Overview of the DifferentialGeometry Package - Maple Programming Help

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Overview of the DifferentialGeometry Package

 

Description

List of the DifferentialGeometry commands and sub-packages

Description

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The DifferentialGeometry package is a comprehensive suite of commands and subpackages featuring a collection of tightly integrated tools for computations in the areas of: calculus on manifolds (vector fields, differential forms and transformations); tensor analysis; calculus on jet spaces; Lie algebras and Lie and transformation groups.

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The software includes a variety of homotopy operators for the deRham and variational bicomplexes; extensive capabilities for computing with the Newman-Penrose and spinor formalisms in general relativity; programs for analyzing the structure of general and semi-simple Lie algebras; programs for finding symmetries of tensor fields and other geometric structures; and programs for construction of a solvable Lie group from its Lie algebra.

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Computations may be performed in user specified frames. One can also compute with abstract differential forms, that is, with differential forms and their structure equations defined without reference to any underlying system of coordinates.

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Also included are extensive tables of Lie algebras, Lie algebras of vectors, differential equations, and space-time metrics taken from the mathematics and mathematical physics literature.

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The DifferentialGeometry package includes a comprehensive collection of Lessons and Tutorials.  The lessons worksheets provide a systematic approach to learning the commands in the DifferentialGeometry, Tensor, LieAlgebras and JetCalculus sub-packages.  Each lesson contains a set of exercises which range in difficult from simple computational exercises to programming exercises.  Solutions are given. The tutorials present specialized applications of the DifferentialGeometry package.

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The DifferentialGeometry package is based upon the Vessiot package developed by I. M. Anderson, Florin Catrina, Sydney Chamberlain, Cinnamon Hillyard, Jeff Humphries, Jamie Jorgensen, Charles Miller, and Charles Torre at Utah State University. The redesign and expansion of Vessiot to DifferentialGeometry for Maple 11 was done by I. M. Anderson and E. S. Cheb-Terrab.

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Each command in the DifferentialGeometry package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

List of the DifferentialGeometry commands and sub-packages

The following is a list of available commands and sub-packages.

&algmult

&minus

&mult

&plus

&tensor

&wedge

Annihilator

ApplyTransformation

ChangeFrame

ComplementaryBasis

ComposeTransformations

Convert

DGIm

DGImageSpace

DGNullSpace

DGRe

DGbasis

DGconjugate

DGsetup

DGsolve

DGzip

DeRhamHomotopy

DualBasis

ExteriorDerivative

ExteriorDifferentialSystems

Flow

FrameData

GetComponents

GroupActions

Hook

InfinitesimalTransformation

IntegrateForm

IntersectSubspaces

InverseTransformation

JetCalculus

Library

LieAlgebras

LieBracket

LieDerivative

Preferences

Pullback

PullbackVector

Pushforward

RemoveFrame

Tensor

Tools

Transformation

evalDG

A brief description of the package's commands is as follows.

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To display the help page for a particular DifferentialGeometry command, see Getting Help with a Command in a Package.

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&minus: find the difference between two vectors, differential forms or tensors.

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&mult: multiply a vector, differential form or tensor by a Maple expression.

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&plus: add two vectors, differential forms or tensors.

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&tensor: calculate the tensor product of two tensors.

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&wedge: calculate the exterior product of two differential forms.

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Annihilator: find the subspace of vectors (or 1-forms) whose interior product with a given list of 1-forms (or vectors) vanish.

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ApplyTransformation: evaluate a transformation at a point.

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ChangeFrame: change the current or active frame.

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ComplementaryBasis: extend a basis for subspace to a basis for larger subspace.

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ComposeTransformations: compose a sequence of two or more transformations.

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Convert: change the presentations or internal representations of various geometric objects.

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DGIm: find the the imaginary part of a vector, a tensor or differential form; find the imaginary part of a quaternion or octonion

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DGImageSpace: find the image space of a linear transformation acting on a vector space of vectors, differential forms, tensors

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DGNullSpace: find the null space of a linear transformation acting on a vector space of vectors, differential forms, tensors

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DGRe: find the the real part of a vector, tensor or differential form; find the real part of a quaternion or octonion

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DGbasis: select a maximal linearly independent list of elements from a list of vectors, forms or tensors.

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DGconjugate: find the the complex conjugate of a vector, tensor or differential form; find the conjugate of a quaternion or octonion

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DGsetup: initialize a coordinate system, frame, or Lie algebra.

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DGsolve: solve a list of tensor equations for an unknown list of tensors

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DGzip: form a linear combination, wedge product or tensor product of a list of vectors, forms or tensors.

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DeRhamHomotopy: the homotopy operator for the exterior derivative operator (the de Rham complex).

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DualBasis: calculate the dual basis to a given basis of vectors or 1-forms.

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evalDG: evaluate a DifferentialGeometry expression.

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ExteriorDerivative: take the exterior derivative of a differential form.

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Flow: calculate the 1-parameter group of diffeomorphisms (the flow) of a vector field.

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FrameData: calculate the structure equations for a generic (anholonomic) frame.

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GroupActions: a package for Lie groups and group actions on manifolds.

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Hook: the interior product of a vector or a list of vectors with a differential form.

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InfinitesimalTransformation: compute the Lie algebra of infinitesimal generators for an action of a Lie group on a manifold.

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IntegrateForm: evaluate a p-fold iterated integral of a differential p-form.

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IntersectSubspaces: find the intersection of a list of vector subspaces of vectors, forms or tensors.

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InverseTransformation: find the inverse of a transformation.

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JetCalculus: a package for the variational calculus on jet spaces.

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Library: a package of databases of Lie algebras, vector field systems, differential equations, and exact solutions in general relativity.

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LieAlgebras: a package for the symbolic analysis of Lie algebras.

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LieBracket: calculate the Lie bracket of two vector fields.

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LieDerivative: calculate the Lie derivative of a vector field, differential form or tensor with respect to a vector field.

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GetComponents: find the coefficients of a vector, differential form or tensor with respect to a list of vectors, differential forms or tensors.

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Preferences: set worksheet preferences for the DifferentialGeometry package.

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Pullback: pullback a differential p-form by the Jacobian of a transformation.

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PullbackVector: find (if possible) a vector field whose pushforward by the Jacobian of a given transformation is a given vector field.

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Pushforward: pushforward a vector or a vector field by the Jacobian of a transformation.

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RemoveFrame: remove a frame from a Maple session.

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Tensor: a package for tensor analysis within the DifferentialGeometry environment.

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Tools: a small utility package for DifferentialGeometry.

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Transformation: create a transformation or mapping from one manifold to another.

See Also

DifferentialGeometry

JetCalculus

Library

LieAlgebras

Tensor

Tools