Try Blinkist to get the key ideas from 7,500+ bestselling nonfiction titles and podcasts. Listen or read in just 15 minutes.
Get started
Blink 3 of 8 - The 5 AM Club
by Robin Sharma
Introduction to Differential Geometry of Space Curves and Surfaces by Taha Sochi offers a comprehensive introduction to the fundamental concepts and techniques of differential geometry, providing a solid foundation for further study in this field.
In Introduction to Differential Geometry of Space Curves and Surfaces by Taha Sochi, the author begins by introducing the fundamental concepts of space curves and surfaces. The book delves into the mathematical representation of curves and surfaces, including parametric equations and vector fields, and explores their intrinsic and extrinsic properties, such as curvature and torsion for curves, and curvature and Gaussian curvature for surfaces.
Sochi then progresses to discuss the Frenet-Serret formulas, which provide a means to describe the kinematic properties of a curve in terms of its curvature and torsion. The author also introduces the first and second fundamental forms, which are essential tools for studying the geometry of surfaces, and demonstrates their significance in characterizing the metric and the shape of a surface.
As the book advances, Sochi delves into the mathematical machinery of tensor calculus and differential forms, which are indispensable for formulating and manipulating geometrical concepts in a coordinate-independent manner. The author provides a detailed explanation of covariant and contravariant vectors, tensor products, and the Einstein summation convention, offering a foundation for understanding the subsequent discussions on curvature and geodesics.
Moreover, Sochi introduces differential forms, a powerful tool for expressing geometrical quantities and equations in a coordinate-independent manner. The exterior derivative, Hodge star operator, and the concept of integration over manifolds are also covered, offering a comprehensive understanding of these mathematical constructs.
Having established the necessary mathematical tools, the book proceeds to explore the curvature of curves and surfaces in more detail. Sochi introduces the concept of geodesics, which are curves that locally minimize distance, and discusses their relationship with the curvature of a surface. The author also provides a comprehensive treatment of the Gauss-Bonnet theorem, a fundamental result linking the curvature of a surface to its topological properties.
Furthermore, Sochi discusses the intrinsic and extrinsic curvatures of surfaces, and their relationship to the first and second fundamental forms. The book also covers the concept of parallel transport, which plays a crucial role in defining the curvature of a surface, and its connection to the Levi-Civita connection, a fundamental component of Riemannian geometry.
In the latter part of the book, Sochi explores various applications of the differential geometry of curves and surfaces. These applications include the study of minimal surfaces, isometric embeddings, and the theory of constant mean curvature surfaces. The author also provides an introduction to the theory of Riemannian manifolds, including the Riemann curvature tensor and the Ricci curvature.
Finally, the book concludes with a discussion of more advanced topics in differential geometry, such as the Gauss-Bonnet-Chern theorem, the concept of parallelism in Riemannian manifolds, and the theory of connections on vector bundles. Throughout these discussions, the author emphasizes the geometric intuition behind the abstract mathematical concepts, making the material accessible and engaging for the reader.
In Introduction to Differential Geometry of Space Curves and Surfaces, Taha Sochi provides a comprehensive and accessible introduction to the differential geometry of curves and surfaces. By combining a rigorous mathematical approach with a focus on geometric intuition, the book equips readers with the tools to understand and analyze the intrinsic and extrinsic properties of curves and surfaces in a coordinate-independent manner, making it an invaluable resource for students and researchers in mathematics and related fields.
Introduction to Differential Geometry of Space Curves and Surfaces by Taha Sochi provides a comprehensive introduction to the fundamental concepts and techniques in the field of differential geometry. It covers topics such as curvature, torsion, geodesics, and the Gauss-Bonnet theorem, offering clear explanations and insightful examples. Whether you're a student or a researcher, this book serves as an invaluable resource for understanding the geometric properties of curves and surfaces in space.
Students studying mathematics, particularly those interested in geometry and calculus
Academics and researchers in the field of differential geometry
Mathematics enthusiasts looking to expand their knowledge and understanding of curved spaces
It's highly addictive to get core insights on personally relevant topics without repetition or triviality. Added to that the apps ability to suggest kindred interests opens up a foundation of knowledge.
Great app. Good selection of book summaries you can read or listen to while commuting. Instead of scrolling through your social media news feed, this is a much better way to spend your spare time in my opinion.
Life changing. The concept of being able to grasp a book's main point in such a short time truly opens multiple opportunities to grow every area of your life at a faster rate.
Great app. Addicting. Perfect for wait times, morning coffee, evening before bed. Extremely well written, thorough, easy to use.
Try Blinkist to get the key ideas from 7,500+ bestselling nonfiction titles and podcasts. Listen or read in just 15 minutes.
Get startedBlink 3 of 8 - The 5 AM Club
by Robin Sharma