Algebraic Calculus

ISBN 978-0-9935483-0-7

Algebraic Calculus puts forward a rather radical new proposal for the teaching of basic undergraduate mathematics aimed especially at students of electronics and computer graphics. The text draws on methods I worked out in my Ph.D. work several years ago which in turn drew on the several books on Differential Geometry that came out largely in the sixties.

The central thesis of the book is that basic Calculus can be developed much more easily and painlessly if the Exterior Derivative is used from the start. This leads naturally to Differential Forms and these in turn can easily be shown to give Integration simply by defining it as a product between an n-form and an n-simplex.

So I’m doing two things:

  • I’m showing that the idea of differential forms, which crystallised around a hundred years ago, allied to the concept of simplexes, suffices as a foundation to develop the entire body of the calculus easily and quickly, and gives a much more coherent line of development.
  • I’m putting it in a way that is clear, readable and, hopefully, entertaining. So I have preferred English readability to mathematical formality wherever reasonably possible.

Along the way, I cover in some depth various supporting fields such as Vector Algebra, with an introduction to the up-and-coming area of Geometric Algebra, and I also give a good, but more critical, introduction to the subject of Generalised Functions. The book ends with a brief introduction to Projective Geometry.

The salient original features of the book can be summarised as these:

  • I introduce differentiation using the exterior derivative on a scalar function to generate a 1-form, so making it multivariate from the start.
  • I define integration as a product between a differential form and a simplex.
  • I use the axioms of a group to show that the addition of angles in the circle leads naturally to the idea of complex numbers.
  • The book incorporates geometric algebra into the presentation of vector algebra and analysis from an early stage.
  • Generalised Functions are introduced fully based on differential forms, and this treatment prepares the way for an advanced coverage of Fourier and Laplace transforms.

In the account of this first edition, I said: “I make no bones about the fact that I’m not happy with any of the existing presentations of Generalised Functions and that I have deep reservations about the Fourier Inversion Theorem. A natural result should eventually have a natural proof. There’s no question that this has been achieved for Stokes’s Theorem, which initially required a somewhat tortuous proof, but when developed in the recast framework of Differential Geometry falls out automatically as the general extension of the Fundamental Theorem of Calculus. In the present treatment I regard both of these as so basic that they are effectively definitions of different grades of Integration and to all intents and purposes need no proof.”

This edition of the book is currently out of print, but the second edition is out as of the end of June 2016, and resolves the questions which dissatisfied me above.

The book will be for sale directly through Amazon.

It is now possible to download full colour copies of the illustrations from this website using the button below.

Download Images

If you have any enquiries in the meantime, contact me directly through the Contact page for further enquiries.

If you’d like a preview, you can browse through some sample Sections of the first edition of the book in PDF format in this page.

Please email me if you should discover any errors! You can use rod “AT”[i.e. the @ symbol], or using the Contact given on this website.

Table of Contents

Chapter 1. 1

Preliminaries. 1

1.1 Functional Notation. 1

1.2 Graphs and Chirality. 3

1.3 Summation. 6

1.4 Predicates and Sets. 7

1.5 A First Look at Axiomatic Systems: Groups. 8

1.6 Equations and Inequalities. 11

1.7 Linear Equations. 13

1.8 Matrices and Determinants. 16

1.9 Polynomials. 21

1.10 Infinite Series. 26

1.11 Colour and Pascal’s Triangle. 28

1.12 Musical Scales. 30

1.13 Simplexes and Topology. 34

1.14 Electrical Circuits. 38

1.15 Euler Characteristics. 47

1.X Exercises. 52

Chapter 2. 55

The Calculus. 55

2. 1 Introduction. 55

2.2 Further Cases. 60

2.3 The Infinitesimal Curse. 64

2.4 Inversion of the Differential Operator 66

2.5 Applications: Minima and Maxima. 68

2.6 A First Look at Integration. 72

2.7 Taylor’s Expansion. 75

2.8 Is there a Second Differential?. 77

2.X Exercises. 79

Chapter 3. 81

Logarithms and Exponentials. 81

3.1 The Natural Logarithm.. 81

3.2 The Exponential Function. 83

3.3 Generalized Exponentiation. 85

3.4 The Number e. 88

3.5 Linear Differential Equations. 88

3.X Exercises. 95

Chapter 4. 97

Angles. 97

4.1 The Basic Notion of Angle. 97

4.2 Addition of Angles. 98

4.3 Ordering Relations. 103

4.4 The Single-Real or Parametric Approach. 105

4.5 Complex Numbers. 111

4.6 Calculus of Complex Numbers. 115

4.7 LDE’s and the Euler Formula. 117

4.8 Further Examples of Angle Algebra. 118

4.9 Hours and ε Notation. 121

4.10 Hyperbolic Angles. 126

4.X Exercises. 131

Chapter 5. 133

Vectors and Geometric Algebra.. 133

5.1 Introduction to Vector Spaces. 133

5.2 The Origin of Vectors: Quaternions. 137

5.3 Axiomatic Formulation. 138

5.4 The Dual Space. 142

5.5 Reciprocal Basis and Tensor Product 147

5.6 Graded Algebras. 157

5.7 Vectors and Angles and the Gibbs Cross Product 158

5.8 The Geometric Product 162

5.9 Geometric Algebra of 3-Vectors. 167

5.10 Rotations. 174

5.11 Eigenvalues and Eigenvectors. 184

5.X Exercises. 189

Chapter 6. 191

Integration.. 191

6.1 Integration. 191

6.2 Line Integrals and the Pullback. 192

6.3 Manifolds. 198

6.4 Areas and Volumes. 202

6.5 Multiple Integrals. 207

6.6 Reinterpretation of the Differential 212

6.7 Surface Integrals. 227

6.8 Integral Theorems. 230

6.A Appendix A: The Volume of a Tet. 240

6.B Appendix B: The Conventional Evaluation of the Volume of
a Pyramid
. 244

6.C Appendix C: The Numerical Interpretation of
. 246

6.D Appendix D: Heuristic Justification of the Alternating

6.X Exercises. 252

Chapter 7. 255

Functional Vector Spaces. 255

7.1 Functions as Vectors. 255

7.2 Elements of Complex Analysis. 259

7.3 Singularities. 266

7.4 A Scalar Product for Differential Forms. 274

7.5 Green’s Formulae and The Functional Derivative. 277

7.6 Tensor Functionals. 284

7.7 Green’s Functions. 286

7.A Appendix A: The Cauchy Integral in Simplex Form.. 291

7.B Appendix B: The Laplacian Green’s Function for n = 2. 292

7.C Appendix C: Distributions on the Unit Circle. 295

7.D Appendix D: Delta Functionals of the Form

7.X Exercises. 302

Chapter 8. 305

Fourier-Laplace Transforms. 305

8.1 Discrete Parallels. 305

8.2 The Fourier Series. 308

8.3 The Fourier Transform.. 317

8.4 The Laplace Transform.. 331

8.5 Convolution. 334

8.6 Applications. 342

8.X Exercises. 352

Chapter 9. 355

Projective Geometry. 355

9.1 Introduction. 355

9.2 On Vanishing Points. 360

9.3 Homogeneous Coordinates. 365

9.4 z-Interpolation and Cross Ratio. 370

9.5 Joins, Meets and Duality. 379

9.6 Desargues’s Theorem.. 387

9.7 Projective Transformations. 402

9.A Appendix A: Demonstration of Acute-angled Triangle
Property for Three Dimensions
. 404

9.X Exercises. 406

Appendix. 409

Methods of Integration.. 409

A.1 Introduction. 409

A.2 Use of Inverse Trigonometric Functions. 410

A.3 Other Trigonometric Integrals. 414

A.4 Use of Partial Fractions. 417

A.5 Integration by Parts. 420

A.6 Contour Integration. 421

Glossary. 429

Select Bibliography. 453

Index. 455