A First Course in Differential Geometry (Paperback)

Woodward, Lyndon, Bolton, John

  • 出版商: Cambridge
  • 出版日期: 2019-01-24
  • 定價: $880
  • 售價: 9.5$836
  • 語言: 英文
  • 頁數: 272
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 1108441025
  • ISBN-13: 9781108441025
  • 相關分類: 微積分 Calculus物理學 Physics
  • 立即出貨 (庫存=1)

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商品描述

Differential geometry is the study of curved spaces using the techniques of calculus. It is a mainstay of undergraduate mathematics education and a cornerstone of modern geometry. It is also the language used by Einstein to express general relativity, and so is an essential tool for astronomers and theoretical physicists. This introductory textbook originates from a popular course given to third year students at Durham University for over twenty years, first by the late L. M. Woodward and later by John Bolton (and others). It provides a thorough introduction by focusing on the beginnings of the subject as studied by Gauss: curves and surfaces in Euclidean space. While the main topics are the classics of differential geometry - the definition and geometric meaning of Gaussian curvature, the Theorema Egregium, geodesics, and the Gauss-Bonnet Theorem - the treatment is modern and student-friendly, taking direct routes to explain, prove and apply the main results. It includes many exercises to test students' understanding of the material, and ends with a supplementary chapter on minimal surfaces that could be used as an extension towards advanced courses or as a source of student projects.

商品描述(中文翻譯)

微分幾何是使用微積分技巧研究曲面的學科。它是大學數學教育的重要組成部分,也是現代幾何學的基石。愛因斯坦用它來表達廣義相對論,因此對於天文學家和理論物理學家來說,它是一個必不可少的工具。這本入門教材源於杜倫大學三年級學生多年來受到歡迎的課程,最初由已故的L. M. Woodward教授授課,後來由約翰·博爾頓(和其他人)接手。它通過專注於高斯研究的主題(歐幾里得空間中的曲線和曲面)提供了徹底的介紹。雖然主要的話題是微分幾何的經典內容 - 高斯曲率的定義和幾何意義、Theorema Egregium、測地線和高斯-博奈特定理 - 但教材的處理方式現代且適合學生,直接解釋、證明和應用主要結果。它包含許多練習題來測試學生對材料的理解,並以一個關於極小曲面的補充章節結束,可以作為進階課程的擴展或學生項目的來源。

目錄大綱

Preface
1. Curves in Rn
2. Surfaces in Rn
3. Tangent planes and the first fundamental form
4. Smooth maps
5. Measuring how surfaces curve
6. The Theorema Egregium
7. Geodesic curvature and geodesics
8. The Gauss–Bonnet theorem
9. Minimal and CMC surfaces
10. Hints or answers to some exercises
Index.

目錄大綱(中文翻譯)

前言
1. R^n中的曲線
2. R^n中的曲面
3. 切平面和第一基本形式
4. 光滑映射
5. 測量曲面彎曲程度
6. Theorema Egregium
7. 流線曲率和測地線
8. Gauss-Bonnet定理
9. 最小曲面和CMC曲面
10. 一些練習的提示或答案
索引。