Calculus from Approximation to Theory takes a fresh and innovative look at the teaching and learning of calculus. One way to describe calculus might be to say it is a suite of techniques that approximate curved things by flat things and through a limiting process applied to those approximations arrive at an exact answer. Standard approaches to calculus focus on that limiting process as the heart of the matter. This text places its emphasis on the approximating processes and thus illuminates the motivating ideas and makes clearer the scientific usefulness, indeed centrality, of the subject while paying careful attention to the theoretical foundations. Limits are defined in terms of sequences, the derivative is defined from the best affine approximation, and greater attention than usual is paid to numerical techniques and the order of an approximation. Access to modern computational tools is presumed throughout and the use of these tools is woven seamlessly into the exposition and problems. All of the central topics of a yearlong calculus course are covered, with the addition of treatment of difference equations, a chapter on the complex plane as the arena for motion in two dimensions, and a much more thorough and modern treatment of differential equations than is standard. Dan Sloughter is Emeritus Professor of Mathematics at Furman University with interests in probability, statistics, and the philosophy of mathematics and statistics. He has been involved in efforts to reform calculus instruction for decades and has published widely on that topic. This book, one of the results of that work, is very well suited for a yearlong introduction to calculus that focuses on ideas over techniques.

This text was produced for the second part of a two-part sequence on advanced calculus, whose aim is to provide a firm logical foundation for analysis. The first part treats analysis in one variable, and the text at hand treats analysis in several variables. After a review of topics from one-variable analysis and linear algebra, the text treats in succession multivariable differential calculus, including systems of differential equations, and multivariable integral calculus. It builds on this to develop calculus on surfaces in Euclidean space and also on manifolds. It introduces differential forms and establishes a general Stokes formula. It describes various applications of Stokes formula, from harmonic functions to degree theory. The text then studies the differential geometry of surfaces, including geodesics and curvature, and makes contact with degree theory, via the Gauss-Bonnet theorem. The text also takes up Fourier analysis, and bridges this with results on surfaces, via Fourier analysis on spheres and on compact matrix groups.