Table of Contents1 Introduction1.1 Background The mathematical representation of physical entities The central concept of the Ausdehnungslehre Comparison with the vector and tensor algebras Algebraicizing the notion of linear dependence Grassmann algebra as a geometric calculus 1.2 The Exterior Product The anti-symmetry of the exterior product Exterior products of vectors in a three-dimensional space Terminology: elements and entities The grade of an element Interchanging the order of the factors in an exterior product A brief summary of the properties of the exterior product 1.3 The Regressive Product The regressive product as a dual product to the exterior product Unions and intersections of spaces A brief summary of the properties of the regressive product The Common Factor Axiom The intersection of two bivectors in a three dimensional space 1.4 Geometric Interpretations Points and vectors Sums and differences of points Determining a mass-centre Lines and planes The intersection of two lines 1.5 The Complement The complement as a correspondence between spaces The Euclidean complement The complement of a complement The Complement Axiom 1.6 The Interior Product The definition of the interior product Inner products and scalar products Sequential interior products Orthogonality Measure and magnitude Calculating interior products Expanding interior products The interior product of a bivector and a vector 2009 9 4 The cross product

Sequential interior products Orthogonality Measure and magnitude Calculating interior products Expanding interior products The interior product of a bivector and a vector The cross product 1.7 Exploring Screw Algebra To be completed 1.8 Exploring Mechanics To be completed 1.9 Exploring Grassmann Algebras To be completed 1.10 Exploring the Generalized Product To be completed 1.11 Exploring Hypercomplex Algebras To be completed 1.12 Exploring Clifford Algebras To be completed 1.13 Exploring Grassmann Matrix Algebras To be completed 1.14 The Various Types of Linear Product Introduction Case 1 The product of an element with itself is zero Case 2 The product of an element with itself is non-zero Examples 1.15 Terminology Terminology 1.16 Summary To be completed

2 The Exterior Product2.1 Introduction 2.2 The Exterior Product Basic properties of the exterior product Declaring scalar and vector symbols in GrassmannAlgebra Entering exterior products 2.3 Exterior Linear Spaces Composing m-elements Composing elements automatically Spaces and congruence The associativity of the exterior product Transforming exterior products 2009 9 4

2.3 Exterior Linear Spaces Composing m-elements Composing elements automatically Spaces and congruence The associativity of the exterior product Transforming exterior products 2.4 Axioms for Exterior Linear Spaces Summary of axioms Grassmann algebras On the nature of scalar multiplication Factoring scalars Grassmann expressions Calculating the grade of a Grassmann expression 2.5 Bases Bases for exterior linear spaces Declaring a basis in GrassmannAlgebra Composing bases of exterior linear spaces Composing palettes of basis elements Standard ordering Indexing basis elements of exterior linear spaces 2.6 Cobases Definition of a cobasis The cobasis of unity Composing palettes of cobasis elements The cobasis of a cobasis 2.7 Determinants Determinants from exterior products Properties of determinants The Laplace expansion technique Calculating determinants 2.8 Cofactors Cofactors from exterior products The Laplace expansion in cofactor form Exploring the calculation of determinants using minors and cofactors Transformations of cobases Exploring transformations of cobases 2.9 Solution of Linear Equations Grassmann's approach to solving linear equations Example solution: 3 equations in 4 unknowns Example solution: 4 equations in 4 unknowns 2.10 Simplicity The concept of simplicity All (n|1)-elements are simple Conditions for simplicity of a 2-element in a 4-space Conditions for simplicity of a 2-element in a 5-space 2.11 Exterior division The definition of an exterior quotient Division by a 1-element Division by a k-element Automating the division process 2009 9 4

2.11 Exterior division The definition of an exterior quotient Division by a 1-element Division by a k-element Automating the division process 2.12 Multilinear forms The span of a simple element Composing spans Example: Refactorizations Multilinear forms Defining m:k-forms Composing m:k-forms Expanding and simplifying m:k-forms Developing invariant forms Properties of m:k-forms The complete span of a simple element 2.13 Unions and intersections Union and intersection as a multilinear form Where the intersection is evident Where the intersections is not evident Intersection with a non-simple element Factorizing simple elements 2.14 Summary

3 The Regressive Product3.1 Introduction 3.2 Duality The notion of duality Examples: Obtaining the dual of an axiom Summary: The duality transformation algorithm 3.3 Properties of the Regressive Product Axioms for the regressive product The unit n-element The inverse of an n-element Grassmann's notation for the regressive product 3.4 The Duality Principle The dual of a dual The Grassmann Duality Principle Using the GrassmannAlgebra function Dual 3.5 The Common Factor Axiom Motivation The Common Factor Axiom Extension of the Common Factor Axiom to general elements Special cases of the Common Factor Axiom Dual versions of the Common Factor Axiom Application of the Common Factor Axiom When the common factor is not simple 2009 9 4

The Common Factor Axiom Extension of the Common Factor Axiom to general elements Special cases of the Common Factor Axiom Dual versions of the Common Factor Axiom Application of the Common Factor Axiom When the common factor is not simple 3.6 The Common Factor Theorem Development of the Common Factor Theorem Proof of the Common Factor Theorem The A and B forms of the Common Factor Theorem Example: The decomposition of a 1-element Example: Applying the Common Factor Theorem Automating the application of the Common Factor Theorem 3.7 The Regressive Product of Simple Elements The regressive product of simple elements The regressive product of (n|1)-elements Regressive products leading to scalar results Expressing an element in terms of another basis Exploration: The cobasis form of the Common Factor Axiom Exploration: The regressive product of cobasis elements 3.8 Factorization of Simple Elements Factorization using the regressive product Factorizing elements expressed in terms of basis elements The factorization algorithm Factorization of (n|1)-elements Factorizing simple m-elements Factorizing contingently simple m-elements Determining if an element is simple 3.9 Product Formulas for Regressive Products The Product Formula Deriving Product Formulas Deriving Product Formulas automatically Computing the General Product Formula Comparing the two forms of the Product Formula The invariance of the General Product Formula Alternative forms for the General Product Formula The Decomposition Formula Exploration: Dual forms of the General Product Formulas 3.10 Summary

4 Geometric Interpretations4.1 Introduction 4.2 Geometrically Interpreted 1-elements Vectors Points Declaring a basis for a bound vector space Composing vectors and points Example: Calculation of the centre of mass 2009 9 4

Vectors Points Declaring a basis for a bound vector space Composing vectors and points Example: Calculation of the centre of mass 4.3 Geometrically Interpreted 2-elements Simple geometrically interpreted 2-elements Bivectors Bound vectors Composing bivectors and bound vectors The sum of two parallel bound vectors The sum of two non-parallel bound vectors Sums of bound vectors Example: Reducing a sum of bound vectors 4.4 Geometrically Interpreted m-Elements Types of geometrically interpreted m-elements The m-vector The bound m-vector Bound simple m-vectors expressed by points Bound simple bivectors Composing m-vectors and bound m-vectors 4.5 Geometrically Interpreted Spaces Vector and point spaces Coordinate spaces Geometric dependence Geometric duality 4.6 m-planes m-planes defined by points m-planes defined by m-vectors m-planes as exterior quotients Computing exterior quotients The m-vector of a bound m-vector 4.7 Line Coordinates Lines in a plane Lines in a 3-plane Lines in a 4-plane Lines in an m-plane 4.8 Plane Coordinates Planes in a 3-plane Planes in a 4-plane Planes in an m-plane The coordinates of geometric entities 4.9 Calculation of Intersections The intersection of two lines in a plane The intersection of a line and a plane in a 3-plane The intersection of two planes in a 3-plane Example: The osculating plane to a curve 4.10 Decomposition into Components The shadow Decomposition in a 2-space Decomposition in a 3-space 2009 9 4 Decomposition in a 4-space Decomposition of a point or vector in an n-space

4.10 Decomposition into Components The shadow Decomposition in a 2-space Decomposition in a 3-space Decomposition in a 4-space Decomposition of a point or vector in an n-space 4.11 Projective Space The intersection of two lines in a plane The line at infinity in a plane Projective 3-space Homogeneous coordinates Duality Desargues' theorem Pappas' theorem Projective n-space 4.12 Regions of Space Regions of space Regions of a plane Regions of a line Planar regions defined by two lines Planar regions defined by three lines Creating a pentagonal region Creating a 5-star region Creating a 5-star pyramid Summary 4.13 Geometric Constructions Geometric expressions Geometric equations for lines and planes The geometric equation of a conic section in the plane The geometric equation as a prescription to construct The algebraic equation of a conic section in the plane An alternative geometric equation of a conic section in the plane Conic sections through five points Dual constructions Constructing conics in space A geometric equation for a cubic in the plane Pascal's Theorem Pascal lines 4.14 Summary

5 The Complement5.1 Introduction 5.2 Axioms for the Complement The grade of a complement The linearity of the complement operation The complement axiom The complement of a complement axiom 2009 9 4 The complement of unity

5.2 Axioms for the Complement The grade of a complement The linearity of the complement operation The complement axiom The complement of a complement axiom The complement of unity 5.3 Defining the Complement The complement of an m-element The complement of a basis m-element Defining the complement of a basis 1-element Constraints on the value of Choosing the value of Defining the complements in matrix form 5.4 The Euclidean Complement Tabulating Euclidean complements of basis elements Formulae for the Euclidean complement of basis elements Products leading to a scalar or n-element 5.5 Complementary Interlude Alternative forms for complements Orthogonality Visualizing the complement axiom The regressive product in terms of complements Glimpses of the inner product 5.6 The Complement of a Complement The complement of a complement axiom The complement of a cobasis element The complement of the complement of a basis 1-element The complement of the complement of a basis m-element The complement of the complement of an m-element Idempotent complements 5.7 Working with Metrics Working with metrics The default metric Declaring a metric Declaring a general metric Calculating induced metrics The metric for a cobasis Creating palettes of induced metrics 5.8 Calculating Complements Entering a complement Creating palettes of complements of basis elements Converting complements of basis elements Simplifying expressions involving complements Converting expressions involving complements to specified forms Converting regressive products of basis elements in a metric space 5.9 Complements in a vector space The Euclidean complement in a vector 2-space The non-Euclidean complement in a vector 2-space 2009 9 4 The Euclidean complement in a vector 3-space The non-Euclidean complement in a vector 3-space

5.9 Complements in a vector space The Euclidean complement in a vector 2-space The non-Euclidean complement in a vector 2-space The Euclidean complement in a vector 3-space The non-Euclidean complement in a vector 3-space 5.10 Complements in a bound space Metrics in a bound space The complement of an m-vector Products of vectorial elements in a bound space The complement of an element bound through the origin The complement of the complement of an m-vector Calculating with vector space complements 5.11 Complements of bound elements The Euclidean complement of a point in the plane The Euclidean complement of a point in a point 3-space The complement of a bound element Euclidean complements of bound elements The regressive product of point complements 5.12 Reciprocal Bases Reciprocal bases The complement of a basis element The complement of a cobasis element The complement of a complement of a basis element The exterior product of basis elements The regressive product of basis elements The complement of a simple element is simple 5.13 Summary

6 The Interior Product6.1 Introduction 6.2 Defining the Interior Product Definition of the inner product Definition of the interior product Implications of the regressive product axioms Orthogonality Example: The interior product of a simple bivector and a vector 6.3 Properties of the Interior Product Implications of the Complement Axiom Extended interior products Converting interior products Example: Orthogonalizing a set of 1-elements 6.4 The Interior Common Factor Theorem The Interior Common Factor Formula The Interior Common Factor Theorem Examples of the Interior Common Factor Theorem The computational form of the Interior Common Factor Theorem 2009 9 4

6.4 The Interior Common Factor Theorem The Interior Common Factor Formula The Interior Common Factor Theorem Examples of the Interior Common Factor Theorem The computational form of the Interior Common Factor Theorem 6.5 The Inner Product Implications of the Common Factor Axiom The symmetry of the inner product The inner product of complements The inner product of simple elements Calculating inner products Inner products of basis elements 6.6 The Measure of an m-element The definition of measure Unit elements Calculating measures The measure of free elements The measure of bound elements Determining the multivector of a bound multivector 6.7 The Induced Metric Tensor Calculating induced metric tensors Using scalar products to construct induced metric tensors Displaying induced metric tensors as a matrix of matrices 6.8 Product Formulae for Interior Products The basic interior Product Formula Deriving interior Product Formulas Deriving interior Product Formulas automatically Computable forms of interior Product Formulas The invariance of interior Product Formulas An alternative form for the interior Product Formula The interior decomposition formula Interior Product Formulas for 1-elements Interior Product Formulas in terms of double sums 6.9 The Zero Interior Sum Theorem The zero interior sum Composing interior sums The Gram-Schmidt process Proving the Zero Interior Sum Theorem 6.10 The Cross Product Defining a generalized cross product Cross products involving 1-elements Implications of the axioms for the cross product The cross product as a universal product Cross product formulae 6.11 The Triangle Formulae Triangle components The measure of the triangle components Equivalent forms for the triangle components 2009 9 4

6.11 The Triangle Formulae Triangle components The measure of the triangle components Equivalent forms for the triangle components 6.12 Angle Defining the angle between elements The angle between a vector and a bivector The angle between two bivectors The volume of a parallelepiped 6.13 Projection To be completed. 6.14 Interior Products of Interpreted Elements To be completed. 6.15 The Closest Approach of Multiplanes To be completed.

7 Exploring Screw Algebra7.1 Introduction 7.2 A Canonical Form for a 2-Entity The canonical form Canonical forms in an n-plane Creating 2-entities 7.3 The Complement of 2-Entity Complements in an n-plane The complement referred to the origin The complement referred to a general point 7.4 The Screw The definition of a screw The unit screw The pitch of a screw The central axis of a screw Orthogonal decomposition of a screw 7.5 The Algebra of Screws To be completed 7.6 Computing with Screws To be completed

2009 9 4

8 Exploring Mechanics8.1 Introduction 8.2 Force Representing force Systems of forces Equilibrium Force in a metric 3-plane 8.3 Momentum The velocity of a particle Representing momentum The momentum of a system of particles The momentum of a system of bodies Linear momentum and the mass centre Momentum in a metric 3-plane 8.4 Newton's Law Rate of change of momentum Newton's second law 8.5 The Angular Velocity of a Rigid Body To be completed. 8.6 The Momentum of a Rigid Body To be completed. 8.7 The Velocity of a Rigid Body To be completed. 8.8 The Complementary Velocity of a Rigid Body To be completed. 8.9 The Infinitesimal Displacement of a Rigid Body To be completed. 8.10 Work, Power and Kinetic Energy To be completed.

9 Grassmann Algebra9.1 Introduction 9.1 Grassmann Numbers Creating Grassmann numbers Body and soul Even and odd components The grades of a Grassmann number Working with complex scalars 2009 9 4

Creating Grassmann numbers Body and soul Even and odd components The grades of a Grassmann number Working with complex scalars 9.3 Operations with Grassmann Numbers The exterior product of Grassmann numbers The regressive product of Grassmann numbers The complement of a Grassmann number The interior product of Grassmann numbers 9.4 Simplifying Grassmann Numbers Elementary simplifying operations Expanding products Factoring scalars Checking for zero terms Reordering factors Simplifying expressions 9.5 Powers of Grassmann Numbers Direct computation of powers Powers of even Grassmann numbers Powers of odd Grassmann numbers Computing positive powers of Grassmann numbers Powers of Grassmann numbers with no body The inverse of a Grassmann number Integer powers of a Grassmann number General powers of a Grassmann number 9.6 Solving Equations Solving for unknown coefficients Solving for an unknown Grassmann number 9.7 Exterior Division Defining exterior quotients Special cases of exterior division The non-uniqueness of exterior division 9.8 Factorization of Grassmann Numbers The non-uniqueness of factorization Example: Factorizing a Grassmann number in 2-space Example: Factorizing a 2-element in 3-space Example: Factorizing a 3-element in 4-space 9.9 Functions of Grassmann Numbers The Taylor series formula The form of a function of a Grassmann number Calculating functions of Grassmann numbers Powers of Grassmann numbers Exponential and logarithmic functions of Grassmann numbers Trigonometric functions of Grassmann numbers Functions of several Grassmann numbers

10 Exploring the Generalized Grassmann Product2009 9 4

10 Exploring the Generalized Grassmann Product10.1 Introduction 10.2 Geometrically Interpreted 1-elements Definition of the generalized product Case l = 0: Reduction to the exterior product Case 0 < l < Min[m, k]: Reduction to exterior and interior products Case l = Min[m, k]: Reduction to the interior product Case Min[m, k] < l < Max[m, k]: Reduction to zero Case l = Max[m, k]: Reduction to zero Case l > Max[m, k]: Undefined 10.3 The Symmetric Form of the Generalized Product Expansion of the generalized product in terms of both factors The quasi-commutativity of the generalized product Expansion in terms of the other factor 10.4 Calculating with Generalized Products Entering a generalized product Reduction to interior products Reduction to inner products Example: Case Min[m, k] < l < Max[m, k]: Reduction to zero 10.5 The Generalized Product Theorem The A and B forms of a generalized product Example: Verification of the Generalized Product Theorem Verification that the B form may be expanded in terms of either factor 10.6 Products with Common Factors Products with common factors Congruent factors Orthogonal factors Generalized products of basis elements Finding common factors 10.7 The Zero Interior Sum Theorem The Zero Interior Sum theorem Composing a zero interior sum 10.8 The Zero Generalized Sum The zero generalized sum conjecture Generating the zero generalized sum Exploring the conjecture 10.9 Nilpotent Generalized Products Nilpotent products of simple elements Nilpotent products of non-simple elements 10.10 Properties of the Generalized Product Summary of properties

2009 9 4

10.10 Properties of the Generalized Product Summary of properties 10.11 The Triple Generalized Sum Conjecture The generalized Grassmann product is not associative The triple generalized sum The triple generalized sum conjecture Exploring the triple generalized sum conjecture An algorithm to test the conjecture 10.12 Exploring Conjectures A conjecture Exploring the conjecture 10.13 The Generalized Product of Intersecting Elements The case l < p The case l p The special case of l = p 10.14 The Generalized Product of Orthogonal Elements The generalized product of totally orthogonal elements The generalized product of partially orthogonal elements 10.15 The Generalized Product of Intersecting Orthogonal Elements The case l < p The case l p 10.16 Generalized Products in Lower Dimensional Spaces Generalized products in 0, 1, and 2-spaces 0-space 1-space 2-space 10.17 Generalized Products in 3-Space To be completed

11 Exploring Hypercomplex Algebra11.1 Introduction 11.2 Some Initial Definitions and Properties The conjugate Distributivity The norm Factorization of scalars Multiplication by scalars The real numbers 11.3 The Complex Numbers Constraints on the hypercomplex signs Complex numbers as Grassmann numbers under the hypercomplex product 11.4 The Hypercomplex Product in a 2-Space Tabulating products in 2-space The hypercomplex product of two 1-elements The hypercomplex product of a 1-element and a 2-element 2009 9 4 The hypercomplex square of a 2-element The product table in terms of exterior and interior products

11.4 The Hypercomplex Product in a 2-Space Tabulating products in 2-space The hypercomplex product of two 1-elements The hypercomplex product of a 1-element and a 2-element The hypercomplex square of a 2-element The product table in terms of exterior and interior products 11.5 The Quaternions The product table for orthonormal elements Generating the quaternions The norm of a quaternion The Cayley-Dickson algebra 11.6 The Norm of a Grassmann number The norm The norm of a simple m-element The skew-symmetry of products of elements of different grade The norm of a Grassmann number in terms of hypercomplex products The norm of a Grassmann number of simple components The norm of a non-simple element 11.7 Products of two different elements of the same grade The symmetrized sum of two m-elements Symmetrized sums for elements of different grades The body of a symmetrized sum The soul of a symmetrized sum Summary of results of this section 11.8 Octonions To be completed

12 Exploring Clifford Algebra12.1 Introduction 12.2 The Clifford Product Definition of the Clifford product Tabulating Clifford products The grade of a Clifford product Clifford products in terms of generalized products Clifford products in terms of interior products Clifford products in terms of inner products Clifford products in terms of scalar products 12.3 The Reverse of an Exterior Product Defining the reverse Computing the reverse 12.4 Special Cases of Clifford Products The Clifford product with scalars The Clifford product of 1-elements The Clifford product of an m-element and a 1-element The Clifford product of an m-element and a 2-element The Clifford product of two 2-elements 2009 9 4 The Clifford product of two identical elements

12.4 Special Cases of Clifford Products The Clifford product with scalars The Clifford product of 1-elements The Clifford product of an m-element and a 1-element The Clifford product of an m-element and a 2-element The Clifford product of two 2-elements The Clifford product of two identical elements 12.5 Alternate Forms for the Clifford Product Alternate expansions of the Clifford product The Clifford product expressed by decomposition of the first factor Alternative expression by decomposition of the first factor The Clifford product expressed by decomposition of the second factor The Clifford product expressed by decomposition of both factors 12.6 Writing Down a General Clifford Product The form of a Clifford product expansion A mnemonic way to write down a general Clifford product 12.7 The Clifford Product of Intersecting Elements General formulae for intersecting elements Special cases of intersecting elements 12.8 The Clifford Product of Orthogonal Elements The Clifford product of totally orthogonal elements The Clifford product of partially orthogonal elements Testing the formulae 12.9 The Clifford Product of Intersecting Orthogonal Elements Orthogonal union Orthogonal intersection 12.10 Summary of Special Cases of Clifford Products Arbitrary elements Arbitrary and orthogonal elements Orthogonal elements Calculating with Clifford products 12.11 Associativity of the Clifford Product Associativity of orthogonal elements A mnemonic formula for products of orthogonal elements Associativity of non-orthogonal elements Testing the general associativity of the Clifford product 12.13 Clifford Algebra Generating Clifford algebras Real algebra Clifford algebras of a 1-space 12.14 Clifford Algebras of a 2-Space The Clifford product table in 2-space Product tables in a 2-space with an orthogonal basis Case 1: 8e1 e1 + 1, e2 e2 + 1< Case 2: 8e1 e1 + 1, e2 e2 - 1< Case 3: 8e1 e1 - 1, e2 e2 - 1