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Differential Geometry

 

For Maple 17, the DifferentialGeometry package contains over 40 new commands or commands with enhanced functionality or improved efficiency.

 

Updates to the DifferentialGeometry package:

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DGsetup has a new keyword argument for declaring certain coordinates to be complex.

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The new command DGconjugate calculates the complex conjugate of a vector, tensor or differential form, quaternion or octonion.

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The new commands DGRe and DGIm calculate the real and imaginary parts of a vector, tensor, differential form, quaternion or octonion.

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The new commands DGsolve, DGNullSpace and DGImageSpace provide useful utilities for solving equations whose unknowns are tensors or differential forms.

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The new command `&algmult` computes non-commutative multiplication in general algebras.

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A new geometric type, "multivector", has been created. This significantly improves computational efficiency when working with contravariant skew-symmetric tensors. The commands convert, Hook, RaiseLowerIndices, and LieDerivative have been extended to accept multi-vector arguments.

 

ExteriorDifferentialSystems

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This is a new package for the study of differential systems.

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The first release of this package contains 5 commands. These commands calculate the Cauchy characteristics, derived flags, first integrals, integral manifolds, and infinitesimal symmetries of Pfaffian differential systems.

 

Library

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Over 100 new metrics have been added to the database of solutions to the Einstein equations.

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A command line version of the MetricSearch maplet is now available.

 

LieAlgebras

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 The Lie algebra package is being extended to work with more general algebras. The new AlgebraData and AlgebraLibraryData commands creates the multiplication structures for quaternions, octonions, Jordan algebras and Clifford algebras.The commands AlgebraNorm and AlgebraInverse calculate the norms and inverses of quaternions and octonions.

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The command LieAlgebraData has been re-written to give massive improvements in efficiency for calculating the structure equations for matrix algebras. The command also has a new calling sequence for working with graded Lie algebras.

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The functionality of the SimpleLieAlgebraData command has been extended to include the exceptional Lie algebras g2 and f4.

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A command for finding the split and compact forms of a complex semi-simple Lie algebra is now available.

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The functionality of the CartanDecomposition and CartanInvolution commands has been extended to abstract semi-simple Lie algebras.

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The new commandChevalleyBasis calculates the Chevalley basis of a semi-simple Lie algebra.

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The Derivations command has been rewritten to give significant improvements in efficiency. The command can now also compute the derivations of a general algebra.

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New functionality has been added for creating representations of Lie algebras. The commands Adjoint and StandardRepresentation now accept a new keyword argument "representationspace", which can be used to create representations.  The new command RestrictedRepresentation calculates the restriction of a representation to a subalgebra. The names of the two commands ChangeBasis and TensorProduct have been changed to ChangeRepresentationBasis and TensorProductOfRepresentations.

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 The Codifferential command has been added to calculat