面向物理学家的拓扑学、微分几何学与广义相对论基础：基于Mathematica的教程式方法
Elements of Topology, Differential Geometry and General Relativity for Physicists: A Mathematica-based Tutorial Approach
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中文总结 AI 辅助
本书是面向物理研究者的拓扑学、微分几何与广义相对论入门教程，采用基于Mathematica的实操方法，配套原创笔记本，兼顾理论严谨性与计算实操性，助力读者掌握并复现相关结果。
中文摘要 AI 辅助
本书是面向物理学专业学生与研究者的自包含教程式入门读物，涵盖拓扑学、微分几何学与广义相对论三大领域。其显著特色是采用基于Mathematica的方法，通过可运行的实体笔记本与实例演算将抽象构造具象化。核心贡献是作者自行编写的大量原创Mathematica笔记本，读者可运行、验证并扩展其中的每一项演示内容。第一部分涵盖点集拓扑、拓扑空间、映射的连续性、同伦与基本群，相关应用凸显了拓扑学在现代物理学中的作用。第二部分（篇幅最大）从零构建微分几何学：包括流形、切空间与余切空间、向量场、拉回与推送、李括号与李代数、局部流及李导数，随后论述张量、微分形式、外微分、体积形式、度量张量与霍奇对偶，强调无坐标表述及其计算实现。这些工具被应用于形式语言表述的麦克斯韦方程与广义斯托克斯定理，而李群、纤维丛、联络与曲率则搭建起几何与规范理论物理之间的桥梁。最后一部分运用该框架呈现广义相对论，通过专用Mathematica包计算标准时空的曲率张量与场方程。全书在数学严谨性与实操计算间取得平衡，使读者既能理解理论，又能自行复现每一项结果。
英文摘要
This book is a self-contained, tutorial introduction to topology, differential geometry, and general relativity for students and researchers in physics. Its distinguishing feature is a Mathematica-based approach that makes abstract constructions concrete through explicit, runnable notebooks and worked examples. A key contribution is a large collection of original Mathematica notebooks, written by the authors themselves, that let readers run, verify, and extend every demonstration. The first part covers point-set topology, topological spaces, continuity of maps, homotopy, and the fundamental group with applications that highlight the role of topology in modern physics. The second and largest part develops differential geometry from the ground up: manifolds, tangent and dual spaces, vector fields, pullbacks and pushforwards, Lie brackets and Lie algebras, local flows, and the Lie derivative. It then treats tensors, differential forms, the exterior derivative, volume forms, the metric tensor, and Hodge duality, emphasizing coordinate-free formulations and their computational realization. These tools are applied to Maxwell's equations in the language of forms and the generalized Stokes theorem, while Lie groups, fiber bundles, connections, and curvature bridge geometry and gauge-theoretic physics. The final part uses this framework to present general relativity, computing curvature tensors and field equations for standard spacetimes with dedicated Mathematica packages. Throughout, the book balances mathematical rigor with hands-on computation, enabling readers both to understand the theory and to reproduce every result themselves.
