Vector and Geometric Calculus (Geometric Algebra & Calculus)

Description

This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus This is the printing of January 2020. It contains several small corrections and improvements. The previous printing, from October 2020, introduced a new Section 10.4 with some recently published results about antiderivatives. Another new Section 11.4 introduced the differential geometry of manifolds of arbitrary dimension. The new sections are only introductions; neither goes into any depth. The book is a sequel to the text Linear and Geometric Algebra by the same author. That text is a prerequisite for this one. Its web page is at&nbsp&nbspfaculty.luther.edu/~macdonal/laga. Linear algebra and vector calculus have provided the basic vocabulary of mathematics in dimensions greater than one for the past one hundred years. Just as geometric algebra generalizes linear algebra in powerful ways, geometric calculus generalizes vector calculus in powerful ways. Traditional vector calculus topics are covered, as they must be, since readers will encounter them in other texts and out in the world. Differential geometry is used today in many disciplines. A final chapter is devoted to it. Download the book's table of contents, preface, and index at the book's web site:&nbsp&nbsp faculty.luther.edu/~macdonal/vagc. From a review of Linear and Geometric Algebra: &nbsp&nbsp&nbsp&nbspAlan Macdonald's text is an excellent resource if you are just beginning the study of geometric algebra and would like to learn or review traditional linear algebra in the process. The clarity and evenness of the writing, as well as the originality of presentation that is evident throughout this text, suggest that the author has been successful as a mathematics teacher in the undergraduate classroom. This carefully crafted text is ideal for anyone learning geometric algebra in relative isolation, which I suspect will be the case for many readers. &nbsp&nbsp&nbsp&nbsp-- Jeffrey Dunham, William R. Kenan Jr. Professor of Natural Sciences, Middlebury College Contents Preface I Preliminaries 1 Curve and Surface Representations 1.1 Curve Representations 1.2 Surface Representations 1.3 Polar, Cylindrical, Spherical Coordinates 2 Limits and Continuity 2.1 Open and Closed Sets 2.2 Limits 2.3 Continuity II Derivatives 3 The Differential 3.1 The Partial Derivative 3.2 The Taylor Expansion 3.3 The Differential 3.4 The Chain Rule 3.5 The Directional Derivative 3.6 Inverse and Implicit Functions 4 Tangent Spaces 4.1 Manifolds 4.2 Tangent Spaces to Curves 4.3 Tangent Spaces to Surfaces 5 The Gradient 5.1 Fields 5.2 The Gradient 5.3 Scalar and Vector Fields 5.4 Curvilinear Coordinates 5.5 The Vector Derivative 6 Extrema 6.1 Extrema 6.2 Constrained Extrema III Integrals 7 Integrals over Curves 7.1 The Scalar Integral 7.2 The Path Integ ral 7.3 The Line Integral 7.4 Conservative Vector Fields 8 Multiple Integrals 8.1 Multiple Integrals 8.2 Change of Variables 9 Integrals over Surfaces 9.1 The Surface Integral 9.2 The Flux Integral IV The Fundamental Theorem of Calculus 10.1 The Fundamental Theorem of Calculus 10.2 The Divergence Theorem 10.3 The Curl Theorem 10.4 Analytic Functions V Differential Geometry 11 Differential Geometry in R³ 11.1 Curves 11.2 Surfaces 11.3 Curves in Surfaces VI Appendices A Review of Geometric Algebra B Software C Formulas D Differential Forms Index