Differential Geometry of Curves and Surfaces (Hardcover)

Through two previous editions, the third edition of this popular and intriguing text takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces.

Requiring only multivariable calculus and linear algebra, it develops students' geometric intuition through interactive graphics applets. Applets are presented in Maple workbook format, which readers can access using the free Maple Player.

The book explains the reasons for various definitions while the interactive applets offer motivation for definitions, allowing students to explore examples further, and give a visual explanation of complicated theorems. The ability to change parametric curves and parametrized surfaces in an applet lets students probe the concepts far beyond what static text permits. Investigative project ideas promote student research.

At users of the previous editions' request, this third edition offers a broader list of exercises. More elementary exercises are added and some challenging problems are moved later in exercise sets to assure more graduated progress. The authors also add hints to motivate students grappling with the more difficult exercises.

This student-friendly and readable approach offers additional examples, well-placed to assist student comprehension. In the presentation of the Gauss-Bonnet Theorem, the authors provide more intuition and stepping-stones to help students grasp phenomena behind it. Also, the concept of a homeomorphism is new to students even though it is a key theoretical component of the definition of a regular surface. Providing more examples show students how to prove certain functions are homeomorphisms.

這本受歡迎且引人入勝的書籍的第三版，延續了前兩版的分析/理論方法和視覺/直觀方法，探討曲線和曲面的局部和全局特性。

本書只需要多變量微積分和線性代數的基礎，通過互動圖形應用程式開發學生的幾何直覺。這些應用程式以Maple工作簿格式呈現，讀者可以使用免費的Maple Player來訪問。

本書解釋了各種定義的原因，而互動應用程式則提供了定義的動機，讓學生進一步探索例子，並對複雜定理進行視覺解釋。在應用程式中更改參數曲線和參數化曲面的能力使學生能夠深入研究概念，超越靜態文本所能提供的範圍。調查項目的想法促進學生的研究。

根據前幾版的用戶要求，第三版提供了更廣泛的習題列表。增加了更多基礎練習題，並將一些具有挑戰性的問題移至練習題集的後部，以確保更有階梯式的進展。作者還提供提示，以激發學生解決更難練習題的動力。

這種學生友好且易讀的方法提供了更多的例子，這些例子放置得很好，以幫助學生理解。在介紹高斯-波內定理時，作者提供了更多直觀和引導，以幫助學生理解其背後的現象。此外，雖然同胚的概念是定義正則曲面的關鍵理論組成部分，但對學生來說是新的。提供更多的例子可以向學生展示如何證明某些函數是同胚。

Thomas F. Banchoff is a geometer and a professor at Brown University. Dr. Banchoff was president of the Mathematical Association of America (MAA) from 1999 to 2000. He has published numerous papers in a variety of journals and has been the recipient of many honors, including the MAA's Deborah and Franklin Tepper Haimo Award and Brown's Teaching with Technology Award. He is the author of several books, including Linear Algebra Through Geometry with John Wermer and Beyond the Third Dimension.

Stephen Lovett is an associate professor of mathematics at Wheaton College. Dr. Lovett has taught introductory courses on differential geometry for many years, including at Eastern Nazarene College. He has given many talks over the past several years on differential and algebraic geometry as well as cryptography. In 2015, he was awarded Wheaton's Senior Scholarship Faculty Award. He is the author of Abstract Algebra: Structures and Applications, Differential Geometry of Manifolds, Second Edition, A Transition to Advanced Mathematics with Danilo Dedrichs (forthcoming), all published by CRC Press.

Thomas F. Banchoff是一位幾何學家，也是布朗大學的教授。Banchoff博士曾擔任美國數學協會（MAA）的主席，任期從1999年到2000年。他在多個期刊上發表了許多論文，並獲得了許多榮譽，包括MAA的Deborah和Franklin Tepper Haimo獎和布朗大學的教學技術獎。他是幾本書的作者，包括與John Wermer合著的《通過幾何學學習線性代數》和《超越第三維度》。

Stephen Lovett是Wheaton College的副教授。Lovett博士多年來一直教授微分幾何的入門課程，包括在Eastern Nazarene College教授。他在過去幾年中就微分幾何和代數幾何以及密碼學等主題發表了許多演講。2015年，他獲得了Wheaton的高級獎學金教職員獎。他是CRC Press出版的幾本書的作者，包括《抽象代數：結構與應用》、《流形的微分幾何，第二版》和即將出版的《進階數學過渡》（與Danilo Dedrichs合著）。