## Manifolds, Tensor Analysis, and Applications, Volume 75The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid me chanics, electromagnetism, plasma dynamics and control thcory arc given in Chapter 8, using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail, and it does not treat Lie groups, principal bundles, or Morse theory. Some of this is planned for a subsequent edition. Meanwhile, the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book's contents and development. Throughout the text supplementary topics are given, marked with the symbols ~ and {l:;J. This device enables the reader to skip various topics without disturbing the main flow of the text. Some of these provide additional background material intended for completeness, to minimize the necessity of consulting too many outside references. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings, the study of infinite-dimensional manifolds can be hard to motivate. |

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### Contents

Topology | 1 |

11 Topological Spaces | 2 |

12 Metric Spaces | 9 |

13 Continuity | 14 |

14 Subspaces Products and Quotients | 18 |

15 Compactness | 24 |

16 Connectedness | 31 |

17 Baire Spaces | 37 |

52 Tensor Bundles and Tensor Fields | 349 |

Algebraic Approach | 359 |

Dynamic Approach | 370 |

55 Partitions of Unity | 377 |

Differential Forms | 392 |

62 Determinants Volumes and the Hodge Star Operator | 402 |

63 Differential Forms | 417 |

64 The Exterior Derivative Interior Product and Lie Derivative | 423 |

BANACH SPACES AND DIFFERENTIAL CALCULUS | 40 |

22 Linear and Multilinear Mappings | 56 |

23 The Derivative | 75 |

24 Properties of the Derivative | 83 |

25 The Inverse and Implicit Function Theorems | 116 |

MANIFOLDS AND VECTOR BUNDLES | 141 |

32 Submanifolds Products and Mappings | 150 |

33 The Tangent Bundle | 157 |

34 Vector Bundles | 167 |

35 Submersions Immersions and Transversality | 196 |

Vector Fields and Dynamical Systems | 238 |

42 Vector Fields as Differential Operators | 265 |

43 An Introduction to Dynamical Systems | 298 |

44 Frobenius Theorem and Foliations | 326 |

Tensors | 338 |

65 Orientation Volume Elements and the Codifferential | 449 |

Integration on Manifolds | 464 |

72 StokesTheorem | 476 |

73 The Classical Theorems of Green Gauss and Stokes | 504 |

74 Induced Flows on Function Spaces and Ergodicity | 513 |

75 Introduction to HodgeDeRham Theory and Topological Applications of Differential Forms | 538 |

Applications | 560 |

82 Fluid Mechanics | 584 |

83 Electromagnetism | 599 |

84 The LiePoisson Bracket in Continuum Mechanics and Plasma Physics | 609 |

85 Constraints and Control | 624 |

631 | |

643 | |

### Common terms and phrases

algebra assume atlas Banach space basis boundary bundle chart bundle map called closed compact support components compute constant continuous converges coordinates Corollary countable curl defined Definition Let denote dense Df(u Df(x diffeomorphism equivalence relation example Exercise exists fiber bundle finite dimensional flow formula given Hausdorff Hence Hilbert space Hint identity implies induced inner product integral curve inverse isomorphism Lemma Let f Lie derivative linear map locally manifold map f Marsden matrix metric space n-manifold norm open neighborhood open set operator paracompact partitions of unity Proof Let Proposition Let prove Riemannian satisfying self-adjoint sequence Show subbundle submanifold submersion subset subspace surjective symmetric tangent bundle tensor Theorem Let topological space topology trivial unique vector bundle vector bundle map vector field vector space volume form