Mathematics | Programming | Visualization | User Interface | Connectivity | Resources

Mathematics
Maple® 8 includes over 3000 computational functions for symbolic and numeric mathematics.

Major topics include:
Algebra

• Exact symbolic arithmetic with real and complex numbers and polynomials
• Factoring, expansion, combination, and simplifica- tions of algebraic expressions and polynomials
• Sequences and series

Calculus and Calculus Education**

• Derivatives, integrals and limits. These can either be computed automatically for applications, or in step-by-step mode for calculus instruction
• Visualization routines for differentiation include animating difference quotients, plotting the derivative, plotting tangent lines, function charts, Taylor series approximation, Rolle's Theorem, the Mean Value Theorem and Newton's method
• Visualization routines for integration include approximating integrals, plotting antiderivatives, and finding average value of a function, volume and surface of revolution, and arc length

Differential Equations

• Exact and numerical solution of ODEs and ODE systems*
• Exact and numerical solution of ODE Initial Value Problems*
• Numerical solution of ODE Boundary Value Problems
• Exact solution of PDEs and PDE systems
• Numerical solutions of PDEs and PDE Boundary Value Problems**
• Differential elimination for ODE and PDE systems
• Structural analysis and order-reduction of ODEs and PDEs

Linear Algebra

• Over 100 functions for constructing, solving, pro- gramming, and querying topics in Linear Algebra
• Construction of Hankel matrices, Hilbert matrices, Identity matrices, Toeplitz matrices, Vandermonde matrices, Bezout matrices, and the Sylvester matrix of two polynomials

Modular linear algebra*

• Exchange data between two mod m matrices and vectors
• Apply forward and backward substitution with a square lower triangular mod m matrix to a mod m matrix and vector
• Compute the determinant of a square mod m matrix
• Gaussian and Gauss-Jordan elimination
  on a matrix
• QR factorization and the Cholesky, PLU, and PLUIR decomposition of a matrix

Vector Calculus**

• Directional derivatives, gradients, Hessian matrices, and Laplacians of a function
• Curl and divergence of a vector field
• Jacobians and Wronskian matrices of a list of functions
• Cross products and dot products of vectors and differential operators

Other topics include:

• Abstract Algebras
• Algebras of Linear Operators
  • Numerous functions to manipulate  d’Alembertian terms
  • Conversions among OrePoly structures, linear ordinary differential equations, linear recurrence equations, and factored OrePoly structures
• Algebraic Curves
  • Holomorphic differentials and genus of an algebraic curve
  • Normal forms for elliptic and hyperelliptic curve
• 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 the system of generating functions equations associated with an attribute grammar
  • Generation of random combinatorial objects and counting the objects of a given size
• Complex Variables
• Curve Fitting
  • B-spline basis functions, polynomial interpolation, least-squares approximation, rational interpolation, and splines*
• Differential Algebra
  • Manipulation and reduction of differential equations, and development of the solutions into formal power series
• Differential Forms
  • Euclidean geometry
  • Close to 300 routines for constructing, computing, plotting, and translating 2-D and 3-D objects
• Financial Mathematics
  • Annuities, cash flows, growing annuities, growing perpetuities, level coupon, and perpetuities
  • Amortization, Black-scholes option pricing, effective rate, future value, present value, and yield to maturity
• Formal Power Series
  • Gaussian Integers
  • Routines include: gcd and lcm of Gaussian integers, Chinese remaindering, Gaussian integer factorization, and extended Euclidean algorithm for Gaussian integers
• Graph Theory
  • Groebner Bases
  • Hilbert dimensions, polynomial and series of an ideal
• Group Theory
  • 34 routines including generation of the elements of a permutation group, order computations and finding a permutation of a group
  • Laplace, inverse Laplace, Fourier sine, Hankel, Hilbert, fast Fourier, inverse Fourier, inverse Mellin, and Z transforms
• Lie Symmetries
  • 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 Programming
  • Linear Recurrence Equations
  • Polynomial, rational and hypergeometric solutions of Linear Recurrence Equations
  • Solutions of divide and conquer recurrence equations
• Logic
  • Numerical Approximations
  • Infinite precision numerical computations
  • Chebyshev-Pade and minimax rational approximation
  • Conversion of a rational function to continued- fraction form and a polynomial 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 fraction
• Orthogonal Polynomials
  • Routines to generate the nth Gegenbauer, Hermite, Laguerre, Legendre, and Jacobi polynomials
• P-adic Numbers
  • Routines for p-adic evaluation, expansion, and functions
  • Computation of the order and the leading coeffi- cient of a p-adic expansion of a rational function
• Rational Generating Functions
  • Rational Normal Forms
  • Computation of the polynomial normal form and the first and second rational canonical forms of a rational function
  • Computation of the first and second minimal representations of a hypergeometric
• Real Domain computations
  • Optional restriction of calculations to the domain of real numbers
• Series Expansions
  • Scientific Constants**
  • Support for 70 scientific constants including the Newtonian constant of gravitation, magnetic flux quantum, conductance quantum, Hartree energy, shielded proton magnetic moment, neutron g factor, and shielded hellion gyromagnetic ratio
  • Properties of all elements and isotopes of the Periodic Table
• Symbolic-Numeric Algorithms for Polynomials**
  • Provides a set of tools for the algebraic manipulation of numerical polynomials
• Statistics
  • Mean, variance, covariance, kurtosis, and many other standard statistical functions
  • Least-squares and least-median-squares linear regression
  • Visualization and transformation features include box plots, histograms, scatter plots, scales, and shifts of the x, y, z coordinates of 2D and 3D plots
• Tensors
  • Routines that deal with tensors, their operations, and their use in General Relativity, both in the natural basis and in a moving frame
• Units and Dimensions*
  • Support for over 500 units and dimensions defined using exact conversions
  • System of units supported are Atomic, CGS, Electromagnetic, Electrostatic, FPS, MKS, MTS, and SI
  • Over 50 base quantities supported, including acceleration, action, area, currency, dynamic viscosity, electric capacity, electric current, electric resistance, energy, heat transfer, length, light, magnetic flux, mass, molar flow, power, radioactivity, and volume
  • Ability to add and remove a system and dimension
• Variational Calculus**
  • Euler-Lagrange equations and first integrals
  • Jacobi differential equation from which conjugate points can be computed
  • Weierstrass excess function

Programming
We give access to the same flexible programming language, tools, and base routines we use ourselves.

Advanced programming language

• Procedural and functional programming
• Operator overloading
• Exception handling
• Debugging and profiling tools
• Analysis of the code complexity of a Maple procedure and module**
• Ability to create new worksheets, programs, packages, modules, and help pages
• Comprehensive word-processing
• Source code of most routines available for viewing
• Assumptions on variables
  
• Restrict computations to real domain
• Many basic tools packages are available for:
  • Definite and Indefinite Sums
  • Differential Equations
  • List manipulation
  • Linear Recurrence Equations
  • MathML
  • Partial Differential Equations
  • Plots
  • Polynomials
  • Randomly generated objects
  • Solving equations
  • Spreadsheets
  • String manipulation
  • XML data manipulation

Visualisation
Maple 8 includes a comprehensive set of visualization tools to make problem exploration easier.

• Predefined plots typically found in a first year calculus course**
• 2-D and 3-D graphs
• 2-D and 3-D animations
• Layering of graphics and animations of different types
• Wide variety of coordinate systems
• Implicit plots in 2-D and 3-D
• Conformal mapping
• Contour plots
• Vector fields
• Complex plots in 2-D and 3-D
• ODE and PDE plots
• Real-time rotation of 3-D plots
• Standard geometric objects, regular solids and polyhedra
• ellipse, hyperbola, polygon, rectangle
• cone, cylinder, hemisphere, sphere, torus
• dodecahedron, hexahedron, icosahedron, octahedron, tetrahedron
• Smart plots in 2-D and 3-D
• Light modeling, legends, styles, axis control, and titles
• Many plot attributes easily set through context-sensitive menus

User Interface
Maple 8 includes many features to automate tasks.

• By using the Maplets™ package, you can create custom Java™-based graphical user interfaces to access the Maple library or user-written Maple functions**
• Perform calculations and display graphs without seeing the Maple code
• Graphical elements include text areas, buttons, equation editors, slider bars, tool tips, and many others
• Numerous built-in dialogs allow the user to select a color, find a file, deliver a message, and pose a question automatically
• Run a maplet by double-clicking the file type .maplet**
• Interactive plot builder**
• Spell check text regions in Maple**
• Context-sensitive menus*
• Palettes for expressions, symbols, matrices, and vectors
• Mathematical expression editor**
• Symbolic spreadsheets
• Command completion
• Structured data browser
• Units converter

Connectivity
Maple 8 adheres to international standards for data communication by enhancing tool interoperability and Web connectivity.

• 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 to Java**, C, and FORTRAN
• Direct links to the Maple Application Center, Maple Student Center, and Maple Corporate Home page**
• E-mail worksheets from Maple**
• Link to MATLAB®
• Compute the Cholesky factorization, determinant, dimensions, eigenvalues, LU decomposition, QR orthogonal-triangular decomposition, size and transpose of a Maple Matrix and a MATLAB Matrix
• Compute the discrete Fourier transform of a vector
• Link to Microsoft® Excel 2000
• Access the Maple kernel from within Excel
• Copy and paste between Maple and Excel
• Access a subset of the mathematics section of Maple online help
• Maple Function Wizard steps through the creation of a Maple function
• Export worksheets to HTML, XML**, MathML, LaTeX, RTF
• Export plots to BMP, CHAR, CPS, DXF, GIF, HPGL, JPEG, PCX, EPS, POV, TEK, WMF, X11
• Import data: ASCII, Matrix Market, MATLAB

Resources
Maple 8 includes access to a large and expanding collection of free supplementary material:The Maple Application Center www.mapleapps.com

• Over 1,000 user-contributed applications from over 100 categories, including Fluid Dynamics, Abstract Algebra, Financial Engineering, Physics Education, and many more Maple® PowerTools™
• PowerTool course materials: Calculus I, II and III, Differential Equations, Complex Analysis, Maple Programming, Linear Algebra, Engineering Math, Dynamics, Maple Tutorials, Diagram Creation
• PowerTool add-on packages: Java View Library, Finite Element Modeling, Nonlinear Programming, Electronic Circuit Analysis, Multibody Dynamics, Vector CalculusThe Maple Student Center
• Maple Student Center provides tutorials in mathematics and Maple to assist studentsMaplePrimes™
• MaplePrimes is a controlled-access Web site from Waterloo Maple that provides exclusive content to customers and end-users who subscribe to the Maple Extended Maintenance Plan (EMP).
   
  Not all features are available on all platforms
  ** New functionality in Maple 8.
  * Enhanced functionality in Maple 8

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