Maple 11 Features

Mathematics

Maple includes over 4000 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 MapletT 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*
Differential elimination for ODE and PDE systems
Structural analysis and order-reduction of ODEs and PDEs
Interactive MapletT Assistant for numeric and symbolic solutions of ODEs and ODE systems*

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

Differential Geometry*

Vector fields, differential forms & transformations; tensor analysis; calculus on jet spaces; Lie algebras and Lie groups, and Transformation groups

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*

Directed graphs, undirected graphs, and edge weights*
More then 150 functions and 35 pre defined special graphs **

Groebner Bases* and Polynomial Ideals*

Groebner bases over commutative and non-commutative domains. Incorporates the FGb library, the worlds fastest Groebner Bases engine** 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 & Discrete Transforms

Laplace, inverse Laplace, Fourier sine, Hankel, Hilbert, inverse Mellin, and Z transforms and their inverses

Lie Symmetries

Lie group symmetry methods for differential forms, ODEs and PDEs **

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

Physics**

Includes 3D non-projected vectors, non cummulative variables, spacetime tensors including the indexed differential operator, Einstein's summation convention for repeated indices, Pauli and Dirac matrices, Dirac notation for bra-ket quantum state vectors, and other mathematical physics objects
Supports the use of standard notation and allows you to define framework of notational conventions and spacetime properties.

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 MapleNetT 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
User level routines from multi-threaded programming on multi-code compueters**
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.

Embedded Components*

Components include buttons, sliders, plots, check boxes, list boxes, toggle buttons, and mathematical expressions boxes for entering and displaying 2-D math. Add components to your documents using the components palette
Easily programmable to accept input, display results and plots, and interact with other elements of the document

Maplet Application

Create custom JavaT-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 tetrahedral
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 OpenMapleTM
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
Link to Microsoft® Excel 2000, Excel XP, and Excel 2003* on Windows
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 CenterT, Maple Student CenterT, MaplesoftT corporate Web site, and more

* Enhanced features in Maple 11
**New features in Maple 11

Maple 11 Features

Related Information

Case Studies

Technical Support

Our Alliance