Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts (Paperback)

Needham, Tristan

  • 出版商: Princeton University
  • 出版日期: 2021-07-13
  • 售價: $1,460
  • 貴賓價: 9.8$1,431
  • 語言: 英文
  • 頁數: 584
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 0691203709
  • ISBN-13: 9780691203706
  • 立即出貨(限量) (庫存=1)

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商品描述

An inviting, intuitive, and visual exploration of differential geometry and forms

Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner.

Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book.

Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.

商品描述(中文翻譯)

《視覺微分幾何與形式》是一本引人入勝、直觀且視覺化的微分幾何和形式探索。

這本書有兩個主要目標。在前四個部分中,Tristan Needham將幾何學重新引入微分幾何。他使用了235個手繪圖解,運用牛頓的幾何方法,以幾何的方式解釋了經典結果。在第五個部分中,他以直觀和幾何的方式提供了微分形式的本科入門課程,涵蓋了高級主題。

前四個部分的獨特特點包括:對於基本重要的全局高斯-波涅定理提供了四種不同的幾何證明,提供了局部幾何和全局拓撲之間驚人的聯繫;對高斯著名的Theorema Egregium提供了一個簡單的幾何證明;對n-流形的黎曼曲率張量進行了完整的幾何處理;以及對愛因斯坦場方程的詳細幾何處理,描述了重力作為彎曲時空(廣義相對論)的情況,以及它對引力波、黑洞和宇宙學的影響。最後一個部分闡明了向量微積分的所有積分定理的統一;以2-形式重新表述麥克斯韋方程組的優雅方法;德拉姆上同調;通過卡爾坦的移動框架方法進行微分幾何;以及使用曲率2-形式計算黎曼張量。第五部分的七個章節中,有六個章節可以完全獨立於本書的其他部分閱讀。

《視覺微分幾何與形式》只需要基礎的微積分和幾何知識,大膽地重新思考了這一重要數學領域的考慮和教學方式。

作者簡介

Tristan Needham is professor of mathematics at the University of San Francisco. He is the author of Visual Complex Analysis.

作者簡介(中文翻譯)

Tristan Needham是舊金山大學的數學教授。他是《Visual Complex Analysis》一書的作者。