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内容简介：

本书试图提供外微分形式、微分几何、代数拓扑、微分拓扑、李群、向量丛、Chern公式等前沿知识，它们对于深入理解经典物理、现代物理以及工程都是必需的。其中包含解析动力学、流体动力学、电磁学（在平坦空间和弯曲空间）、热力学、弹性理论、Kirchhoff电路定律的几何及拓扑、肥皂泡薄膜、狭义相对论和广义相对论、Dirac算子和旋量、Yang-Mills规范场、Aharonov-Bohm效应、Berry相、瞬子绕数、夸克、介子的夸克模型。在讨论抽象的微分几何概念前，通过大量的关于常规空间中曲面的研究培养几何直观，因此，学数学的学生对此书也会很感兴趣。

    本书对于物理、工程、数学的高年级学生和研究生是非常有益的，可供他们作为自学教材。

内容截图：

preface to the second edition
perface to the revised printing
perface to the first edition

Ⅰ Manifolds, Tensors, and Exterior Forms
1 Manifolds and Vector Fields        3
2 Tensors and Exterior Forms        37
3 Integration of Differential Forms        95
4 The Lie Derivative        125
5 The Poincaré Lemma and Potentials        155
6 Holonomic and Nonholonomic Constraints        165

Ⅱ Geometry and Topology
7 R3 and Minkowski Space        191
8 The Geometry of Surfaces in R3        201
9 Covariant Differentiation and Curvature        241
10 Geodesics        269
11 Relativity, Tensors, and Curvature        291
12 Curvature and Topology: Synge’s Theorem        323
13 Betti Numbers and De Rham’s Theorem        333
14 Harmonic Forms        361

Ⅲ Lie Groups, Bundles, and Chern Forms
15 Lie Groups        391
16 Vector Bundles in Geometry and Physics        413
17 Fiber Bundles, Gauss\|Bonnet, and Topological Quantization        451
18 Connections and Associated Bundles        475
19 The Dirac Equation        491
20 Yang\|Mills Fields        523
21 Betti Numbers and Covering Spaces        561
22 Chern Forms and Homotopy Groups        583

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