Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Color is even used within the text to highlight logical relationships.Īpplications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. This is a textbook on differential geometry well-suited to a variety of courses on this topic. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Green’s Theorem makes possible a drafting tool called a planimeter. Clairaut’s Theorem is presented as a conservation law for angular momentum. Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Color is even used within the text to highlight logical relationships. The general theory is illustrated and expanded using the examples of curves and surfaces.This is a textbook on differential geometry well-suited to a variety of courses on this topic. The authors begin with the necessary tools from analysis and topology, including Sard's theorem, de Rham cohomology, calculus on manifolds, and a degree theory. This book is an introduction to modern differential geometry. Differential Geometry: Manifolds, Curves, and Surfacesīackground - Differential Equations - Differentiable Manifolds - Partitions of Unity, Densities and Curves - Critical Points - Differential Forms - Integration of Differential Forms - Degree Theory - Curves: The Local Theory - Plane Curves: The Global Theory - A Guide to the Local Theory of Surfaces in R3 - A Guide to the Global Theory of Surfaces - Bibliography - Index of Symbols and Notations - Index
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