Differential Geometry of Curves and Surfaces

Viac o knihe

This textbook on differential geometry caters to a range of courses, offering minimal prerequisites for beginners while including numerous examples and detailed proofs. It serves as a clear, concise, and rigorous resource for those pursuing graduate studies in math or physics, encompassing deep global theorems. The author employs various techniques to make complex abstract ideas more accessible and engaging. Over 300 color illustrations enhance understanding, clarifying concepts that grayscale images cannot. Green-boxed definitions and purple-boxed theorems visually organize the content, with color used to highlight logical relationships. Applications are plentiful, such as the study of conformal and equiareal functions in cartography. The narrative of Christiaan Huygens introduces evolutes, involutes, and cycloids while addressing the longitude problem, leading to advancements in optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum, while Green’s Theorem facilitates the creation of a planimeter. Foucault’s Pendulum illustrates a parallel vector field along Earth's latitude, and a south-pointing chariot visualizes this concept along any curve on a surface. Although the book does not delve into the profound applications of differential geometry in modern physics, it emphasizes that the GPS system relies on general relativity, articulated through this mathematical framework. Throughout, a