Introduction to Tensor Analysis and the Calculus of Moving Surfaces (Hardcover)

This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds.

Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations.

The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20^th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author’s skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.

The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.

這本教科書在該主題上與其他教材有所不同，其深度的呈現和對移動曲面微積分的討論使其成為將張量微積分擴展到變形流形的一種方法。

該教材針對高年級本科生和研究生設計，邀請讀者通過張量微積分的透鏡重新審視之前學過的材料。一旦掌握了框架，讀者將被介紹到包括流形上的微分幾何、形狀優化、邊界擾動和動態流體薄膜方程等新材料。

張量的語言最初由愛因斯坦提倡，與微積分和線性代數的語言一樣基礎，是每個技術科學家都應該掌握的。這種在20世紀初發明的張量技術現在被認為是經典的。然而，正如作者所展示的，它仍然非常重要和相關。作者傑出的講授能力通過深入的例子和大量的練習題得以體現。大部分內容都致力於幾何基礎、變量變換的力學、張量符號的正確使用以及代數和幾何之間的相互作用的討論。早期章節有很多文字但很少方程式。張量的定義僅在第6章中提出，當讀者準備好時。儘管這本教材保持了一致的嚴謹性，但它非常謹慎地避免形式化主題。

教科書的最後一部分專門介紹了移動曲面微積分。這是對這一重要技術的首次教科書闡述，也是本教材的亮點之一。書中介紹了一些令人興奮的應用，包括形狀優化、邊界值問題的邊界擾動和作者近年來開發的動態流體薄膜方程。此外，移動曲面框架被用來提供對經典結果的新推導，如测地線方程和著名的高斯-波涅特定理。