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:
- 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.
- 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.
- 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.
- 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.
- 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