Advanced Calculus and its Applications in Variational Quantum Mechanics and Relativity Theory

The first part of this book reviews some key topics on multi-variable advanced calculus. The approach presented includes detailed and rigorous studies on surfaces in Rn which comprises items such as differential forms and an abstract version of the Stokes Theorem in Rn. The conclusion section introduces readers to Riemannian geometry, which is used in the subsequent chapters.

The second part reviews applications, specifically in variational quantum mechanics and relativity theory. Topics such as a variational formulation for the relativistic Klein-Gordon equation, the derivation of a variational formulation for relativistic mechanics firstly through (semi)-Riemannian geometry are covered. The second part has a more general context. It includes fundamentals of differential geometry.

The later chapters describe a new interpretation for the Bohr atomic model through a semi-classical approach. The book concludes with a classical description of the radiating cavity model in quantum mechanics.

本書的第一部分回顧了一些關於多變數高等微積分的重要主題。所介紹的方法包括對Rn中的曲面進行詳細而嚴謹的研究，其中包括微分形式和Rn中斯托克斯定理的抽象版本等內容。結論部分向讀者介紹了黎曼幾何，該主題在後續章節中有所應用。

第二部分回顧了應用領域，特別是在變分量子力學和相對論理論中的應用。涵蓋的主題包括相對論克萊因-戈登方程的變分形式，以及通過（半）黎曼幾何首次導出相對論力學的變分形式。第二部分具有更廣泛的背景，包括微分幾何的基礎知識。

後面的章節描述了一種半經典方法對波耳原子模型進行的新解釋。本書以量子力學中輻射腔模型的經典描述作為結尾。