The new DifferentialGeometry package provides an integrated approach for differential geometry and its applications, based upon the modern differential geometry formalism (the so-called index free approach). Special care has been taken in providing an interface consistent with standard textbooks and references.

DifferentialGeometry includes a suite of commands and twelve application subpackages featuring a collection of tightly integrated tools for computations with vector fields, differential forms, and transformations; tensor analysis; calculus on jet spaces; Lie algebras; and Lie and transformation groups. The software includes a variety of homotopy operators for the de Rham and variational bicomplexes; programs for the decomposition of Lie algebras, and programs for construction of a solvable Lie group from its Lie algebra. Computations can also be performed in user specified frames.

DifferentialGeometry is added to the library together with a course on DifferentialGeometry and its applications, presented as worksheets, with the topics developed using the packages' commands, and with a searchable table of Lie algebras, Lie algebras of vectors, and differential equations taken from the mathematics and mathematical physics literature.

The new GraphTheory package, intended for both research and teaching, provides a comprehensive set of tools for working with graphs.

Both directed and undirected graphs are supported, as are graphs with edge weights.

Sophisticated algorithms for visualizing graphs in both two and three dimensions have been developed and implemented in this package. See GraphTheory[DrawGraph] and the demonstration worksheet examples,GraphTheory.

Graphs may be exported to, and imported from, a variety of graph data file formats. See GraphTheory[ExportGraph] and GraphTheory[ImportGraph].

A large number of special graph constructors allow you to easily create examples of graphs with interesting properties.

Examples

with( GraphTheory ):

with( SpecialGraphs ):

G := DodecahedronGraph();

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 20\; vertices\; and\; 30\; edge(s)}$

IsHamiltonian( G );

$\mathrm{true}$

IsPlanar( G );

$\mathrm{true}$

The new Physics package implements computational representations and related operations for most of the objects used in mathematical physics computations, for example, the spacetime metrics g_[mu, nu], the Kronecker and Levi-Civita symmetric and antisymmetric symbols delta[mu, nu] and epsilon[mu, nu, ...], the Pauli and Dirac matrices sigma[mu] and gamma[mu], the differentiation operators d_[mu] and d'Alembertian, a quantum state space with Bras, Kets and quantum operators, etc.

With Physics, the standard operations and computational domain (input/output) are also extended with representations for anticommutative and noncommutative variables and functions, tensor indices of spacetime, spinor and/or gauge types, differentiation with respect to anticommutative variables, functional differentiation, and differentiation and simplification of tensorial expressions using the Einstein sum rule for repeated indices. You can take advantage of the computational power of the Maple environment without having to change the flexible notation used when computing with paper and pencil.

Physics includes a subpackage, Vectors, to perform abstract vector calculus, implementing representations for non project 3-D vectors, including inert and active representations for the non projected differential operators nabla, gradient, divergent, curl and the Laplacian, as well as algebraic (non-matricial) representations for projected 3-D vectors in the cartesian, cylindrical and spherical vector basis. It is then possible to compute using coordinates free vectorial formulations, exploring the coordinates free properties of the vectors and vectorial operations involved, that is to work with vectorial expressions involving both nonprojected and projected vectors, using essentially the same notation you use when computing by hand.

The LinearAlgebra package includes a new subpackage, LinearAlgebra[Generic], for computing with generic implementations of algorithms for linear algebra over fields, Euclidean domains, integral domains, and rings.

The PDEtools package includes a new subpackage for computing with symmetries for PDE systems.

Note: This package is only available when running a multi-threaded engine. For more information on the multi-threaded engine see the Multithreaded help page.

The Threads package provides user level routines for multi-threaded programming. Multi-threaded routines allow the Maple Engine to utilize more than one CPU. This will allow for higher performance on machines with multiple CPUs. For a detailed overview, see the Threads package overview page.

The Threads package provides the Create function that allows users to execute an expression in a new thread. By passing a function as the expression to be executed in the new thread, a user can create complex multi-threaded programs.

The Threads package also provides mutexes. A mutex is a programming object that can be used to control access to data structures when multiple threads are running.

Another programming object provided by the Threads package is the Condition Variable. Condition variables allow for threads to synchronize their execution.