# J. Browne: Updated free book and Mathematica application for Grassmann algebra

Updated version of free draft book and Mathematica application for doing Grassmann algebra

Book “Grassmann Algebra: Exploring extended vector algebra with Mathematica” (Mathematica and .pdf formats)
Application “GrassmannAlgebra” (requires Mathematica)

The book chapters: 1. Introduction || 2. The Exterior Product || 3. The Regressive Product || 4. Geometric Interpretations || 5. The Complement || 6. The Interior Product || 7. Exploring Screw Algebra || 8. Exploring Mechanics || 9. Exploring Grassmann Algebra || 10. Exploring the Generalized Grassmann Product || 11. Exploring Hypercomplex Algebra || 12. Exploring Clifford Algebra || 13. Exploring Grassmann Matrix Algebra || A Guide to GrassmannAlgebra || A Brief Biography of Grassmann || Notation || Glossary || Bibliography || Index

Some of the things you can do with the application:

Preferences

• Set up your own space of any dimension and metric. The default is a three-dimensional Euclidean space.
• Work basis-free or with a basis as appropriate.
• Work metric-free or with a metric as appropriate.
• Declare your own scalar symbols: symbols or symbol patterns you want specially interpreted as scalars.
• Declare your own vector symbols: symbols or symbol patterns you want specially interpreted as vectors.
• Distinguish between points and vectors for an easy approach to projective space.

Operations

• Work in metric or metric-free spaces with the exterior or regressive products.
• Work with the complement operation, Grassmann’s version of the Hodge Star.
• Work in a metric space with the interior product, a generalization of the inner and scalar products.
• Apply higher order products (Generalized Grassmann, Hypercomplex, and Clifford) defined in terms of the exterior, regressive and interior products.
• Manipulate Grassmann expressions constructed from sums or products of symbols.
• Manipulate lists and matrices of Grassmann expressions (where applicable) as easily as single expressions.

Expression Composition

• Compose bases and cobases for the algebra or any of its graded linear spaces.
• Compose metrics for any of the graded linear spaces.
• Create palettes of the bases, cobases or metrics for any of the graded linear spaces.
• Compose elements of the algebra or any of its graded linear spaces.
• Attach a grade to a symbol by using an underscript.
• Compose complex Grassmann expressions with a minimal number of keystrokes.

Expression Analysis

• Query the attributes of any expression. For example, is it: a Grassmann expression? a scalar? a Grassmann variable? a basis element? a metric element? of grade m? of even grade? of odd grade? an interior product? an inner product? a scalar product? factorizable?
• Determine the grade of any Grassmann expression.
• Extract components of different types from Grassmann expressions.

Expression Transformation

• Expand Grassmann expressions containing products of sums.
• Simplify Grassmann expressions using a recursive multi-rule process tailored to the dimension of the space.
• Use a Grassmann rule database for simplifying or transforming your own expressions.
• Convert Grassmann expressions from one form to another. For example you can convert:
• complements of elements according to the declared metric
• regressive products to congruent exterior products
• Clifford and hypercomplex products to sums of generalized Grassmann products
• generalized Grassmann products to sums involving exterior and interior products
• interior product into sums involving exterior and inner products
• inner products into sums involving scalar products
• regressive, interior, generalized, hypercomplex and Clifford products into sums involving exterior and scalar products.

NOTE carefully that the Mathematica application is currently valid for computations in book chapters 1 to 6 only.

Source: Email by John Browne (john.browne_at_alumni.unimelb.edu.au) of 2010/05/11 14:53