Manifolds, Tensor Analysis, and Applications

1. Front Matter

   Pages i-xi

   PDF

2. Topology

   • Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

   Pages 1-39

3. Banach Spaces and Differential Calculus

   • Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

   Pages 40-140

4. Manifolds and Vector Bundles

   • Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

   Pages 141-237

5. Vector Fields and Dynamical Systems

   • Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

   Pages 238-337

6. Tensors

   • Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

   Pages 338-391

7. Differential Forms

   • Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

   Pages 392-463

8. Integration on Manifolds

   • Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

   Pages 464-559

9. Applications

   • Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu

   Pages 560-630

10. Back Matter

    Pages 631-656

    PDF

The 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.

• Derivative
• Electromagnetism
• Hodge star operator
• Implicit function
• Riemannian geometry
• calculus
• compactness
• exterior derivative
• manifold

• Department of Mathematics, University of California—Santa Cruz, Santa Cruz, USA

  Ralph Abraham, Tudor Ratiu

• Control and Dynamical Systems, 107–81, California Institute of Technology, Pasadena, USA

  Jerrold E. Marsden

Policies and ethics