Grassmann Algebra

Exploring applications of extended vector algebra with Mathematica

Incomplete Book Draft

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The new address of this page (February 2007) is http://grassmannalgebra.info/grassmannalgebra/book. If you used an old link you will have been automatically re-directed. Please update any old links, as they will soon be dead.

The Grassmann Algebra Book

File format This page enables you to read the current draft text of the Grassmann Algebra book by clicking on the chapters below. The chapters are in Portable Document Format (.pdf), and provided your browser is enabled with Adobe Acrobat Reader or your computer is running Mac OS X, you should be able to read them on screen and print them if you wish. Please note that several sections of the text and the Mathematica based software package which accompanies the book are still to be completed. These files were produced with Mathematica version 4.

Copyright agreement All the chapters you see available here are of course protected by copyright. It is a condition of your accessing the files that you agree to use them for your own individual private study only. They may not otherwise be copied or distributed in any way in whole or in part without the permission of the author. You may reference or quote small sections of the work as long as due acknowledgment is made.

Precedence claims The publication of this draft on this website on this date of October 25, 2001 establishes claim to precedence for any new or original previously unpublished results it may contain. Any reference to elements of this work should follow standard journal publication practice.

If you wish to contact me, please email me using the address at the bottom of the page.

Title Page [36 KB]

Table of Contents  [85 KB]

Preface  [91 KB]

Chapter 1: Introduction  [182 KB]
Background. The exterior product. The regressive product. Geometric interpretations. The complement. The interior product. Exploring screw algebra. Exploring mechanics. Exploring Grassmann algebra. Exploring the generalized product. Exploring hypercomplex numbers. Exploring Clifford algebra. Exploring Grassmann matrix algebra.

Chapter 2: The Exterior Product  [276 KB]
Introduction. The exterior product Exterior linear spaces. Axioms for exterior linear spaces. Bases. Cobases. Determinants. Cofactors. Solutions of linear equations. Simplicity. Exterior division.

Chapter 3: The Regressive Product  [310 KB]
Introduction. Duality. Properties of the regressive product. The duality principle. The common factor axiom. The common factor theorem. The regressive product of simple elements. Factorization of simple elements. Product formulae for regressive products.

Chapter 4: Geometric Interpretations  [244 KB]
Introduction. Geometrically interpreted 1-elements. Geometrically interpreted 2-elements. Geometrically interpreted m-elements. Decomposition into components. Geometrically interpreted spaces. m-planes. Line coordinates. Plane coordinates. Calculation of intersections.

Chapter 5: The Complement  [282 KB]
Introduction. Axioms for the complement. Defining the complement. The Euclidean complement. Complementary interlude. The complement of a complement. Working with metrics. Calculating complements. Geometric interpretations. Complements in a vector subspace of a multiplane. Reciprocal bases. Summary.

Chapter 6: The Interior Product  [286 KB]
Introduction. Defining the interior product. Properties of the interior product. The interior common factor theorem. The inner product. The measure of an m-element. The induced metric tensor. Product formulae for interior products. The cross product. The triangle formulae. Angle. Projection. Interior products of interpreted elements. The closest approach of multiplanes.

Chapter 7: Exploring Screw Algebra  [128 KB]
Introduction. A canonical form for a 2-entity. The complement of a 2-entity. The screw. The algebra of screws. Computing with screws.

Chapter 8: Exploring Mechanics  [110 KB]
Introduction. Force. Momentum. Newton's law. The angular velocity of a rigid body. The momentum of a rigid body. The velocity of a rigid body. The complementary velocity of a rigid body. The infinitesimal displacement of a rigid body. Work, power and kinetic energy.

Chapter 9: Exploring Grassmann Algebra  [307 KB]
Introduction. Grassmann numbers. Operations with Grassmann numbers. Simplifying Grassmann numbers. Powers of Grassmann numbers. Solving equations. Exterior division. Factorization of Grassmann numbers. Functions of Grassmann numbers.

Chapter 10: Exploring the Generalized Product  [192 KB]
Introduction. Geometrically interpreted 1-elements. The symmetric form of the generalized product. Calculating with generalized products. The generalized product theorem. The zero interior sum theorem. The zero generalized sum. Nilpotent generalized products. Properties of the generalized product. The triple generalized sum conjecture. Exploring conjectures. The generalized product of intersecting elements. The generalized product of orthogonal elements. The generalized product of intersecting orthogonal elements. Generalized products in lower dimensional spaces. Generalized products in 3-space.

Chapter 11: Exploring Hypercomplex Algebra  [98 KB]
Introduction. Some initial definitions and properties. The complex numbers. The hypercomplex product in a 2-space. The quaternions. The norm of a Grassmann number. The product of two different elements of the same grade. Octonions.

Chapter 12: Exploring Clifford Algebra  [293 KB]
Introduction. The Clifford product. The reverse of an exterior product. Special cases of Clifford products. Alternate forms for the Clifford product. Writing down a general Clifford product. The Clifford product of intersecting elements. The Clifford product of orthogonal elements. The Clifford product of intersecting orthogonal elements. Summary of special cases of Clifford products. Associativity of the Clifford product. Clifford algebra. Clifford algebras of a 2-space. Clifford algebras of a 3-space. Clifford algebras of a 4-space. Rotations.

Chapter 13: Exploring Grassmann Matrix Algebra  [143 KB]
Introduction. Creating matrices. Manipulating matrices. Matrix algebra. Transpose and determinant. Matrix powers. Matrix inverses. Matrix equations. Matrix eigensystems. Matrix functions. Supermatrices.

Appendix 1: A Guide to the GrassmannAlgebra Software

Appendix 2: A Brief Biography of Hermann Grassmann  [20 KB]

Appendix 3: Notation  [17 KB]

Appendix 4: Glossary  [40 KB]

Appendix 5: Bibliography  [14 KB]

The GrassmannAlgebra Software

When the book is published it will include a Mathematica package called GrassmannAlgebra, for doing a range of manipulations on numeric or symbolic Grassmann expressions in spaces of any dimension and metric.

If you wish to be advised when the software becomes available, or you are interested in beta-testing it, please email me using the address at the bottom of the page. You can get some idea of its current capabilities from the book and the summary below. It would be helpful in finalizing these capabilities if you could also let me know what types of computations you would like the software to do.

Some of the things you can do with the GrassmannAlgebra software. You can:

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.
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.
Apply Grassmann operations. A Grassmann operation is any of: the complement operation and the six product operations: the exterior, regressive, interior, generalized Grassmann, hypercomplex and Clifford products.
Manipulate Grassmann expressions and numbers. A Grassmann expression is either a scalar, a Grassmann variable, or the result of a sequence of Grassmann operations or sums on Grassmann expressions. A Grassmann number is a Grassmann expression expressed as a linear combination of basis elements.
Compute the grade of any Grassmann expression. For example the grade of an expression which reduces to the sum of a scalar, vector and bivector would be computed as {0, 1, 2}.
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? the soul of a Grassmann expression? an interior product? an inner product? a scalar product?
Extract components of different types from Grassmann expressions. For example extract: scalars, basis elements, metric elements, Grassmann variables, elements of even grade, elements of odd grade, elements of grade m, the body, the soul.
Convert Grassmann expressions from one form to another. For example you can:
- Expand products of sums
- Factor out any scalar factors
- Factorize an expression into a desired form (if possible)
- Reorder factors into a standard order
- Put any manifestly zero elements to zero
- Put any expressions of grade greater than the dimension of the declared space to zero.
- Convert complements of elements according to the declared metric
- Convert regressive products to congruent exterior products
- Convert Clifford and hypercomplex products to sums of generalized Grassmann products
- Convert generalized Grassmann products to sums involving exterior and interior products
- Convert interior product into sums involving exterior and inner products
- Convert inner products into sums involving scalar products
- Convert regressive, interior, generalized, hypercomplex and Clifford products into sums involving exterior and scalar products.
Simplify Grassmann expressions using multi-purpose simplification functions.
Create Grassmann expressions of any complexity with a minimal number of keystrokes. For example, a matrix of different symbolic Grassmann numbers in a 4-space, or a list of expressions whose parameters are represented by placeholders, ready for tabbed entry of their values.
Create tables and palettes of the basis, cobasis and metric elements of a Grassmann algebra.
Compute functions of a Grassmann number, where the product operation is taken to be the exterior product, and the function is sufficiently well-behaved.
Compute functions of several Grassmann numbers (with caveats).
Perform exterior division of one Grassmann number by another yielding a space of solutions.
Solve systems of equations amongst Grassmann numbers in certain cases.
Perform Grassmann matrix algebra with the elements being Grassmann expressions and the underlying (element by element) product operation being any of the Grassmann product operations.
Compute functions of a matrix of Grassmann numbers with distinct eigenvalues (with caveats).
Manipulate lists and matrices of Grassmann expressions (where applicable) as easily as single expressions.

Contact Details

John M Browne
http://grassmannalgebra.info
To avoid spam spiders stealing my email address, please construct it from "jbrowne at grassmannalgebra.info".