Algebraic Calculus

Algebraic Calculus advocates a radically new approach to the introduction of Higher Mathematics at Freshman level. I adopt a slightly polemical tone because I'm aiming to stimulate debate. The methods, and some of the terminology, that I propose may appear unconventional, but they have sound roots in mathematical history and translate exceptionally well into digital practice, so I'll start by reviewing this background. The mathematical methods introduced by Elie Cartan the better part of a hundred years ago are now widespread in research-level work. But what is not fully acknowledged is that they can revolutionize the teaching of the subject too. All that is needed is a readable, informal account of them. Bringing in these methods, suitably simplified, right at the start, in a simple, engaging style, transforms the clarity and comprehensibility of the subject. The true meaning of so many aspects of intermediate mathematics falls naturally into place. 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, which were more the fashion in Europe in the fifties. And to enrich the readability of the text, there are digressions into fields that are not obviously mathematical, especially if they relate to computer graphics or are particularly relevant to digital practice. I would hope the book's groundbreaking approach will be especially interesting to teachers working in digital applications at this level. [From the Preface to the book]

Table of Contents:
Prefaceiii Table of Contentsix 1Preliminaries1 1.1Functional Notation1 1.2Graphs and Chirality3 1.3Summation6 1.4Predicates and Sets7 1.5A First Look at Axiomatic Systems: Groups8 1.6Equations and Inequalities11 1.7Linear Equations13 1.8Matrices and Determinants16 1.9Polynomials20 1.10Infinite Series25 1.11 Colour and Pascal's Triangle27 1.12Musical Scales29 1.13Simplexes and Topology33 1.14Electrical Circuits37 1.15Euler Characteristics43 2The Calculus51 2.1Introduction51 2.2Further Cases55 2.3The Infinitesimal Curse59 2.4Inversion of the Differential Operator61 2.5Applications: Maxima and Minima63 2.6A First Look at Integration67 2.7Taylor's Expansion69 2.8Is there a Second Differential?71 3Logarithms and Exponentials73 3.1The Natural Logarithm73 3.2The Exponential Function75 3.3Generalised Exponentiation77 3.4The Number e80 3.5Linear Differential Equations81 4Angles87 4.1The Basic Notion of Angle87 4.2Addition of Angles88 4.3Ordering Relations93 4.4The Single-Real or Parametric Approach95 4.5Complex Numbers101 4.6Calculus of Complex Numbers105 4.7LDE's and the Euler Formula106 4.8Further Examples of Angle Algebra108 5Vectors and Geometric Algebra111 5.1Introduction to Vector Spaces111 5.2The Origin of Vectors: Quaternions115 5.3Axiomatic Formulation116 5.4The Dual Space117 5.5Reciprocal Basis and Tensor Product121 5.6Graded Algebras127 5.7Vectors and Angles128 5.8The Geometric Product130 5.9Geometric Algebra of 3-Vectors135 5.10Rotations141 5.11Eigenvalues and Eigenvectors148 6Integration153 6.1Integration153 6.2Line Integrals and the Pullback154 6.3 Manifolds160 6.4Areas and Volumes163 6.5Multiple Integrals166 6.6Reinterpretation of the Differential171 6.7Surface Integrals181 6.8Integral Theorems185 6.AAppendix A: The Subdivision of a Parallelepiped into Tets195 6.BAppendix B: The Volume of a Tet197 6.CAppendix C: Heuristic Justification of the Alternating Product200 7Functional Vector Spaces207 7.1Functions as Vectors207 7.2Elements of Complex Analysis210 7.3Singularities216 7.4A Scalar Product for Differential Forms223 7.5Green's Formulae and The Functional Derivative226 7.6Tensor Functionals 232 7.7Green's Functions234 7.AAppendix A: The Cauchy Integral in Simplex Form236 7.BAppendix B: The Laplacian Green's Function for n = 2237 7.CAppendix C: Distributions on the Unit Circle240 7.DAppendix D: Delta Functionals of the Form ;(966;(x)).245 8Fourier-Laplace Transforms247 8.1Discrete Parallels247 8.2 8.3 8.4 8.5 8.6 8.7 8.A 8.B 8.CThe Fourier Inversion Theorem The Fourier Transform Fourier Analysis on the Unit Circle The Laplace Transform Convolution Applications Appendix A: Details of the Proof of The Inversion Theorem Appendix B: The Inversion Theorem in Complex Arguments Appendix C: The Fourier Transform for a Periodic Function is a Dirac Comb250 253 258 264 267 274 283 286 287 9Projective Geometry289 9.1Introduction289 9.2 On Vanishing Points294 9.3Homogeneous Coordinates300 9.4z-Interpolation and Cross Ratio305 9.5Joins, Meets and Duality313 9.6Desargues's Theorem321 9.7Projective Transformations332 9.AAppendix A: Demonstration of Acute-angled Triangle Property for Three Dimensions334 Glossary337 Bibliography361 Index363

About the Author :
Roderick Lumsden, b. 1951, studied Natural Sciences and then Computer Science at St. John's College, Cambridge, and he has a Ph.D. with the Department of Electrical and Electronic