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Mathematics Maple includes over 3,500 computational functions to deliver the richest set of computation tools for any area in mathematics, science, or engineering. MAJOR TOPICS INCLUDE: Algebra • Exact symbolic arithmetic with real and complex numbers • Factor, expand, combine*, and simplify* algebraic expressions • Sequences and series Calculus* • Compute derivatives, integrals, and limits • Continuity testing • Asymptotic expansion, and directional and multidirectional limits Differential Equations • Exact and numerical solutions of ODEs and ODE systems* • Exact and numerical solutions of PDEs and PDE systems* • Exact and numerical solutions of Boundary Value Problems for PDEs • Numerical solutions for Differential Algebraic ODE initial value problems (DAEs) • Differential elimination for ODE and PDE systems • Structural analysis and order-reduction of ODEs and PDEs • Interactive Maplet™ Assistant for numeric and symbolic solutions of ODEs and ODE systems Linear Algebra* • Over 100 commands for constructing, solving, programming, and querying topics in linear algebra • Symbolic and numeric computations • Eigenvalues and eigenvectors, both classical and generalized • Sophisticated algorithms for modular linear algebra • Many matrix factorizations and system solvers • State-of-the-art numerical methods for dense and sparse systems with a high degree of user control Solvers • Solve equations and systems of algebraic equations symbolically and numerically for closed form and approximate solutions* • Inequalities and systems of inequalities • ODEs, PDEs, and DAEs • Numerous specialized solvers including routines for equations over the integers, equations over the integers mod m, recurrence equations, differential equations, series solutions, and q-difference equations Statistics** • Wide range of common statistical tasks, such as quantitative and graphical data analysis, simulation, and curve fitting • Over 35 continuous and discrete probability distributions and tools for creating custom distributions • Interactive data analysis assistant • Statistical plots including box plots, bar charts, histograms, probability plots, and scatter plots • Tools for fitting linear and nonlinear models to data points and performing regression analysis • Statistical Process Control (SPC) package with 11 different statistical control charts for process control variables and attributes Vector Calculus* • Directional derivatives, gradients, Hessian matrices, and Laplacians of a function • Curl and divergence of a vector field • Jacobian and Wronskian matrices of a list of functions • Cross products and dot products of vectors and differential operators OTHER TOPICS INCLUDE: Abstract Algebra • Groups, polynomials ideals, greatest common divisors, finite rings and fields Algebraic Curves • Holomorphic differentials and genus of an algebraic curve • Normal forms for elliptic and hyperelliptic curves Combinatorial Functions • Permutations and combinations • Construction of random combinations, partitions, and permutations • Stirling numbers of the first and second kind, polynomials, and Fibonacci numbers Combinatorial Structures • Computation and solution of a system of generating function equation sassociated with an attribute grammar • Generation of random combinatorial objects and counting the objects of a given size Complex Arithmetic and Functions • Complex numbers, fractions, constants, and variables Curve Fitting • B-spline basis functions, polynomial interpolation, least-squares approximation, rational interpolation, and splines Differential Algebra • Manipulation and reduction of differential equations • Development of the solutions into formal power series Differential Forms • Create, manipulate, and compute with differential forms Discrete Transforms • Fast Fourier transform and inverse transform of single or multidimensional data Euclidean Geometry • Close to 300 commands for constructing, computing, plotting, and translating 2-D and 3-D objects Financial Mathematics • Annuities and growing annuities, cash flows, perpetuities and growing perpetuities, and level coupons • Amortization, Black-Scholes option pricing, effective rate, future value, present value, and yield to maturity Gaussian Integers • Chinese remainder, GCD, and LCM of Gaussian integers • Gaussian integer factorization • Extended Euclidean algorithm for Gaussian integers Generating Functions* • Tools for determining and manipulating generating functions, including ordinary and exponential generating functions • Tools for determining a probable generating function from initial terms Graph Theory • Over 75 commands to create and study networks Groebner Bases* and Polynomial Ideals** • Groebner bases over commutative and non-commutative domains • Ideal membership and containment • Radical, prime and primary decompositions • Hilbert series, polynomials and dimensions Group Theory • 34 commands, including calculating the elements of a permutation group, order computations, and finding a permutation of a group Integral Transforms • Laplace, inverse Laplace, Fourier sine, Hankel, Hilbert, inverse Mellin, and Z transforms Lie Symmetries • Lie group symmetry methods for differential forms Linear Functional Systems of Equations* • Transformations of a matrix recurrence system into an equivalent system with nonsingular leading and trailing matrices • Rational and formal power series solutions of a linear functional system of equations with polynomial coefficients Linear Operators • Numerous commands to manipulate d’Alembertian terms • Conversions among Ore polynomial structures, linear ODEs, linear recurrence equations, and factored OrePoly structures Linear Programming • Solve linear programs using symbolic and numeric methods, including simplex Linear Recurrence Equations • Polynomial, rational, and hypergeometric solutions of linear recurrence equations • Solutions of divide and conquer recurrence equations Logic • Commands for manipulating Boolean expressions Numerical Approximations • Infinite precision numerical computations • Chebyshev-Pade and minimax rational approximation • Conversion of rational functions to continued-fraction form and Polynomials to Horner form Number Theory • Primality testing • Computation of the nth Fermat number and the nth Mersenne prime • Computation of the nth convergent, denominator, and numerator of simple and regular continued fractions Optimization* • Numeric methods for the solution of optimization problems • Interactive Optimization assistant for defining, solving and visualizing** problems • Solvers for linear, quadratic, and nonlinear programs, including constrained and unconstrained problems • Solvers for linear and nonlinear least-squares problems • Solvers for integer linear programs** Orthogonal Polynomials • Commands to generate the nth Gegenbauer, Hermite, Laguerre, Legendre, and Jacobi polynomials P-adic Numbers • Commands for p-adic evaluation, expansion, and functions • Computation of the order and the leading coefficient of a p-adic expansion of a rational function Polynomials • Factor, expand, manipulate, and compute properties of polynomials Q-Difference Equations* • Solve linear q-difference equations with polynomial coefficients Rational Normal Forms • Computation of polynomial and rational normal forms, canonical forms of rational functions, and minimal representations of hypergeometric terms Real Domain Computations • Restrict calculations to the domain of real numbers Series Expansions • Compute Taylor, Puiseux and asymptotic series expansions Scientific Constants • Support for over 70 scientific constants including the Newtonian constant of gravitation, magnetic flux quantum, and conductance quantum • Properties of all elements and isotopes of the Periodic Table • Tools for building custom scientific constants Scientific Error Analysis • Representation and construction of numerical quantities that have a central value and associated error Special Functions* • Over 100 special functions • Includes Airy Ai and Bi wave functions, Bessel, Chebyshev, Ei, Si, Ci, Li, error function, Gamma, Zeta, Heun**, Legendre, Jacobi functions Statistical Process Control** • Supports 11 different control charts for process control variables and attributes, including EWMA, S, MA and X-bar charts • Visualize both numeric and non-numeric quality characteristics Symbolic-Numeric Algorithms for Polynomials • Algebraic manipulation of numeric polynomials Tensors • Routines that deal with tensors, their operations, and their use in General Relativity, both in the natural basis and in a moving frame Tolerances** • Perform best- and worst-case computations with quantities involving tolerances, using standard notation • Tolerances understood by arithmetic operators and by many built-in functions, including square roots, exponentials, trigonometric and special functions Units and Dimensions • Support for over 500 units and dimensions defined using exact conversions • Systems of units include Atomic, CGS, Electromagnetic, Electrostatic, FPS, MKS, MTS, and SI • Over 50 base quantities include acceleration, area, dynamic viscosity, electric resistance, energy, heat transfer, light, magnetic flux, and mass • Ability to add and remove systems and dimensions Variational Calculus • Euler-Lagrange equations and first integrals • Solve Jacobi differential equation for conjugate points • Weierstrass excess function Mathematics Education Maple includes educational packages and resources to assist students by furthering their understanding of concepts presented in mathematics courses. Precalculus* • Study lines, polynomials, rational functions, compositions of functions, and more using interactive tutors and visualization routines Calculus • Single step through differentiation, integration, and limit problems • Visualization routines display a function and its derivative, Taylor series approximation, Newton’s method, approximating integrals, antiderivatives, surfaces of revolution, and more • Explore these Calculus 1 concepts using interactive tutors* Linear Algebra • Single step through Gaussian elimination; Gauss- Jordan elimination; and the computation of the inverse, eigenvalues, and eigenvectors of a matrix using interactive tutors* • Visualization routines display the cross product of two vectors, a system of 2-D or 3-D linear equations, the projection of a vector onto a subspace, and more • Over 65 routines for computing with matrices and vectors Multivariate Calculus • Interactive tutors* allow you to work through the standard problems of multivariate calculus in a visually directed manner • Visualization routines to aid in the understanding of concepts including Taylor approximation, change of variables, center of mass, gradient, Jacobian, surface area Vector Calculus** • Computations, visualizations and interactive problem solving involving the calculus of functions from Rn to Rm • Interactive tutors for exploring space curves and vector fields through plots and animations • Supports 5 predefined coordinate systems Dictionary* • Over 5,000 definitions of mathematical and engineering terms and concepts incorporated into the Help system • Includes over 300 figures to clearly explain the concepts Graphing Calculator** • Traditional graphing calculator interface • Standalone or over a MapleNet™ installation Maple T.A. Assessment Tools** • Question authoring environment provides templates for creating common question types, such as fill-in- the-blank, Maple-graded, and multiple-choice questions. • Questions can include algorithmically generated content • Questions can contain plots, hints, feedback and algorithmically-generated variables. • Assignments are used inside Maple, printed, or exported to Maple T.A. Programming Flexible programming language, tools, and base routines. • Advanced programming language • Procedural and functional programming • Operator overloading • Exception handling • Debugging, profiling, and library management tools • Analysis of the code complexity of a Maple procedure and module • Ability to create new worksheets, programs, packages, modules, and help pages • Source code of most routines available for viewing • Assumptions on variables • Create and manipulate many kinds of objects, including sets, strings, lists, arrays*, and libraries • Tools for manipulating mathematical objects, including polynomials**, integrals** and sums* • Extend existing types • Generate and manipulate Maple worksheets through its XML representation • Compiler package for dramatic speed-up of numerical computations** Custom Interfaces Maple lets you create custom user interfaces to access built-in and user-written routines. Maplet Application • Create custom Java™-based graphical user interfaces to access the Maple kernel and libraries, or user-written Maple functions • Graphical elements include text areas, buttons, equation editors, slider bars, tool tips, plotting windows, and numerous built-in dialogs • Can be created programmatically or through the point-and-click Maplet Builder** • Perform calculations and display graphs without seeing the Maple code Context-Sensitive Menu • Design a context menu or edit an existing menu Visualization Maple includes a comprehensive set of visualization tools to make problem exploration easier. • 2-D and 3-D graphs • 2-D and 3-D animations • 2-D and 3-D implicit plots • Conformal mapping • 2-D and 3-D contour plots • Vector fields • 2-D and 3-D complex plots • ODE and PDE plots • Light modeling, legends, axis control*, titles, glossiness**, gridlines**, and transparency • Scale and pan 2-D and 3-D plots and animations • Real-time rotation of 3-D plots • Interactive control of parameters through slider controls • Wide variety of coordinate systems • Layering of graphics and animations of different types • Standard geometric objects, regular solids, and polyhedra, including ellipses, hyperbolae, polygons, cones, spheres, torii, dodecahedra, icosahedra, and tetrahedra • Over 40 visualization routines display concepts presented in calculus, precalculus, multivariate calculus, and linear algebra courses • Plot builder assistant for creating and modifying plots and animations User Interface Maple includes many features to automate tasks and interact with Maple without knowledge of Maple commands and syntax. • Sophisticated 2-D Math editing** • Context menus* • Over 20 palettes** for expressions, symbols, matrices, operators and more • Comprehensive word-processing** for creating professional reports • Spell-checker containing standard math terms • Command completion* • Automatic equation labeling** • Tables**, sketch regions, symbolic spreadsheets, embedded GUI components** • Over 40 interactive tutors* display concepts presented in precalculus, calculus, multivariate calculus, vector calculus**, and linear algebra courses • Over 200 task templates for fill-in-the-blank problem solving** • Task assistants for many tasks including analyzing ODEs and ODE systems, creating plots and matrices, and converting units • Document blocks** to write content with live calculations while hiding computation details • Help Navigator* for searching help pages, tasks, tutors, definitions, and manuals Connectivity Maple adheres to international standards for data communication by enhancing tool interoperability and Web connectivity. • Access Maple algorithms and data structures in compiled C, Java and Visual Basic® programs using OpenMaple™ • MathML 2.0 presentation and content support • Import and export of XML documents • TCP/IP socket connectivity • External calling to Java, C, and Fortran • Code Generation      ° Generate Visual Basic, MATLAB®, Java, C, and Fortran code      ° Override or add to code translations specified in the existing definition or specify an entirely new language definition • Link to MATLAB      ° From within Maple, use MATLAB to compute Cholesky factorization, determinants, dimensions, eigenvalues, LU decompositions      ° Call out to MATLAB to compute and retrieve the results of MATLAB expressions. • Link to Microsoft® Excel 2000, Excel XP, and Excel 2003* on Windows      ° Access the Maple kernel from within Excel      ° Copy and paste between Maple and Excel      ° Function Wizard steps through the creation of a Maple function • Mathematica® Notebook conversion and command translation tools • Export worksheets to HTML, XML, MathML, LaTeX, and RTF • Export plots to BMP, DXF, EPS, GIF, HPGL, JPEG, PCX, POV, TEK, and WMF • Import, manipulate and export WAV, JPEG and TIFF data files** • Import data* from ASCII, CSV, Matrix Market, MATLAB, and more • Direct links to the Maple Application Center™, Maple Student Center™, Maplesoft™ corporate Web site, and more * Enhanced features in Maple 10 **New features in Maple 10