**Graduate Studies in Mathematics**

Volume: 52;
2002;
343 pp;
Hardcover

MSC: Primary 53;
Secondary 57; 58; 22; 74; 78; 80; 35

Print ISBN: 978-0-8218-2951-6

Product Code: GSM/52

List Price: $72.00

AMS Member Price: $57.60

MAA member Price: $64.80

**Electronic ISBN: 978-1-4704-2101-4
Product Code: GSM/52.E**

List Price: $72.00

AMS Member Price: $57.60

MAA member Price: $64.80

#### Supplemental Materials

# Global Analysis: Differential Forms in Analysis, Geometry and Physics

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*Ilka Agricola; Thomas Friedrich*

This book is an introduction to differential geometry through
differential forms, emphasizing their applications in various areas of
mathematics and physics. Well-written and with plenty of examples,
this textbook originated from courses on geometry and
analysis and presents a widely-used mathematical technique in a lucid
and very readable style. The authors introduce readers to the world of
differential forms while covering relevant topics from analysis,
differential geometry, and mathematical physics.

The book begins with a self-contained introduction to the calculus
of differential forms in Euclidean space and on manifolds. Next, the
focus is on Stokes' theorem, the classical integral formulas and their
applications to harmonic functions and topology. The authors then
discuss the integrability conditions of a Pfaffian system (Frobenius's
theorem). Chapter 5 is a thorough exposition of the theory of curves
and surfaces in Euclidean space in the spirit of Cartan. The following
chapter covers Lie groups and homogeneous spaces. Chapter 7 addresses
symplectic geometry and classical mechanics. The basic tools for the
integration of the Hamiltonian equations are the moment map and
completely integrable systems (Liouville-Arnold Theorem). The authors discuss Newton, Lagrange,
and Hamilton formulations of mechanics. Chapter 8 contains an introduction to statistical
mechanics and thermodynamics. The final chapter deals with
electrodynamics. The material in the book is carefully illustrated
with figures and examples, and there are over 100 exercises.

Readers should be familiar with first-year algebra and advanced calculus. The
book is intended for graduate students and researchers interested in delving
into geometric analysis and its applications to mathematical physics.

#### Readership

Graduate students, research mathematicians, and mathematical physicists.

#### Table of Contents

# Table of Contents

## Global Analysis: Differential Forms in Analysis, Geometry and Physics

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents xi12
- Preface v6 free
- Chapter 1. Elements of Multilinear Algebra 116 free
- Chapter 2. Differential Forms in R[sup(n)] 1126
- §2.1. Vector Fields and Differential Forms 1126
- §2.2. Closed and Exact Differential Forms 1833
- §2.3. Gradient, Divergence and Curl 2338
- §2.4. Singular Cubes and Chains 2641
- §2.5. Integration of Differential Forms and Stokes' Theorem 3045
- §2.6. The Classical Formulas of Green and Stokes 3550
- §2.7. Complex Differential Forms and Holomorphic Functions 3651
- §2.8. Brouwer's Fixed Point Theorem 3853
- Exercises 4358

- Chapter 3. Vector Analysis on Manifolds 4762
- §3.1. Submanifolds of M[sup(n)] 4762
- §3.2. Differential Calculus on Manifolds 5469
- §3.3. Differential Forms on Manifolds 6782
- §3.4. Orientable Manifolds 6984
- §3.5. Integration of Differential Forms over Manifolds 7691
- §3.6. Stokes' Theorem for Manifolds 7994
- §3.7. The Hedgehog Theorem (Hairy Sphere Theorem) 8196
- §3.8. The Classical Integral Formulas 8297
- §3.9. The Lie Derivative and the Interpretation of the Divergence 87102
- §3.10. Harmonic Functions 94109
- §3.11. The Laplacian on Differential Forms 100115
- Exercises 105120

- Chapter 4. Pfaffian Systems 111126
- Chapter 5. Curves and Surfaces in Euclidean 3-Space 129144
- §5.1. Curves in Euclidean 3-Space 129144
- §5.2. The Structural Equations of a Surface 141156
- §5.3. The First and Second Fundamental Forms of a Surface 147162
- §5.4. Gaussian and Mean Curvature 155170
- §5.5. Curves on Surfaces and Geodesic Lines 172187
- §5.6. Maps between Surfaces 180195
- §5.7. Higher-Dimensional Riemannian Manifolds 183198
- Exercises 198213

- Chapter 6. Lie Groups and Homogeneous Spaces 207222
- Chapter 7. Symplectic Geometry and Mechanics 229244
- Chapter 8. Elements of Statistical Mechanics and Thermodynamics 271286
- Chapter 9. Elements of Electrodynamics 295310
- Bibliography 333348
- Symbols 337352
- Index 339354
- Back Cover Back Cover1362