2D .D2DArea2D . . . .D2DCircle2D . . .D2DConic2D . . . .D2DConicArc2D . .D2DEllipse2D . .D2DEquations2D .D2DExpressions2DD2DGeometry2D . .385.387395399405411415421427429437
ContentsD2DHyperbola2D . . .D2DIntersect2D . . .D2DLine2D . . . . . .D2DLoci2D . . . . . .D2DMaster2D . . . . .D2DMedial2D . . . . .D2DNumbers2D . . . .D2DParabola2D . . . .D2DPencil2D . . . . .D2DPoint2D . . . . . .D2DQuadratic2D . . .D2DSegment2D . . . .D2DSketch2D . . . . .D2DSolve2D . . . . . nes2D .D2DTangentPoints2DD2DTransform2D . . .D2DTriangle2D . . . .xv.apollon.nb, Circle of Apollonius . . . . . . . . . . . . . . .arccent.nb, Centroid of Semicircular Arc . . . . . . . . . .arcentry.nb, Arc from Bounding Points and Entry Directionarcexit.nb, Arc from Bounding Points and Exit Direction .archimed.nb, Archimedes’ Circles . . . . . . . . . . . . . . .arcmidpt.nb, Midpoint of an Arc . . . . . . . . . . . . . . .caarclen.nb, Arc Length of a Parabolic Conic Arc . . . . . .caarea1.nb, Area of a Conic Arc (General) . . . . . . . . .caarea2.nb, Area of a Conic Arc (Parabola) . . . . . . . . .cacenter.nb, Center of a Conic Arc . . . . . . . . . . . . . .cacircle.nb, Circular Conic Arc . . . . . . . . . . . . . . . .camedian.nb, Shoulder Point on Median . . . . . . . . . . . .caparam.nb, Parametric Equations of a Conic Arc . . . . .carlyle.nb, Carlyle Circle . . . . . . . . . . . . . . . . . . .castill.nb, Castillon’s Problem . . . . . . . . . . . . . . .catnln.nb, Tangent Line at Shoulder Point . . . . . . . . .center.nb, Center of a Quadratic . . . . . . . . . . . . . .chdlen.nb, Chord Length of Intersecting Circles . . . . . .cir3pts.nb, Circle Through Three Points . . . . . . . . . .circarea.nb, One-Third of a Circle’s Area . . . . . . . . . 69571573575577579581583585589591593595597
xviContentscirptmid.nb, Circle–Point Midpoint Theorem . . . . . . . .cramer2.nb, Cramer’s Rule (Two Equations) . . . . . . . .cramer3.nb, Cramer’s Rule (Three Equations) . . . . . . .deter.nb,Determinants . . . . . . . . . . . . . . . . . . .elfocdir.nb, Focus of Ellipse is Pole of Directrix . . . . . .elimlin.nb, Eliminate Linear Terms . . . . . . . . . . . . .elimxy1.nb, Eliminate Cross-Term by Rotation . . . . . . .elimxy2.nb, Eliminate Cross-Term by Change in Variableselimxy3.nb, Eliminate Cross-Term by Change in Variableselldist.nb, Ellipse Locus, Distance from Two Lines . . . .ellfd.nb,Ellipse from Focus and Directrix . . . . . . . .ellips2a.nb, Sum of Focal Distances of an Ellipse . . . . . .elllen.nb, Length of Ellipse Focal Chord . . . . . . . . .ellrad.nb, Apoapsis and Periapsis of an Ellipse . . . . . .ellsim.nb, Similar Ellipses . . . . . . . . . . . . . . . . . .ellslp.nb, Tangent to an Ellipse with Slope . . . . . . . .eqarea.nb, Equal Areas Point . . . . . . . . . . . . . . . .eyeball.nb, Eyeball Theorem . . . . . . . . . . . . . . . . .gergonne.nb, Gergonne Point of a Triangle . . . . . . . . . .heron.nb,Heron’s Formula . . . . . . . . . . . . . . . . .hyp2a.nb,Focal Distances of a Hyperbola . . . . . . . . .hyp4pts.nb, Equilateral Hyperbolas . . . . . . . . . . . . .hyparea.nb, Areas Related to Hyperbolas . . . . . . . . . .hypeccen.nb, Eccentricities of Conjugate Hyperbolas . . . .hypfd.nb,Hyperbola from Focus and Directrix . . . . . .hypinv.nb, Rectangular Hyperbola Distances . . . . . . .hyplen.nb, Length of Hyperbola Focal Chord . . . . . . .hypslp.nb, Tangent to a Hyperbola with Given Slope . . .hyptrig.nb, Trigonometric Parametric Equations . . . . . .intrsct.nb, Intersection of Lines in Intercept Form . . . . .inverse.nb, Inversion . . . . . . . . . . . . . . . . . . . . .johnson.nb, Johnson’s Congruent Circle Theorem . . . . .knotin.nb, Incenter on Knot Circle . . . . . . . . . . . . .lndet.nb,Line General Equation Determinant . . . . . .lndist.nb, Vertical/Horizontal Distance to a Line . . . . .lnlndist.nb, Line Segment Cut by Two Lines . . . . . . . .lnquad.nb, Line Normal to a Quadratic . . . . . . . . . . .lnsdst.nb, Distance Between Parallel Lines . . . . . . . .lnsegint.nb, Intersection Parameters of Two Line Segmentslnsegpt.nb, Intersection Point of Two Line Segments . . .lnsperp.nb, Equations of Perpendicular Lines . . . . . . . .lntancir.nb, Line Tangent to a Circle . . . . . . . . . . . . .lntancon.nb, Line Tangent to a Conic . . . . . . . . . . . . 677679681685687689691693695697
Contentsmdcircir.nb, Medial Curve, Circle–Circle . . . . . . . . .mdlncir.nb, Medial Curve, Line–Circle . . . . . . . . . .mdlnln.nb, Medial Curve, Line–Line . . . . . . . . . .mdptcir.nb, Medial Curve, Point–Circle . . . . . . . . .mdptln.nb, Medial Curve, Point–Line . . . . . . . . . .mdptpt.nb, Medial Curve, Point–Point . . . . . . . . .mdtype.nb, Medial Curve Type . . . . . . . . . . . . .monge.nb,Monge’s Theorem . . . . . . . . . . . . . .narclen.nb, Approximate Arc Length of a Curve . . . .normal.nb, Normals and Minimum Distance . . . . . .pb3pts.nb, Parabola Through Three Points . . . . . .pb4pts.nb, Parabola Through Four Points . . . . . . .pbang.nb,Parabola Intersection Angle . . . . . . . . .pbarch.nb, Parabolic Arch . . . . . . . . . . . . . . . .pbarclen.nb, Arc Length of a Parabola . . . . . . . . . .pbdet.nb,Parabola Determinant . . . . . . . . . . . .pbfocchd.nb, Length of Parabola Focal Chord . . . . . .pbslp.nb,Tangent to a Parabola with a Given Slope .pbtancir.nb, Circle Tangent to a Parabola . . . . . . . .pbtnlns.nb, Perpendicular Tangents to a Parabola . . .polarcir.nb, Polar Equation of a Circle . . . . . . . . . .polarcol.nb, Collinear Polar Coordinates . . . . . . . . .polarcon.nb, Polar Equation of a Conic . . . . . . . . . .polardis.nb, Distance Using Polar Coordinates . . . . .polarell.nb, Polar Equation of an Ellipse . . . . . . . .polareqn.nb, Polar Equations . . . . . . . . . . . . . . .polarhyp.nb, Polar Equation of a Hyperbola . . . . . . .polarpb.nb, Polar Equation of a Parabola . . . . . . . .polarunq.nb, Non-uniqueness of Polar Coordinates . . .pquad.nb,Parameterization of a Quadratic . . . . . .ptscol.nb, Collinear Points . . . . . . . . . . . . . . .radaxis.nb, Radical Axis of Two Circles . . . . . . . . .radcntr.nb, Radical Center . . . . . . . . . . . . . . . .raratio.nb, Radical Axis Ratio . . . . . . . . . . . . . .reccir.nb, Reciprocal of a Circle . . . . . . . . . . . .recptln.nb, Reciprocals of Points and Lines . . . . . . .recquad.nb, Reciprocal of a Quadratic . . . . . . . . . .reflctpt.nb, Reflection in a Point . . . . . . . . . . . . .rtangcir.nb, Angle Inscribed in a Semicircle . . . . . . .rttricir.nb, Circle Inscribed in a Right Triangle . . . .shoulder.nb, Coordinates of Shoulder Point . . . . . . .stewart.nb, Stewart’s Theorem . . . . . . . . . . . . . .tancir1.nb, Circle Tangent to Circle, Given Center . . 67769771773775777779781783785787789
xviiiContents
Preface vii Descarta2D Descarta2D provides a full-scale Mathematica implementation of the concepts developed in Exploring Analytic Geometry with Mathematica.A reference manual section explains in detail the usage of over100 new commands that are providedbyDescarta2D for creating, manipulat- ing and querying geometric o
1 Mathematica Basics This chapter is an introduction to Mathematica.We briefly describe many of the most important and basic elements of Mathematica and discuss a few of the more common technical issues related to using Mathematica.Since our primary goal is to use Mathematica to help us understand calculus, you should not initially spend a great amount of time pouring
Introduction.NET/Link Welcome to .NET/Link, a product that integrates Mathematica and Microsoft's .NET platform.NET/Link lets you call .NET from Mathematica in a completely transparent way, and allows you to use and control the Mathematica kernel from a .NET program. For Mathematica users,.NET/Link makes the entire .NET world an automatic extension to the Mathematica environ-
Dec 09, 2005 · Beginner’s Mathematica Tutorial Introduction This document is designed to act as a tutorial for an individual who has had no prior experience with Mathematica. For a more advanced tutorial, walk through the Mathematica built in tutorial located at Help Tutorial on the Mathematica Task Bar.
Online Help from Mathematica 2 Mathematica Tutorial.nb. Introduction Mathematica is a system for doing mathematics on the computer.It can do numerics,symbolics,graphics and is also a programming language.Mathematica has infinite precision.It can plot functions of a single variable; make
mathematica Remarks This section provides an overview of what wolfram-mathematica is, and why a developer might want to use it. It should also mention any large subjects within wolfram-mathematica, and link out to the related topics. Since the Documentation for wolfram-mathematica is new, you may need to create initial versions of those related .
PROGRAMMING IN MATHEMATICA, A PROBLEM-CENTRED APPROACH 7 1.3. Algebraic computations. One of the abilities of Mathematica is to handle symbolic com-putations. Consider the expression (x 1)2. One can use Mathematica to expand this expression: Expand[(x 1) 2] 1 2x x2 Mathematica can also do the inverse of this task, namely to factorize an expression:
Start up Mathematica from a Linux desktop terminal window. Parallel Mathematica jobs can be submitted from with the Mathematica notebook interface as well as using PBS command files and the example scripts show how to setup and submit the jobs Documentation:Submitting Mathematica Parallel Jobs (UVACSE) October 8, 2014 44 / 46
Level 1 1–2 Isolated elements of knowledge and understanding – recall based. Weak or no relevant application to business examples. Generic assertions may be presented. Level 2 3–4 Elements of knowledge and understanding, which are applied to the business example. Chains of reasoning are presented, but may be assertions or incomplete.
