Synopsis
From the Preface of the First Edition:This book 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 Élie 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.So for those teaching the subject, I’ll first give a brief summary of what I see as the salient original features of the book.1)I introduce differentiation using the exterior derivative on a scalar function to generate a 1-form, so making it multivariate from the start.2)I define integration as a product between a differential form and a simplex.3)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.4)The book incorporates geometric algebra into the presentation of vector algebra and analysis from an early stage.5)Generalised Functions are introduced fully based on differential forms, and this treatment prepares the way for an advanced coverage of Fourier and Laplace transforms.
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